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• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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How to Write a Great Hypothesis

Hypothesis Format, Examples, and Tips

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk,  "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis, operational definitions, types of hypotheses, hypotheses examples.

• Collecting Data

Frequently Asked Questions

A hypothesis is a tentative statement about the relationship between two or more  variables. It is a specific, testable prediction about what you expect to happen in a study.

One hypothesis example would be a study designed to look at the relationship between sleep deprivation and test performance might have a hypothesis that states: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. It is only at this point that researchers begin to develop a testable hypothesis. Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore a number of factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk wisdom that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis.   In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in a number of different ways. One of the basic principles of any type of scientific research is that the results must be replicable.   By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. How would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

In order to measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming other people. In this situation, the researcher might utilize a simulated task to measure aggressiveness.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests that there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type of hypothesis suggests a relationship between three or more variables, such as two independent variables and a dependent variable.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative sample of the population and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• Complex hypothesis: "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "Children who receive a new reading intervention will have scores different than students who do not receive the intervention."
• "There will be no difference in scores on a memory recall task between children and adults."

Examples of an alternative hypothesis:

• "Children who receive a new reading intervention will perform better than students who did not receive the intervention."
• "Adults will perform better on a memory task than children." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when it would be impossible or difficult to  conduct an experiment . These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a correlational study can then be used to look at how the variables are related. This type of research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

A Word From Verywell

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Some examples of how to write a hypothesis include:

• "Staying up late will lead to worse test performance the next day."
• "People who consume one apple each day will visit the doctor fewer times each year."
• "Breaking study sessions up into three 20-minute sessions will lead to better test results than a single 60-minute study session."

The four parts of a hypothesis are:

• The research question
• The independent variable (IV)
• The dependent variable (DV)
• The proposed relationship between the IV and DV

Castillo M. The scientific method: a need for something better? . AJNR Am J Neuroradiol. 2013;34(9):1669-71. doi:10.3174/ajnr.A3401

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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Home » What is a Hypothesis – Types, Examples and Writing Guide

What is a Hypothesis – Types, Examples and Writing Guide

Table of Contents

Definition:

Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation.

Hypothesis is often used in scientific research to guide the design of experiments and the collection and analysis of data. It is an essential element of the scientific method, as it allows researchers to make predictions about the outcome of their experiments and to test those predictions to determine their accuracy.

Types of Hypothesis

Types of Hypothesis are as follows:

Research Hypothesis

A research hypothesis is a statement that predicts a relationship between variables. It is usually formulated as a specific statement that can be tested through research, and it is often used in scientific research to guide the design of experiments.

Null Hypothesis

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables. It is often used as a starting point for testing the research hypothesis, and if the results of the study reject the null hypothesis, it suggests that there is a significant difference or relationship between variables.

Alternative Hypothesis

An alternative hypothesis is a statement that assumes there is a significant difference or relationship between variables. It is often used as an alternative to the null hypothesis and is tested against the null hypothesis to determine which statement is more accurate.

Directional Hypothesis

A directional hypothesis is a statement that predicts the direction of the relationship between variables. For example, a researcher might predict that increasing the amount of exercise will result in a decrease in body weight.

Non-directional Hypothesis

A non-directional hypothesis is a statement that predicts the relationship between variables but does not specify the direction. For example, a researcher might predict that there is a relationship between the amount of exercise and body weight, but they do not specify whether increasing or decreasing exercise will affect body weight.

Statistical Hypothesis

A statistical hypothesis is a statement that assumes a particular statistical model or distribution for the data. It is often used in statistical analysis to test the significance of a particular result.

Composite Hypothesis

A composite hypothesis is a statement that assumes more than one condition or outcome. It can be divided into several sub-hypotheses, each of which represents a different possible outcome.

Empirical Hypothesis

An empirical hypothesis is a statement that is based on observed phenomena or data. It is often used in scientific research to develop theories or models that explain the observed phenomena.

Simple Hypothesis

A simple hypothesis is a statement that assumes only one outcome or condition. It is often used in scientific research to test a single variable or factor.

Complex Hypothesis

A complex hypothesis is a statement that assumes multiple outcomes or conditions. It is often used in scientific research to test the effects of multiple variables or factors on a particular outcome.

Applications of Hypothesis

Hypotheses are used in various fields to guide research and make predictions about the outcomes of experiments or observations. Here are some examples of how hypotheses are applied in different fields:

• Science : In scientific research, hypotheses are used to test the validity of theories and models that explain natural phenomena. For example, a hypothesis might be formulated to test the effects of a particular variable on a natural system, such as the effects of climate change on an ecosystem.
• Medicine : In medical research, hypotheses are used to test the effectiveness of treatments and therapies for specific conditions. For example, a hypothesis might be formulated to test the effects of a new drug on a particular disease.
• Psychology : In psychology, hypotheses are used to test theories and models of human behavior and cognition. For example, a hypothesis might be formulated to test the effects of a particular stimulus on the brain or behavior.
• Sociology : In sociology, hypotheses are used to test theories and models of social phenomena, such as the effects of social structures or institutions on human behavior. For example, a hypothesis might be formulated to test the effects of income inequality on crime rates.
• Business : In business research, hypotheses are used to test the validity of theories and models that explain business phenomena, such as consumer behavior or market trends. For example, a hypothesis might be formulated to test the effects of a new marketing campaign on consumer buying behavior.
• Engineering : In engineering, hypotheses are used to test the effectiveness of new technologies or designs. For example, a hypothesis might be formulated to test the efficiency of a new solar panel design.

How to write a Hypothesis

Here are the steps to follow when writing a hypothesis:

Identify the Research Question

The first step is to identify the research question that you want to answer through your study. This question should be clear, specific, and focused. It should be something that can be investigated empirically and that has some relevance or significance in the field.

Conduct a Literature Review

Before writing your hypothesis, it’s essential to conduct a thorough literature review to understand what is already known about the topic. This will help you to identify the research gap and formulate a hypothesis that builds on existing knowledge.

Determine the Variables

The next step is to identify the variables involved in the research question. A variable is any characteristic or factor that can vary or change. There are two types of variables: independent and dependent. The independent variable is the one that is manipulated or changed by the researcher, while the dependent variable is the one that is measured or observed as a result of the independent variable.

Formulate the Hypothesis

Based on the research question and the variables involved, you can now formulate your hypothesis. A hypothesis should be a clear and concise statement that predicts the relationship between the variables. It should be testable through empirical research and based on existing theory or evidence.

Write the Null Hypothesis

The null hypothesis is the opposite of the alternative hypothesis, which is the hypothesis that you are testing. The null hypothesis states that there is no significant difference or relationship between the variables. It is important to write the null hypothesis because it allows you to compare your results with what would be expected by chance.

Refine the Hypothesis

After formulating the hypothesis, it’s important to refine it and make it more precise. This may involve clarifying the variables, specifying the direction of the relationship, or making the hypothesis more testable.

Examples of Hypothesis

Here are a few examples of hypotheses in different fields:

• Psychology : “Increased exposure to violent video games leads to increased aggressive behavior in adolescents.”
• Biology : “Higher levels of carbon dioxide in the atmosphere will lead to increased plant growth.”
• Sociology : “Individuals who grow up in households with higher socioeconomic status will have higher levels of education and income as adults.”
• Education : “Implementing a new teaching method will result in higher student achievement scores.”
• Marketing : “Customers who receive a personalized email will be more likely to make a purchase than those who receive a generic email.”
• Physics : “An increase in temperature will cause an increase in the volume of a gas, assuming all other variables remain constant.”
• Medicine : “Consuming a diet high in saturated fats will increase the risk of developing heart disease.”

Purpose of Hypothesis

The purpose of a hypothesis is to provide a testable explanation for an observed phenomenon or a prediction of a future outcome based on existing knowledge or theories. A hypothesis is an essential part of the scientific method and helps to guide the research process by providing a clear focus for investigation. It enables scientists to design experiments or studies to gather evidence and data that can support or refute the proposed explanation or prediction.

The formulation of a hypothesis is based on existing knowledge, observations, and theories, and it should be specific, testable, and falsifiable. A specific hypothesis helps to define the research question, which is important in the research process as it guides the selection of an appropriate research design and methodology. Testability of the hypothesis means that it can be proven or disproven through empirical data collection and analysis. Falsifiability means that the hypothesis should be formulated in such a way that it can be proven wrong if it is incorrect.

In addition to guiding the research process, the testing of hypotheses can lead to new discoveries and advancements in scientific knowledge. When a hypothesis is supported by the data, it can be used to develop new theories or models to explain the observed phenomenon. When a hypothesis is not supported by the data, it can help to refine existing theories or prompt the development of new hypotheses to explain the phenomenon.

When to use Hypothesis

Here are some common situations in which hypotheses are used:

• In scientific research , hypotheses are used to guide the design of experiments and to help researchers make predictions about the outcomes of those experiments.
• In social science research , hypotheses are used to test theories about human behavior, social relationships, and other phenomena.
• I n business , hypotheses can be used to guide decisions about marketing, product development, and other areas. For example, a hypothesis might be that a new product will sell well in a particular market, and this hypothesis can be tested through market research.

Characteristics of Hypothesis

Here are some common characteristics of a hypothesis:

• Testable : A hypothesis must be able to be tested through observation or experimentation. This means that it must be possible to collect data that will either support or refute the hypothesis.
• Falsifiable : A hypothesis must be able to be proven false if it is not supported by the data. If a hypothesis cannot be falsified, then it is not a scientific hypothesis.
• Clear and concise : A hypothesis should be stated in a clear and concise manner so that it can be easily understood and tested.
• Based on existing knowledge : A hypothesis should be based on existing knowledge and research in the field. It should not be based on personal beliefs or opinions.
• Specific : A hypothesis should be specific in terms of the variables being tested and the predicted outcome. This will help to ensure that the research is focused and well-designed.
• Tentative: A hypothesis is a tentative statement or assumption that requires further testing and evidence to be confirmed or refuted. It is not a final conclusion or assertion.
• Relevant : A hypothesis should be relevant to the research question or problem being studied. It should address a gap in knowledge or provide a new perspective on the issue.

Advantages of Hypothesis

Hypotheses have several advantages in scientific research and experimentation:

• Guides research: A hypothesis provides a clear and specific direction for research. It helps to focus the research question, select appropriate methods and variables, and interpret the results.
• Predictive powe r: A hypothesis makes predictions about the outcome of research, which can be tested through experimentation. This allows researchers to evaluate the validity of the hypothesis and make new discoveries.
• Facilitates communication: A hypothesis provides a common language and framework for scientists to communicate with one another about their research. This helps to facilitate the exchange of ideas and promotes collaboration.
• Efficient use of resources: A hypothesis helps researchers to use their time, resources, and funding efficiently by directing them towards specific research questions and methods that are most likely to yield results.
• Provides a basis for further research: A hypothesis that is supported by data provides a basis for further research and exploration. It can lead to new hypotheses, theories, and discoveries.
• Increases objectivity: A hypothesis can help to increase objectivity in research by providing a clear and specific framework for testing and interpreting results. This can reduce bias and increase the reliability of research findings.

Limitations of Hypothesis

Some Limitations of the Hypothesis are as follows:

• Limited to observable phenomena: Hypotheses are limited to observable phenomena and cannot account for unobservable or intangible factors. This means that some research questions may not be amenable to hypothesis testing.
• May be inaccurate or incomplete: Hypotheses are based on existing knowledge and research, which may be incomplete or inaccurate. This can lead to flawed hypotheses and erroneous conclusions.
• May be biased: Hypotheses may be biased by the researcher’s own beliefs, values, or assumptions. This can lead to selective interpretation of data and a lack of objectivity in research.
• Cannot prove causation: A hypothesis can only show a correlation between variables, but it cannot prove causation. This requires further experimentation and analysis.
• Limited to specific contexts: Hypotheses are limited to specific contexts and may not be generalizable to other situations or populations. This means that results may not be applicable in other contexts or may require further testing.
• May be affected by chance : Hypotheses may be affected by chance or random variation, which can obscure or distort the true relationship between variables.

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Definition of hypothesis

Did you know.

The Difference Between Hypothesis and Theory

A hypothesis is an assumption, an idea that is proposed for the sake of argument so that it can be tested to see if it might be true.

In the scientific method, the hypothesis is constructed before any applicable research has been done, apart from a basic background review. You ask a question, read up on what has been studied before, and then form a hypothesis.

A hypothesis is usually tentative; it's an assumption or suggestion made strictly for the objective of being tested.

A theory , in contrast, is a principle that has been formed as an attempt to explain things that have already been substantiated by data. It is used in the names of a number of principles accepted in the scientific community, such as the Big Bang Theory . Because of the rigors of experimentation and control, it is understood to be more likely to be true than a hypothesis is.

In non-scientific use, however, hypothesis and theory are often used interchangeably to mean simply an idea, speculation, or hunch, with theory being the more common choice.

Since this casual use does away with the distinctions upheld by the scientific community, hypothesis and theory are prone to being wrongly interpreted even when they are encountered in scientific contexts—or at least, contexts that allude to scientific study without making the critical distinction that scientists employ when weighing hypotheses and theories.

The most common occurrence is when theory is interpreted—and sometimes even gleefully seized upon—to mean something having less truth value than other scientific principles. (The word law applies to principles so firmly established that they are almost never questioned, such as the law of gravity.)

This mistake is one of projection: since we use theory in general to mean something lightly speculated, then it's implied that scientists must be talking about the same level of uncertainty when they use theory to refer to their well-tested and reasoned principles.

The distinction has come to the forefront particularly on occasions when the content of science curricula in schools has been challenged—notably, when a school board in Georgia put stickers on textbooks stating that evolution was "a theory, not a fact, regarding the origin of living things." As Kenneth R. Miller, a cell biologist at Brown University, has said , a theory "doesn’t mean a hunch or a guess. A theory is a system of explanations that ties together a whole bunch of facts. It not only explains those facts, but predicts what you ought to find from other observations and experiments.”

While theories are never completely infallible, they form the basis of scientific reasoning because, as Miller said "to the best of our ability, we’ve tested them, and they’ve held up."

• proposition
• supposition

hypothesis , theory , law mean a formula derived by inference from scientific data that explains a principle operating in nature.

hypothesis implies insufficient evidence to provide more than a tentative explanation.

theory implies a greater range of evidence and greater likelihood of truth.

law implies a statement of order and relation in nature that has been found to be invariable under the same conditions.

Examples of hypothesis in a Sentence

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'hypothesis.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

Greek, from hypotithenai to put under, suppose, from hypo- + tithenai to put — more at do

1641, in the meaning defined at sense 1a

Phrases Containing hypothesis

• null hypothesis
• Whorfian hypothesis
• nebular hypothesis
• counter - hypothesis
• planetesimal hypothesis

Articles Related to hypothesis

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This is the Difference Between a Hypothesis and a Theory

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Dictionary Entries Near hypothesis

hypothermia

hypothesize

Cite this Entry

“Hypothesis.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/hypothesis. Accessed 14 Apr. 2024.

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• How to Write a Strong Hypothesis | Guide & Examples

How to Write a Strong Hypothesis | Guide & Examples

Published on 6 May 2022 by Shona McCombes .

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more variables . An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1: ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2: Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalise more complex constructs.

Step 3: Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

Step 4: Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

Step 5: Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

Step 6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis. The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis is not just a guess. It should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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General Education

Think about something strange and unexplainable in your life. Maybe you get a headache right before it rains, or maybe you think your favorite sports team wins when you wear a certain color. If you wanted to see whether these are just coincidences or scientific fact, you would form a hypothesis, then create an experiment to see whether that hypothesis is true or not.

But what is a hypothesis, anyway? If you’re not sure about what a hypothesis is--or how to test for one!--you’re in the right place. This article will teach you everything you need to know about hypotheses, including: 

• Defining the term “hypothesis” 
• Providing hypothesis examples 
• Giving you tips for how to write your own hypothesis

So let’s get started!

What Is a Hypothesis?

Merriam Webster defines a hypothesis as “an assumption or concession made for the sake of argument.” In other words, a hypothesis is an educated guess . Scientists make a reasonable assumption--or a hypothesis--then design an experiment to test whether it’s true or not. Keep in mind that in science, a hypothesis should be testable. You have to be able to design an experiment that tests your hypothesis in order for it to be valid. 

As you could assume from that statement, it’s easy to make a bad hypothesis. But when you’re holding an experiment, it’s even more important that your guesses be good...after all, you’re spending time (and maybe money!) to figure out more about your observation. That’s why we refer to a hypothesis as an educated guess--good hypotheses are based on existing data and research to make them as sound as possible.

Hypotheses are one part of what’s called the scientific method .  Every (good) experiment or study is based in the scientific method. The scientific method gives order and structure to experiments and ensures that interference from scientists or outside influences does not skew the results. It’s important that you understand the concepts of the scientific method before holding your own experiment. Though it may vary among scientists, the scientific method is generally made up of six steps (in order):

• Observation
• Asking questions
• Forming a hypothesis
• Analyze the data
• Communicate your results

You’ll notice that the hypothesis comes pretty early on when conducting an experiment. That’s because experiments work best when they’re trying to answer one specific question. And you can’t conduct an experiment until you know what you’re trying to prove!

Independent and Dependent Variables 

After doing your research, you’re ready for another important step in forming your hypothesis: identifying variables. Variables are basically any factor that could influence the outcome of your experiment . Variables have to be measurable and related to the topic being studied.

There are two types of variables:  independent variables and dependent variables. I ndependent variables remain constant . For example, age is an independent variable; it will stay the same, and researchers can look at different ages to see if it has an effect on the dependent variable. 

Speaking of dependent variables... dependent variables are subject to the influence of the independent variable , meaning that they are not constant. Let’s say you want to test whether a person’s age affects how much sleep they need. In that case, the independent variable is age (like we mentioned above), and the dependent variable is how much sleep a person gets. 

Variables will be crucial in writing your hypothesis. You need to be able to identify which variable is which, as both the independent and dependent variables will be written into your hypothesis. For instance, in a study about exercise, the independent variable might be the speed at which the respondents walk for thirty minutes, and the dependent variable would be their heart rate. In your study and in your hypothesis, you’re trying to understand the relationship between the two variables.

Elements of a Good Hypothesis

The best hypotheses start by asking the right questions . For instance, if you’ve observed that the grass is greener when it rains twice a week, you could ask what kind of grass it is, what elevation it’s at, and if the grass across the street responds to rain in the same way. Any of these questions could become the backbone of experiments to test why the grass gets greener when it rains fairly frequently.

As you’re asking more questions about your first observation, make sure you’re also making more observations . If it doesn’t rain for two weeks and the grass still looks green, that’s an important observation that could influence your hypothesis. You'll continue observing all throughout your experiment, but until the hypothesis is finalized, every observation should be noted.

Finally, you should consult secondary research before writing your hypothesis . Secondary research is comprised of results found and published by other people. You can usually find this information online or at your library. Additionally, m ake sure the research you find is credible and related to your topic. If you’re studying the correlation between rain and grass growth, it would help you to research rain patterns over the past twenty years for your county, published by a local agricultural association. You should also research the types of grass common in your area, the type of grass in your lawn, and whether anyone else has conducted experiments about your hypothesis. Also be sure you’re checking the quality of your research . Research done by a middle school student about what minerals can be found in rainwater would be less useful than an article published by a local university.

Writing Your Hypothesis

Once you’ve considered all of the factors above, you’re ready to start writing your hypothesis. Hypotheses usually take a certain form when they’re written out in a research report.

When you boil down your hypothesis statement, you are writing down your best guess and not the question at hand . This means that your statement should be written as if it is fact already, even though you are simply testing it.

The reason for this is that, after you have completed your study, you'll either accept or reject your if-then or your null hypothesis. All hypothesis testing examples should be measurable and able to be confirmed or denied. You cannot confirm a question, only a statement! 

In fact, you come up with hypothesis examples all the time! For instance, when you guess on the outcome of a basketball game, you don’t say, “Will the Miami Heat beat the Boston Celtics?” but instead, “I think the Miami Heat will beat the Boston Celtics.” You state it as if it is already true, even if it turns out you’re wrong. You do the same thing when writing your hypothesis.

Additionally, keep in mind that hypotheses can range from very specific to very broad.  These hypotheses can be specific, but if your hypothesis testing examples involve a broad range of causes and effects, your hypothesis can also be broad.  

The Two Types of Hypotheses

Now that you understand what goes into a hypothesis, it’s time to look more closely at the two most common types of hypothesis: the if-then hypothesis and the null hypothesis.

#1: If-Then Hypotheses

First of all, if-then hypotheses typically follow this formula:

If ____ happens, then ____ will happen.

The goal of this type of hypothesis is to test the causal relationship between the independent and dependent variable. It’s fairly simple, and each hypothesis can vary in how detailed it can be. We create if-then hypotheses all the time with our daily predictions. Here are some examples of hypotheses that use an if-then structure from daily life: 

• If I get enough sleep, I’ll be able to get more work done tomorrow.
• If the bus is on time, I can make it to my friend’s birthday party. 
• If I study every night this week, I’ll get a better grade on my exam. 

In each of these situations, you’re making a guess on how an independent variable (sleep, time, or studying) will affect a dependent variable (the amount of work you can do, making it to a party on time, or getting better grades). 

You may still be asking, “What is an example of a hypothesis used in scientific research?” Take one of the hypothesis examples from a real-world study on whether using technology before bed affects children’s sleep patterns. The hypothesis read s:

“We hypothesized that increased hours of tablet- and phone-based screen time at bedtime would be inversely correlated with sleep quality and child attention.”

It might not look like it, but this is an if-then statement. The researchers basically said, “If children have more screen usage at bedtime, then their quality of sleep and attention will be worse.” The sleep quality and attention are the dependent variables and the screen usage is the independent variable. (Usually, the independent variable comes after the “if” and the dependent variable comes after the “then,” as it is the independent variable that affects the dependent variable.) This is an excellent example of how flexible hypothesis statements can be, as long as the general idea of “if-then” and the independent and dependent variables are present.

#2: Null Hypotheses

Your if-then hypothesis is not the only one needed to complete a successful experiment, however. You also need a null hypothesis to test it against. In its most basic form, the null hypothesis is the opposite of your if-then hypothesis . When you write your null hypothesis, you are writing a hypothesis that suggests that your guess is not true, and that the independent and dependent variables have no relationship .

One null hypothesis for the cell phone and sleep study from the last section might say: 

“If children have more screen usage at bedtime, their quality of sleep and attention will not be worse.” 

In this case, this is a null hypothesis because it’s asking the opposite of the original thesis! 

Conversely, if your if-then hypothesis suggests that your two variables have no relationship, then your null hypothesis would suggest that there is one. So, pretend that there is a study that is asking the question, “Does the amount of followers on Instagram influence how long people spend on the app?” The independent variable is the amount of followers, and the dependent variable is the time spent. But if you, as the researcher, don’t think there is a relationship between the number of followers and time spent, you might write an if-then hypothesis that reads:

“If people have many followers on Instagram, they will not spend more time on the app than people who have less.”

In this case, the if-then suggests there isn’t a relationship between the variables. In that case, one of the null hypothesis examples might say:

“If people have many followers on Instagram, they will spend more time on the app than people who have less.”

You then test both the if-then and the null hypothesis to gauge if there is a relationship between the variables, and if so, how much of a relationship. 

4 Tips to Write the Best Hypothesis

If you’re going to take the time to hold an experiment, whether in school or by yourself, you’re also going to want to take the time to make sure your hypothesis is a good one. The best hypotheses have four major elements in common: plausibility, defined concepts, observability, and general explanation.

#1: Plausibility

At first glance, this quality of a hypothesis might seem obvious. When your hypothesis is plausible, that means it’s possible given what we know about science and general common sense. However, improbable hypotheses are more common than you might think. 

Imagine you’re studying weight gain and television watching habits. If you hypothesize that people who watch more than  twenty hours of television a week will gain two hundred pounds or more over the course of a year, this might be improbable (though it’s potentially possible). Consequently, c ommon sense can tell us the results of the study before the study even begins.

Improbable hypotheses generally go against  science, as well. Take this hypothesis example: 

“If a person smokes one cigarette a day, then they will have lungs just as healthy as the average person’s.” 

This hypothesis is obviously untrue, as studies have shown again and again that cigarettes negatively affect lung health. You must be careful that your hypotheses do not reflect your own personal opinion more than they do scientifically-supported findings. This plausibility points to the necessity of research before the hypothesis is written to make sure that your hypothesis has not already been disproven.

#2: Defined Concepts

The more advanced you are in your studies, the more likely that the terms you’re using in your hypothesis are specific to a limited set of knowledge. One of the hypothesis testing examples might include the readability of printed text in newspapers, where you might use words like “kerning” and “x-height.” Unless your readers have a background in graphic design, it’s likely that they won’t know what you mean by these terms. Thus, it’s important to either write what they mean in the hypothesis itself or in the report before the hypothesis.

Here’s what we mean. Which of the following sentences makes more sense to the common person?

If the kerning is greater than average, more words will be read per minute.

If the space between letters is greater than average, more words will be read per minute.

For people reading your report that are not experts in typography, simply adding a few more words will be helpful in clarifying exactly what the experiment is all about. It’s always a good idea to make your research and findings as accessible as possible. 

Good hypotheses ensure that you can observe the results. 

#3: Observability

In order to measure the truth or falsity of your hypothesis, you must be able to see your variables and the way they interact. For instance, if your hypothesis is that the flight patterns of satellites affect the strength of certain television signals, yet you don’t have a telescope to view the satellites or a television to monitor the signal strength, you cannot properly observe your hypothesis and thus cannot continue your study.

Some variables may seem easy to observe, but if you do not have a system of measurement in place, you cannot observe your hypothesis properly. Here’s an example: if you’re experimenting on the effect of healthy food on overall happiness, but you don’t have a way to monitor and measure what “overall happiness” means, your results will not reflect the truth. Monitoring how often someone smiles for a whole day is not reasonably observable, but having the participants state how happy they feel on a scale of one to ten is more observable. 

In writing your hypothesis, always keep in mind how you'll execute the experiment.

#4: Generalizability 

Perhaps you’d like to study what color your best friend wears the most often by observing and documenting the colors she wears each day of the week. This might be fun information for her and you to know, but beyond you two, there aren’t many people who could benefit from this experiment. When you start an experiment, you should note how generalizable your findings may be if they are confirmed. Generalizability is basically how common a particular phenomenon is to other people’s everyday life.

Let’s say you’re asking a question about the health benefits of eating an apple for one day only, you need to realize that the experiment may be too specific to be helpful. It does not help to explain a phenomenon that many people experience. If you find yourself with too specific of a hypothesis, go back to asking the big question: what is it that you want to know, and what do you think will happen between your two variables?

Hypothesis Testing Examples

We know it can be hard to write a good hypothesis unless you’ve seen some good hypothesis examples. We’ve included four hypothesis examples based on some made-up experiments. Use these as templates or launch pads for coming up with your own hypotheses.

Experiment #1: Students Studying Outside (Writing a Hypothesis)

You are a student at PrepScholar University. When you walk around campus, you notice that, when the temperature is above 60 degrees, more students study in the quad. You want to know when your fellow students are more likely to study outside. With this information, how do you make the best hypothesis possible?

You must remember to make additional observations and do secondary research before writing your hypothesis. In doing so, you notice that no one studies outside when it’s 75 degrees and raining, so this should be included in your experiment. Also, studies done on the topic beforehand suggested that students are more likely to study in temperatures less than 85 degrees. With this in mind, you feel confident that you can identify your variables and write your hypotheses:

If-then: “If the temperature in Fahrenheit is less than 60 degrees, significantly fewer students will study outside.”

Null: “If the temperature in Fahrenheit is less than 60 degrees, the same number of students will study outside as when it is more than 60 degrees.”

These hypotheses are plausible, as the temperatures are reasonably within the bounds of what is possible. The number of people in the quad is also easily observable. It is also not a phenomenon specific to only one person or at one time, but instead can explain a phenomenon for a broader group of people.

To complete this experiment, you pick the month of October to observe the quad. Every day (except on the days where it’s raining)from 3 to 4 PM, when most classes have released for the day, you observe how many people are on the quad. You measure how many people come  and how many leave. You also write down the temperature on the hour. 

After writing down all of your observations and putting them on a graph, you find that the most students study on the quad when it is 70 degrees outside, and that the number of students drops a lot once the temperature reaches 60 degrees or below. In this case, your research report would state that you accept or “failed to reject” your first hypothesis with your findings.

Experiment #2: The Cupcake Store (Forming a Simple Experiment)

Let’s say that you work at a bakery. You specialize in cupcakes, and you make only two colors of frosting: yellow and purple. You want to know what kind of customers are more likely to buy what kind of cupcake, so you set up an experiment. Your independent variable is the customer’s gender, and the dependent variable is the color of the frosting. What is an example of a hypothesis that might answer the question of this study?

Here’s what your hypotheses might look like: 

If-then: “If customers’ gender is female, then they will buy more yellow cupcakes than purple cupcakes.”

Null: “If customers’ gender is female, then they will be just as likely to buy purple cupcakes as yellow cupcakes.”

This is a pretty simple experiment! It passes the test of plausibility (there could easily be a difference), defined concepts (there’s nothing complicated about cupcakes!), observability (both color and gender can be easily observed), and general explanation ( this would potentially help you make better business decisions ).

Experiment #3: Backyard Bird Feeders (Integrating Multiple Variables and Rejecting the If-Then Hypothesis)

While watching your backyard bird feeder, you realized that different birds come on the days when you change the types of seeds. You decide that you want to see more cardinals in your backyard, so you decide to see what type of food they like the best and set up an experiment. 

However, one morning, you notice that, while some cardinals are present, blue jays are eating out of your backyard feeder filled with millet. You decide that, of all of the other birds, you would like to see the blue jays the least. This means you'll have more than one variable in your hypothesis. Your new hypotheses might look like this: 

If-then: “If sunflower seeds are placed in the bird feeders, then more cardinals will come than blue jays. If millet is placed in the bird feeders, then more blue jays will come than cardinals.”

Null: “If either sunflower seeds or millet are placed in the bird, equal numbers of cardinals and blue jays will come.”

Through simple observation, you actually find that cardinals come as often as blue jays when sunflower seeds or millet is in the bird feeder. In this case, you would reject your “if-then” hypothesis and “fail to reject” your null hypothesis . You cannot accept your first hypothesis, because it’s clearly not true. Instead you found that there was actually no relation between your different variables. Consequently, you would need to run more experiments with different variables to see if the new variables impact the results.

Experiment #4: In-Class Survey (Including an Alternative Hypothesis)

You’re about to give a speech in one of your classes about the importance of paying attention. You want to take this opportunity to test a hypothesis you’ve had for a while: 

If-then: If students sit in the first two rows of the classroom, then they will listen better than students who do not.

Null: If students sit in the first two rows of the classroom, then they will not listen better or worse than students who do not.

You give your speech and then ask your teacher if you can hand out a short survey to the class. On the survey, you’ve included questions about some of the topics you talked about. When you get back the results, you’re surprised to see that not only do the students in the first two rows not pay better attention, but they also scored worse than students in other parts of the classroom! Here, both your if-then and your null hypotheses are not representative of your findings. What do you do?

This is when you reject both your if-then and null hypotheses and instead create an alternative hypothesis . This type of hypothesis is used in the rare circumstance that neither of your hypotheses is able to capture your findings . Now you can use what you’ve learned to draft new hypotheses and test again! 

Key Takeaways: Hypothesis Writing

The more comfortable you become with writing hypotheses, the better they will become. The structure of hypotheses is flexible and may need to be changed depending on what topic you are studying. The most important thing to remember is the purpose of your hypothesis and the difference between the if-then and the null . From there, in forming your hypothesis, you should constantly be asking questions, making observations, doing secondary research, and considering your variables. After you have written your hypothesis, be sure to edit it so that it is plausible, clearly defined, observable, and helpful in explaining a general phenomenon.

Writing a hypothesis is something that everyone, from elementary school children competing in a science fair to professional scientists in a lab, needs to know how to do. Hypotheses are vital in experiments and in properly executing the scientific method . When done correctly, hypotheses will set up your studies for success and help you to understand the world a little better, one experiment at a time.

What’s Next?

If you’re studying for the science portion of the ACT, there’s definitely a lot you need to know. We’ve got the tools to help, though! Start by checking out our ultimate study guide for the ACT Science subject test. Once you read through that, be sure to download our recommended ACT Science practice tests , since they’re one of the most foolproof ways to improve your score. (And don’t forget to check out our expert guide book , too.)

If you love science and want to major in a scientific field, you should start preparing in high school . Here are the science classes you should take to set yourself up for success.

If you’re trying to think of science experiments you can do for class (or for a science fair!), here’s a list of 37 awesome science experiments you can do at home

Ashley Sufflé Robinson has a Ph.D. in 19th Century English Literature. As a content writer for PrepScholar, Ashley is passionate about giving college-bound students the in-depth information they need to get into the school of their dreams.

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The Craft of Writing a Strong Hypothesis

Table of Contents

Writing a hypothesis is one of the essential elements of a scientific research paper. It needs to be to the point, clearly communicating what your research is trying to accomplish. A blurry, drawn-out, or complexly-structured hypothesis can confuse your readers. Or worse, the editor and peer reviewers.

A captivating hypothesis is not too intricate. This blog will take you through the process so that, by the end of it, you have a better idea of how to convey your research paper's intent in just one sentence.

What is a Hypothesis?

The first step in your scientific endeavor, a hypothesis, is a strong, concise statement that forms the basis of your research. It is not the same as a thesis statement , which is a brief summary of your research paper .

The sole purpose of a hypothesis is to predict your paper's findings, data, and conclusion. It comes from a place of curiosity and intuition . When you write a hypothesis, you're essentially making an educated guess based on scientific prejudices and evidence, which is further proven or disproven through the scientific method.

The reason for undertaking research is to observe a specific phenomenon. A hypothesis, therefore, lays out what the said phenomenon is. And it does so through two variables, an independent and dependent variable.

The independent variable is the cause behind the observation, while the dependent variable is the effect of the cause. A good example of this is “mixing red and blue forms purple.” In this hypothesis, mixing red and blue is the independent variable as you're combining the two colors at your own will. The formation of purple is the dependent variable as, in this case, it is conditional to the independent variable.

Different Types of Hypotheses‌

Types of hypotheses

Some would stand by the notion that there are only two types of hypotheses: a Null hypothesis and an Alternative hypothesis. While that may have some truth to it, it would be better to fully distinguish the most common forms as these terms come up so often, which might leave you out of context.

Apart from Null and Alternative, there are Complex, Simple, Directional, Non-Directional, Statistical, and Associative and casual hypotheses. They don't necessarily have to be exclusive, as one hypothesis can tick many boxes, but knowing the distinctions between them will make it easier for you to construct your own.

1. Null hypothesis

A null hypothesis proposes no relationship between two variables. Denoted by H 0 , it is a negative statement like “Attending physiotherapy sessions does not affect athletes' on-field performance.” Here, the author claims physiotherapy sessions have no effect on on-field performances. Even if there is, it's only a coincidence.

2. Alternative hypothesis

Considered to be the opposite of a null hypothesis, an alternative hypothesis is donated as H1 or Ha. It explicitly states that the dependent variable affects the independent variable. A good  alternative hypothesis example is “Attending physiotherapy sessions improves athletes' on-field performance.” or “Water evaporates at 100 °C. ” The alternative hypothesis further branches into directional and non-directional.

• Directional hypothesis: A hypothesis that states the result would be either positive or negative is called directional hypothesis. It accompanies H1 with either the ‘<' or ‘>' sign.
• Non-directional hypothesis: A non-directional hypothesis only claims an effect on the dependent variable. It does not clarify whether the result would be positive or negative. The sign for a non-directional hypothesis is ‘≠.'

3. Simple hypothesis

A simple hypothesis is a statement made to reflect the relation between exactly two variables. One independent and one dependent. Consider the example, “Smoking is a prominent cause of lung cancer." The dependent variable, lung cancer, is dependent on the independent variable, smoking.

4. Complex hypothesis

In contrast to a simple hypothesis, a complex hypothesis implies the relationship between multiple independent and dependent variables. For instance, “Individuals who eat more fruits tend to have higher immunity, lesser cholesterol, and high metabolism.” The independent variable is eating more fruits, while the dependent variables are higher immunity, lesser cholesterol, and high metabolism.

5. Associative and casual hypothesis

Associative and casual hypotheses don't exhibit how many variables there will be. They define the relationship between the variables. In an associative hypothesis, changing any one variable, dependent or independent, affects others. In a casual hypothesis, the independent variable directly affects the dependent.

6. Empirical hypothesis

Also referred to as the working hypothesis, an empirical hypothesis claims a theory's validation via experiments and observation. This way, the statement appears justifiable and different from a wild guess.

Say, the hypothesis is “Women who take iron tablets face a lesser risk of anemia than those who take vitamin B12.” This is an example of an empirical hypothesis where the researcher  the statement after assessing a group of women who take iron tablets and charting the findings.

7. Statistical hypothesis

The point of a statistical hypothesis is to test an already existing hypothesis by studying a population sample. Hypothesis like “44% of the Indian population belong in the age group of 22-27.” leverage evidence to prove or disprove a particular statement.

Characteristics of a Good Hypothesis

Writing a hypothesis is essential as it can make or break your research for you. That includes your chances of getting published in a journal. So when you're designing one, keep an eye out for these pointers:

• A research hypothesis has to be simple yet clear to look justifiable enough.
• It has to be testable — your research would be rendered pointless if too far-fetched into reality or limited by technology.
• It has to be precise about the results —what you are trying to do and achieve through it should come out in your hypothesis.
• A research hypothesis should be self-explanatory, leaving no doubt in the reader's mind.
• If you are developing a relational hypothesis, you need to include the variables and establish an appropriate relationship among them.
• A hypothesis must keep and reflect the scope for further investigations and experiments.

Separating a Hypothesis from a Prediction

Outside of academia, hypothesis and prediction are often used interchangeably. In research writing, this is not only confusing but also incorrect. And although a hypothesis and prediction are guesses at their core, there are many differences between them.

A hypothesis is an educated guess or even a testable prediction validated through research. It aims to analyze the gathered evidence and facts to define a relationship between variables and put forth a logical explanation behind the nature of events.

Predictions are assumptions or expected outcomes made without any backing evidence. They are more fictionally inclined regardless of where they originate from.

For this reason, a hypothesis holds much more weight than a prediction. It sticks to the scientific method rather than pure guesswork. "Planets revolve around the Sun." is an example of a hypothesis as it is previous knowledge and observed trends. Additionally, we can test it through the scientific method.

Whereas "COVID-19 will be eradicated by 2030." is a prediction. Even though it results from past trends, we can't prove or disprove it. So, the only way this gets validated is to wait and watch if COVID-19 cases end by 2030.

Finally, How to Write a Hypothesis

Quick tips on writing a hypothesis

1.  Be clear about your research question

A hypothesis should instantly address the research question or the problem statement. To do so, you need to ask a question. Understand the constraints of your undertaken research topic and then formulate a simple and topic-centric problem. Only after that can you develop a hypothesis and further test for evidence.

2. Carry out a recce

Once you have your research's foundation laid out, it would be best to conduct preliminary research. Go through previous theories, academic papers, data, and experiments before you start curating your research hypothesis. It will give you an idea of your hypothesis's viability or originality.

Making use of references from relevant research papers helps draft a good research hypothesis. SciSpace Discover offers a repository of over 270 million research papers to browse through and gain a deeper understanding of related studies on a particular topic. Additionally, you can use SciSpace Copilot , your AI research assistant, for reading any lengthy research paper and getting a more summarized context of it. A hypothesis can be formed after evaluating many such summarized research papers. Copilot also offers explanations for theories and equations, explains paper in simplified version, allows you to highlight any text in the paper or clip math equations and tables and provides a deeper, clear understanding of what is being said. This can improve the hypothesis by helping you identify potential research gaps.

3. Create a 3-dimensional hypothesis

Variables are an essential part of any reasonable hypothesis. So, identify your independent and dependent variable(s) and form a correlation between them. The ideal way to do this is to write the hypothetical assumption in the ‘if-then' form. If you use this form, make sure that you state the predefined relationship between the variables.

In another way, you can choose to present your hypothesis as a comparison between two variables. Here, you must specify the difference you expect to observe in the results.

4. Write the first draft

Now that everything is in place, it's time to write your hypothesis. For starters, create the first draft. In this version, write what you expect to find from your research.

Clearly separate your independent and dependent variables and the link between them. Don't fixate on syntax at this stage. The goal is to ensure your hypothesis addresses the issue.

5. Proof your hypothesis

After preparing the first draft of your hypothesis, you need to inspect it thoroughly. It should tick all the boxes, like being concise, straightforward, relevant, and accurate. Your final hypothesis has to be well-structured as well.

Research projects are an exciting and crucial part of being a scholar. And once you have your research question, you need a great hypothesis to begin conducting research. Thus, knowing how to write a hypothesis is very important.

Now that you have a firmer grasp on what a good hypothesis constitutes, the different kinds there are, and what process to follow, you will find it much easier to write your hypothesis, which ultimately helps your research.

Now it's easier than ever to streamline your research workflow with SciSpace Discover . Its integrated, comprehensive end-to-end platform for research allows scholars to easily discover, write and publish their research and fosters collaboration.

It includes everything you need, including a repository of over 270 million research papers across disciplines, SEO-optimized summaries and public profiles to show your expertise and experience.

If you found these tips on writing a research hypothesis useful, head over to our blog on Statistical Hypothesis Testing to learn about the top researchers, papers, and institutions in this domain.

Frequently Asked Questions (FAQs)

1. what is the definition of hypothesis.

According to the Oxford dictionary, a hypothesis is defined as “An idea or explanation of something that is based on a few known facts, but that has not yet been proved to be true or correct”.

2. What is an example of hypothesis?

The hypothesis is a statement that proposes a relationship between two or more variables. An example: "If we increase the number of new users who join our platform by 25%, then we will see an increase in revenue."

3. What is an example of null hypothesis?

A null hypothesis is a statement that there is no relationship between two variables. The null hypothesis is written as H0. The null hypothesis states that there is no effect. For example, if you're studying whether or not a particular type of exercise increases strength, your null hypothesis will be "there is no difference in strength between people who exercise and people who don't."

4. What are the types of research?

• Fundamental research

• Applied research

• Qualitative research

• Quantitative research

• Mixed research

• Exploratory research

• Longitudinal research

• Cross-sectional research

• Field research

• Laboratory research

• Fixed research

• Flexible research

• Action research

• Policy research

• Classification research

• Comparative research

• Causal research

• Inductive research

• Deductive research

5. How to write a hypothesis?

• Your hypothesis should be able to predict the relationship and outcome.

• Avoid wordiness by keeping it simple and brief.

• Your hypothesis should contain observable and testable outcomes.

• Your hypothesis should be relevant to the research question.

6. What are the 2 types of hypothesis?

• Null hypotheses are used to test the claim that "there is no difference between two groups of data".

• Alternative hypotheses test the claim that "there is a difference between two data groups".

7. Difference between research question and research hypothesis?

A research question is a broad, open-ended question you will try to answer through your research. A hypothesis is a statement based on prior research or theory that you expect to be true due to your study. Example - Research question: What are the factors that influence the adoption of the new technology? Research hypothesis: There is a positive relationship between age, education and income level with the adoption of the new technology.

8. What is plural for hypothesis?

The plural of hypothesis is hypotheses. Here's an example of how it would be used in a statement, "Numerous well-considered hypotheses are presented in this part, and they are supported by tables and figures that are well-illustrated."

9. What is the red queen hypothesis?

The red queen hypothesis in evolutionary biology states that species must constantly evolve to avoid extinction because if they don't, they will be outcompeted by other species that are evolving. Leigh Van Valen first proposed it in 1973; since then, it has been tested and substantiated many times.

10. Who is known as the father of null hypothesis?

The father of the null hypothesis is Sir Ronald Fisher. He published a paper in 1925 that introduced the concept of null hypothesis testing, and he was also the first to use the term itself.

11. When to reject null hypothesis?

You need to find a significant difference between your two populations to reject the null hypothesis. You can determine that by running statistical tests such as an independent sample t-test or a dependent sample t-test. You should reject the null hypothesis if the p-value is less than 0.05.

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What Is A Research (Scientific) Hypothesis? A plain-language explainer + examples

By:  Derek Jansen (MBA)  | Reviewed By: Dr Eunice Rautenbach | June 2020

If you’re new to the world of research, or it’s your first time writing a dissertation or thesis, you’re probably noticing that the words “research hypothesis” and “scientific hypothesis” are used quite a bit, and you’re wondering what they mean in a research context .

“Hypothesis” is one of those words that people use loosely, thinking they understand what it means. However, it has a very specific meaning within academic research. So, it’s important to understand the exact meaning before you start hypothesizing. 

Research Hypothesis 101

• What is a hypothesis ?
• What is a research hypothesis (scientific hypothesis)?
• Requirements for a research hypothesis
• Definition of a research hypothesis
• The null hypothesis

What is a hypothesis?

Let’s start with the general definition of a hypothesis (not a research hypothesis or scientific hypothesis), according to the Cambridge Dictionary:

Hypothesis: an idea or explanation for something that is based on known facts but has not yet been proved.

In other words, it’s a statement that provides an explanation for why or how something works, based on facts (or some reasonable assumptions), but that has not yet been specifically tested . For example, a hypothesis might look something like this:

Hypothesis: sleep impacts academic performance.

This statement predicts that academic performance will be influenced by the amount and/or quality of sleep a student engages in – sounds reasonable, right? It’s based on reasonable assumptions , underpinned by what we currently know about sleep and health (from the existing literature). So, loosely speaking, we could call it a hypothesis, at least by the dictionary definition.

But that’s not good enough…

Unfortunately, that’s not quite sophisticated enough to describe a research hypothesis (also sometimes called a scientific hypothesis), and it wouldn’t be acceptable in a dissertation, thesis or research paper . In the world of academic research, a statement needs a few more criteria to constitute a true research hypothesis .

What is a research hypothesis?

A research hypothesis (also called a scientific hypothesis) is a statement about the expected outcome of a study (for example, a dissertation or thesis). To constitute a quality hypothesis, the statement needs to have three attributes – specificity , clarity and testability .

Let’s take a look at these more closely.

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Hypothesis Essential #1: Specificity & Clarity

A good research hypothesis needs to be extremely clear and articulate about both what’ s being assessed (who or what variables are involved ) and the expected outcome (for example, a difference between groups, a relationship between variables, etc.).

Let’s stick with our sleepy students example and look at how this statement could be more specific and clear.

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.

As you can see, the statement is very specific as it identifies the variables involved (sleep hours and test grades), the parties involved (two groups of students), as well as the predicted relationship type (a positive relationship). There’s no ambiguity or uncertainty about who or what is involved in the statement, and the expected outcome is clear.

Contrast that to the original hypothesis we looked at – “Sleep impacts academic performance” – and you can see the difference. “Sleep” and “academic performance” are both comparatively vague , and there’s no indication of what the expected relationship direction is (more sleep or less sleep). As you can see, specificity and clarity are key.

Hypothesis Essential #2: Testability (Provability)

A statement must be testable to qualify as a research hypothesis. In other words, there needs to be a way to prove (or disprove) the statement. If it’s not testable, it’s not a hypothesis – simple as that.

For example, consider the hypothesis we mentioned earlier:

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.  

We could test this statement by undertaking a quantitative study involving two groups of students, one that gets 8 or more hours of sleep per night for a fixed period, and one that gets less. We could then compare the standardised test results for both groups to see if there’s a statistically significant difference. 

Again, if you compare this to the original hypothesis we looked at – “Sleep impacts academic performance” – you can see that it would be quite difficult to test that statement, primarily because it isn’t specific enough. How much sleep? By who? What type of academic performance?

So, remember the mantra – if you can’t test it, it’s not a hypothesis 🙂

Defining A Research Hypothesis

You’re still with us? Great! Let’s recap and pin down a clear definition of a hypothesis.

A research hypothesis (or scientific hypothesis) is a statement about an expected relationship between variables, or explanation of an occurrence, that is clear, specific and testable.

So, when you write up hypotheses for your dissertation or thesis, make sure that they meet all these criteria. If you do, you’ll not only have rock-solid hypotheses but you’ll also ensure a clear focus for your entire research project.

What about the null hypothesis?

You may have also heard the terms null hypothesis , alternative hypothesis, or H-zero thrown around. At a simple level, the null hypothesis is the counter-proposal to the original hypothesis.

For example, if the hypothesis predicts that there is a relationship between two variables (for example, sleep and academic performance), the null hypothesis would predict that there is no relationship between those variables.

At a more technical level, the null hypothesis proposes that no statistical significance exists in a set of given observations and that any differences are due to chance alone.

And there you have it – hypotheses in a nutshell. 

If you have any questions, be sure to leave a comment below and we’ll do our best to help you. If you need hands-on help developing and testing your hypotheses, consider our private coaching service , where we hold your hand through the research journey.

Psst… there’s more (for free)

This post is part of our dissertation mini-course, which covers everything you need to get started with your dissertation, thesis or research project. 

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16 Comments

Very useful information. I benefit more from getting more information in this regard.

Very great insight,educative and informative. Please give meet deep critics on many research data of public international Law like human rights, environment, natural resources, law of the sea etc

In a book I read a distinction is made between null, research, and alternative hypothesis. As far as I understand, alternative and research hypotheses are the same. Can you please elaborate? Best Afshin

This is a self explanatory, easy going site. I will recommend this to my friends and colleagues.

Very good definition. How can I cite your definition in my thesis? Thank you. Is nul hypothesis compulsory in a research?

It’s a counter-proposal to be proven as a rejection

Please what is the difference between alternate hypothesis and research hypothesis?

It is a very good explanation. However, it limits hypotheses to statistically tasteable ideas. What about for qualitative researches or other researches that involve quantitative data that don’t need statistical tests?

In qualitative research, one typically uses propositions, not hypotheses.

could you please elaborate it more

I’ve benefited greatly from these notes, thank you.

This is very helpful

well articulated ideas are presented here, thank you for being reliable sources of information

Excellent. Thanks for being clear and sound about the research methodology and hypothesis (quantitative research)

I have only a simple question regarding the null hypothesis. – Is the null hypothesis (Ho) known as the reversible hypothesis of the alternative hypothesis (H1? – How to test it in academic research?

this is very important note help me much more

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Research Hypothesis In Psychology: Types, & Examples

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

Some key points about hypotheses:

• A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
• It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
• A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
• Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
• For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
• Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.

Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

• Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.

• Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
• However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

How to Write a Hypothesis

• Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
• Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
• Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
• Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
• Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

• The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
• The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

More Examples

• Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
• Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
• Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
• Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
• Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
• Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
• Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
• Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

Definition of a Hypothesis

What it is and how it's used in sociology

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A hypothesis is a prediction of what will be found at the outcome of a research project and is typically focused on the relationship between two different variables studied in the research. It is usually based on both theoretical expectations about how things work and already existing scientific evidence.

Within social science, a hypothesis can take two forms. It can predict that there is no relationship between two variables, in which case it is a null hypothesis . Or, it can predict the existence of a relationship between variables, which is known as an alternative hypothesis.

In either case, the variable that is thought to either affect or not affect the outcome is known as the independent variable, and the variable that is thought to either be affected or not is the dependent variable.

Researchers seek to determine whether or not their hypothesis, or hypotheses if they have more than one, will prove true. Sometimes they do, and sometimes they do not. Either way, the research is considered successful if one can conclude whether or not a hypothesis is true. 

Null Hypothesis

A researcher has a null hypothesis when she or he believes, based on theory and existing scientific evidence, that there will not be a relationship between two variables. For example, when examining what factors influence a person's highest level of education within the U.S., a researcher might expect that place of birth, number of siblings, and religion would not have an impact on the level of education. This would mean the researcher has stated three null hypotheses.

Alternative Hypothesis

Taking the same example, a researcher might expect that the economic class and educational attainment of one's parents, and the race of the person in question are likely to have an effect on one's educational attainment. Existing evidence and social theories that recognize the connections between wealth and cultural resources , and how race affects access to rights and resources in the U.S. , would suggest that both economic class and educational attainment of the one's parents would have a positive effect on educational attainment. In this case, economic class and educational attainment of one's parents are independent variables, and one's educational attainment is the dependent variable—it is hypothesized to be dependent on the other two.

Conversely, an informed researcher would expect that being a race other than white in the U.S. is likely to have a negative impact on a person's educational attainment. This would be characterized as a negative relationship, wherein being a person of color has a negative effect on one's educational attainment. In reality, this hypothesis proves true, with the exception of Asian Americans , who go to college at a higher rate than whites do. However, Blacks and Hispanics and Latinos are far less likely than whites and Asian Americans to go to college.

Formulating a Hypothesis

Formulating a hypothesis can take place at the very beginning of a research project , or after a bit of research has already been done. Sometimes a researcher knows right from the start which variables she is interested in studying, and she may already have a hunch about their relationships. Other times, a researcher may have an interest in ​a particular topic, trend, or phenomenon, but he may not know enough about it to identify variables or formulate a hypothesis.

Whenever a hypothesis is formulated, the most important thing is to be precise about what one's variables are, what the nature of the relationship between them might be, and how one can go about conducting a study of them.

Updated by Nicki Lisa Cole, Ph.D

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Meaning of hypothesis in English

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• abstraction
• afterthought
• anthropocentrism
• anti-Darwinian
• exceptionalism
• foundation stone
• great minds think alike idiom
• non-dogmatic
• non-empirical
• non-material
• non-practical
• social Darwinism
• supersensible
• the domino theory

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5.2 - writing hypotheses.

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

• At this point we can write hypotheses for a single mean (\(\mu\)), paired means(\(\mu_d\)), a single proportion (\(p\)), the difference between two independent means (\(\mu_1-\mu_2\)), the difference between two proportions (\(p_1-p_2\)), a simple linear regression slope (\(\beta\)), and a correlation (\(\rho\)). 
• The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
• The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., \(\mu_0\) and \(p_0\)). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., \(p\) and \(\mu\)).  The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

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Hypothesis is a testable statement that explains what is happening or observed. It proposes the relation between the various participating variables. Hypothesis is also called Theory, Thesis, Guess, Assumption, or Suggestion. Hypothesis creates a structure that guides the search for knowledge.

In this article, we will learn what is hypothesis, its characteristics, types, and examples. We will also learn how hypothesis helps in scientific research.

What is Hypothesis?

A hypothesis is a suggested idea or plan that has little proof, meant to lead to more study. It’s mainly a smart guess or suggested answer to a problem that can be checked through study and trial. In science work, we make guesses called hypotheses to try and figure out what will happen in tests or watching. These are not sure things but rather ideas that can be proved or disproved based on real-life proofs. A good theory is clear and can be tested and found wrong if the proof doesn’t support it.

Hypothesis Meaning

A hypothesis is a proposed statement that is testable and is given for something that happens or observed.

• It is made using what we already know and have seen, and it’s the basis for scientific research.
• A clear guess tells us what we think will happen in an experiment or study.
• It’s a testable clue that can be proven true or wrong with real-life facts and checking it out carefully.
• It usually looks like a “if-then” rule, showing the expected cause and effect relationship between what’s being studied.

Characteristics of Hypothesis

Here are some key characteristics of a hypothesis:

• Testable: An idea (hypothesis) should be made so it can be tested and proven true through doing experiments or watching. It should show a clear connection between things.
• Specific: It needs to be easy and on target, talking about a certain part or connection between things in a study.
• Falsifiable: A good guess should be able to show it’s wrong. This means there must be a chance for proof or seeing something that goes against the guess.
• Logical and Rational: It should be based on things we know now or have seen, giving a reasonable reason that fits with what we already know.
• Predictive: A guess often tells what to expect from an experiment or observation. It gives a guide for what someone might see if the guess is right.
• Concise: It should be short and clear, showing the suggested link or explanation simply without extra confusion.
• Grounded in Research: A guess is usually made from before studies, ideas or watching things. It comes from a deep understanding of what is already known in that area.
• Flexible: A guess helps in the research but it needs to change or fix when new information comes up.
• Relevant: It should be related to the question or problem being studied, helping to direct what the research is about.
• Empirical: Hypotheses come from observations and can be tested using methods based on real-world experiences.

Sources of Hypothesis

Hypotheses can come from different places based on what you’re studying and the kind of research. Here are some common sources from which hypotheses may originate:

• Existing Theories: Often, guesses come from well-known science ideas. These ideas may show connections between things or occurrences that scientists can look into more.
• Observation and Experience: Watching something happen or having personal experiences can lead to guesses. We notice odd things or repeat events in everyday life and experiments. This can make us think of guesses called hypotheses.
• Previous Research: Using old studies or discoveries can help come up with new ideas. Scientists might try to expand or question current findings, making guesses that further study old results.
• Literature Review: Looking at books and research in a subject can help make guesses. Noticing missing parts or mismatches in previous studies might make researchers think up guesses to deal with these spots.
• Problem Statement or Research Question: Often, ideas come from questions or problems in the study. Making clear what needs to be looked into can help create ideas that tackle certain parts of the issue.
• Analogies or Comparisons: Making comparisons between similar things or finding connections from related areas can lead to theories. Understanding from other fields could create new guesses in a different situation.
• Hunches and Speculation: Sometimes, scientists might get a gut feeling or make guesses that help create ideas to test. Though these may not have proof at first, they can be a beginning for looking deeper.
• Technology and Innovations: New technology or tools might make guesses by letting us look at things that were hard to study before.
• Personal Interest and Curiosity: People’s curiosity and personal interests in a topic can help create guesses. Scientists could make guesses based on their own likes or love for a subject.

Types of Hypothesis

Here are some common types of hypotheses:

Simple Hypothesis

Complex hypothesis, directional hypothesis.

• Non-directional Hypothesis

Null Hypothesis (H0)

Alternative hypothesis (h1 or ha), statistical hypothesis, research hypothesis, associative hypothesis, causal hypothesis.

Simple Hypothesis guesses a connection between two things. It says that there is a connection or difference between variables, but it doesn’t tell us which way the relationship goes.

Complex Hypothesis tells us what will happen when more than two things are connected. It looks at how different things interact and may be linked together.

Directional Hypothesis says how one thing is related to another. For example, it guesses that one thing will help or hurt another thing.

Non-Directional Hypothesis

Non-Directional Hypothesis are the one that don’t say how the relationship between things will be. They just say that there is a connection, without telling which way it goes.

Null hypothesis is a statement that says there’s no connection or difference between different things. It implies that any seen impacts are because of luck or random changes in the information.

Alternative Hypothesis is different from the null hypothesis and shows that there’s a big connection or gap between variables. Scientists want to say no to the null hypothesis and choose the alternative one.

Statistical Hypotheis are used in math testing and include making ideas about what groups or bits of them look like. You aim to get information or test certain things using these top-level, common words only.

Research Hypothesis comes from the research question and tells what link is expected between things or factors. It leads the study and chooses where to look more closely.

Associative Hypotheis guesses that there is a link or connection between things without really saying it caused them. It means that when one thing changes, it is connected to another thing changing.

Causal Hypothesis are different from other ideas because they say that one thing causes another. This means there’s a cause and effect relationship between variables involved in the situation. They say that when one thing changes, it directly makes another thing change.

Hypothesis Examples

Following are the examples of hypotheses based on their types:

Simple Hypothesis Example

• Studying more can help you do better on tests.
• Getting more sun makes people have higher amounts of vitamin D.

Complex Hypothesis Example

• How rich you are, how easy it is to get education and healthcare greatly affects the number of years people live.
• A new medicine’s success relies on the amount used, how old a person is who takes it and their genes.

Directional Hypothesis Example

• Drinking more sweet drinks is linked to a higher body weight score.
• Too much stress makes people less productive at work.

Non-directional Hypothesis Example

• Drinking caffeine can affect how well you sleep.
• People often like different kinds of music based on their gender.
• The average test scores of Group A and Group B are not much different.
• There is no connection between using a certain fertilizer and how much it helps crops grow.

Alternative Hypothesis (Ha)

• Patients on Diet A have much different cholesterol levels than those following Diet B.
• Exposure to a certain type of light can change how plants grow compared to normal sunlight.
• The average smarts score of kids in a certain school area is 100.
• The usual time it takes to finish a job using Method A is the same as with Method B.
• Having more kids go to early learning classes helps them do better in school when they get older.
• Using specific ways of talking affects how much customers get involved in marketing activities.
• Regular exercise helps to lower the chances of heart disease.
• Going to school more can help people make more money.
• Playing violent video games makes teens more likely to act aggressively.
• Less clean air directly impacts breathing health in city populations.

Functions of Hypothesis

Hypotheses have many important jobs in the process of scientific research. Here are the key functions of hypotheses:

• Guiding Research: Hypotheses give a clear and exact way for research. They act like guides, showing the predicted connections or results that scientists want to study.
• Formulating Research Questions: Research questions often create guesses. They assist in changing big questions into particular, checkable things. They guide what the study should be focused on.
• Setting Clear Objectives: Hypotheses set the goals of a study by saying what connections between variables should be found. They set the targets that scientists try to reach with their studies.
• Testing Predictions: Theories guess what will happen in experiments or observations. By doing tests in a planned way, scientists can check if what they see matches the guesses made by their ideas.
• Providing Structure: Theories give structure to the study process by arranging thoughts and ideas. They aid scientists in thinking about connections between things and plan experiments to match.
• Focusing Investigations: Hypotheses help scientists focus on certain parts of their study question by clearly saying what they expect links or results to be. This focus makes the study work better.
• Facilitating Communication: Theories help scientists talk to each other effectively. Clearly made guesses help scientists to tell others what they plan, how they will do it and the results expected. This explains things well with colleagues in a wide range of audiences.
• Generating Testable Statements: A good guess can be checked, which means it can be looked at carefully or tested by doing experiments. This feature makes sure that guesses add to the real information used in science knowledge.
• Promoting Objectivity: Guesses give a clear reason for study that helps guide the process while reducing personal bias. They motivate scientists to use facts and data as proofs or disprovals for their proposed answers.
• Driving Scientific Progress: Making, trying out and adjusting ideas is a cycle. Even if a guess is proven right or wrong, the information learned helps to grow knowledge in one specific area.

How Hypothesis help in Scientific Research?

Researchers use hypotheses to put down their thoughts directing how the experiment would take place. Following are the steps that are involved in the scientific method:

• Initiating Investigations: Hypotheses are the beginning of science research. They come from watching, knowing what’s already known or asking questions. This makes scientists make certain explanations that need to be checked with tests.
• Formulating Research Questions: Ideas usually come from bigger questions in study. They help scientists make these questions more exact and testable, guiding the study’s main point.
• Setting Clear Objectives: Hypotheses set the goals of a study by stating what we think will happen between different things. They set the goals that scientists want to reach by doing their studies.
• Designing Experiments and Studies: Assumptions help plan experiments and watchful studies. They assist scientists in knowing what factors to measure, the techniques they will use and gather data for a proposed reason.
• Testing Predictions: Ideas guess what will happen in experiments or observations. By checking these guesses carefully, scientists can see if the seen results match up with what was predicted in each hypothesis.
• Analysis and Interpretation of Data: Hypotheses give us a way to study and make sense of information. Researchers look at what they found and see if it matches the guesses made in their theories. They decide if the proof backs up or disagrees with these suggested reasons why things are happening as expected.
• Encouraging Objectivity: Hypotheses help make things fair by making sure scientists use facts and information to either agree or disagree with their suggested reasons. They lessen personal preferences by needing proof from experience.
• Iterative Process: People either agree or disagree with guesses, but they still help the ongoing process of science. Findings from testing ideas make us ask new questions, improve those ideas and do more tests. It keeps going on in the work of science to keep learning things.

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Summary – Hypothesis

A hypothesis is a testable statement serving as an initial explanation for phenomena, based on observations, theories, or existing knowledge. It acts as a guiding light for scientific research, proposing potential relationships between variables that can be empirically tested through experiments and observations. The hypothesis must be specific, testable, falsifiable, and grounded in prior research or observation, laying out a predictive, if-then scenario that details a cause-and-effect relationship. It originates from various sources including existing theories, observations, previous research, and even personal curiosity, leading to different types, such as simple, complex, directional, non-directional, null, and alternative hypotheses, each serving distinct roles in research methodology. The hypothesis not only guides the research process by shaping objectives and designing experiments but also facilitates objective analysis and interpretation of data, ultimately driving scientific progress through a cycle of testing, validation, and refinement.

FAQs on Hypothesis

What is a hypothesis.

A guess is a possible explanation or forecast that can be checked by doing research and experiments.

What are Components of a Hypothesis?

The components of a Hypothesis are Independent Variable, Dependent Variable, Relationship between Variables, Directionality etc.

What makes a Good Hypothesis?

Testability, Falsifiability, Clarity and Precision, Relevance are some parameters that makes a Good Hypothesis

Can a Hypothesis be Proven True?

You cannot prove conclusively that most hypotheses are true because it’s generally impossible to examine all possible cases for exceptions that would disprove them.

How are Hypotheses Tested?

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data

Can Hypotheses change during Research?

Yes, you can change or improve your ideas based on new information discovered during the research process.

What is the Role of a Hypothesis in Scientific Research?

Hypotheses are used to support scientific research and bring about advancements in knowledge.

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What is Hypothesis?

We have heard of many hypotheses which have led to great inventions in science. Assumptions that are made on the basis of some evidence are known as hypotheses. In this article, let us learn in detail about the hypothesis and the type of hypothesis with examples.

A hypothesis is an assumption that is made based on some evidence. This is the initial point of any investigation that translates the research questions into predictions. It includes components like variables, population and the relation between the variables. A research hypothesis is a hypothesis that is used to test the relationship between two or more variables.

Characteristics of Hypothesis

Following are the characteristics of the hypothesis:

• The hypothesis should be clear and precise to consider it to be reliable.
• If the hypothesis is a relational hypothesis, then it should be stating the relationship between variables.
• The hypothesis must be specific and should have scope for conducting more tests.
• The way of explanation of the hypothesis must be very simple and it should also be understood that the simplicity of the hypothesis is not related to its significance.

Sources of Hypothesis

Following are the sources of hypothesis:

• The resemblance between the phenomenon.
• Observations from past studies, present-day experiences and from the competitors.
• Scientific theories.
• General patterns that influence the thinking process of people.

Types of Hypothesis

There are six forms of hypothesis and they are:

• Simple hypothesis
• Complex hypothesis
• Directional hypothesis
• Non-directional hypothesis
• Null hypothesis
• Associative and casual hypothesis

Simple Hypothesis

It shows a relationship between one dependent variable and a single independent variable. For example – If you eat more vegetables, you will lose weight faster. Here, eating more vegetables is an independent variable, while losing weight is the dependent variable.

Complex Hypothesis

It shows the relationship between two or more dependent variables and two or more independent variables. Eating more vegetables and fruits leads to weight loss, glowing skin, and reduces the risk of many diseases such as heart disease.

Directional Hypothesis

It shows how a researcher is intellectual and committed to a particular outcome. The relationship between the variables can also predict its nature. For example- children aged four years eating proper food over a five-year period are having higher IQ levels than children not having a proper meal. This shows the effect and direction of the effect.

Non-directional Hypothesis

It is used when there is no theory involved. It is a statement that a relationship exists between two variables, without predicting the exact nature (direction) of the relationship.

Null Hypothesis

It provides a statement which is contrary to the hypothesis. It’s a negative statement, and there is no relationship between independent and dependent variables. The symbol is denoted by “H O ”.

Associative and Causal Hypothesis

Associative hypothesis occurs when there is a change in one variable resulting in a change in the other variable. Whereas, the causal hypothesis proposes a cause and effect interaction between two or more variables.

Examples of Hypothesis

Following are the examples of hypotheses based on their types:

• Consumption of sugary drinks every day leads to obesity is an example of a simple hypothesis.
• All lilies have the same number of petals is an example of a null hypothesis.
• If a person gets 7 hours of sleep, then he will feel less fatigue than if he sleeps less. It is an example of a directional hypothesis.

Functions of Hypothesis

Following are the functions performed by the hypothesis:

• Hypothesis helps in making an observation and experiments possible.
• It becomes the start point for the investigation.
• Hypothesis helps in verifying the observations.
• It helps in directing the inquiries in the right direction.

How will Hypothesis help in the Scientific Method?

Researchers use hypotheses to put down their thoughts directing how the experiment would take place. Following are the steps that are involved in the scientific method:

• Formation of question
• Doing background research
• Creation of hypothesis
• Designing an experiment
• Collection of data
• Result analysis
• Summarizing the experiment
• Communicating the results

Frequently Asked Questions – FAQs

What is hypothesis.

A hypothesis is an assumption made based on some evidence.

Give an example of simple hypothesis?

What are the types of hypothesis.

Types of hypothesis are:

• Associative and Casual hypothesis

State true or false: Hypothesis is the initial point of any investigation that translates the research questions into a prediction.

Define complex hypothesis..

A complex hypothesis shows the relationship between two or more dependent variables and two or more independent variables.

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Learn Statistics for Data Science, Machine Learning, and AI – Full Handbook

Karl Pearson was a British mathematician who once said "Statistics is the grammar of science". This holds true especially for Computer and Information Sciences, Physical Science, and Biological Science.

When you are getting started with your journey in Data Science, Data Analytics, Machine Learning, or AI (including Generative AI) having statistical knowledge will help you better leverage data insights and actually understand all the algorithms beyond their implementation approach.

I can't overstate the importance of statistics in data science and Artificial Intelligence. Statistics provides tools and methods to find structure and give deeper data insights. Both Statistics and Mathematics love facts and hate guesses. Knowing the fundamentals of these two important subjects will allow you to think critically, and be creative when using the data to solve business problems and make data-driven decisions.

Key statistical concepts for your data science or data analysis journey with Python Code

In this handbook, I will cover the following Statistics topics for data science, machine learning, and artificial intelligence (including GenAI):

• Random variables

Mean, Variance, Standard Deviation

• Covariance and Correlation
• Probability distribution functions (PDFs)
• Bayes Theorem
• Linear Regression and Ordinary Least Squares (OLS)

Gauss-Markov Theorem

• Parameter properties (Bias, Consistency, Efficiency)
• Confidence intervals
• Hypothesis testing
• Statistical significance
• Type I & Type II Error
• Statistical tests (Student's t-test, F-test, 2-Sample T-Test, 2-Sample Z-Test, Chi-Square Test)
• p-value and its limitations

Inferential Statistics

• Central Limit Theorem & Law of Large Numbers
• Dimensionality reduction techniques (PCA, FA)
• Interview Prep - Top 7 Statistics Questions with Answers
• About The Author

How Can You Dive Deeper?

If you have no prior Statistical knowledge and you want to identify and learn the essential statistical concepts from the scratch and prepare for your job interviews, then this handbook is for you. It will also be a good read for anyone who wants to refresh their statistical knowledge.

Prerequisites

Before you start reading this handbook about key concepts in Statistics for Data Science, Machine Learning, and Artificial Intelligence, there are a few prerequisites that will help you make the most out of it.

This list is designed to ensure you are well-prepared and can fully grasp the statistical concepts discussed:

• Basic Mathematical Skills : Comfort with high school level mathematics, including algebra and basic calculus, is essential. These skills are crucial for understanding statistical formulas and methods.
• Logical Thinking : Ability to think logically and methodically to solve problems will aid in understanding statistical reasoning and applying these concepts to data-driven scenarios.
• Computer Literacy : Basic knowledge of using computers and the internet is necessary since many examples and exercises might require the use of statistical software or coding.
• Basic knowledge of Python, such as the creation of variables and working with some basic data structures and coding is also required (if you are not familiar with these concepts, check out my Python for Data Science 2024 -Full Course for Beginners here).
• Curiosity and Willingness to Learn : A keen interest in learning and exploring data is perhaps the most important prerequisite. The field of data science is constantly evolving, and a proactive approach to learning will be incredibly beneficial.

This handbook assumes no prior knowledge of statistics, making it accessible to beginners. Still, familiarity with the above concepts will greatly enhance your understanding and ability to apply statistical methods effectively in various domains.

If you want to learn Mathematics, Statistics, Machine Learning or AI check out our YouTube Channel and LunarTech.ai for free resources.

Random Variables

Random variables form the cornerstone of many statistical concepts. It might be hard to digest the formal mathematical definition of a random variable, but simply put, it's a way to map the outcomes of random processes, such as flipping a coin or rolling a dice, to numbers.

For instance, we can define the random process of flipping a coin by random variable X which takes a value 1 if the outcome is heads and 0 if the outcome is tails.

In this example, we have a random process of flipping a coin where this experiment can produce two possible outcomes : {0,1}. This set of all possible outcomes is called the sample space of the experiment. Each time the random process is repeated, it is referred to as an event .

In this example, flipping a coin and getting a tail as an outcome is an event. The chance or the likelihood of this event occurring with a particular outcome is called the probability of that event.

A probability of an event is the likelihood that a random variable takes a specific value of x which can be described by P(x). In the example of flipping a coin, the likelihood of getting heads or tails is the same, that is 0.5 or 50%. So we have the following setting:

where the probability of an event, in this example, can only take values in the range [0,1].

To understand the concepts of mean, variance, and many other statistical topics, it is important to learn the concepts of population and sample.

The population is the set of all observations (individuals, objects, events, or procedures) and is usually very large and diverse. On the other hand, a sample is a subset of observations from the population that ideally is a true representation of the population.

Given that experimenting with an entire population is either impossible or simply too expensive, researchers or analysts use samples rather than the entire population in their experiments or trials.

To make sure that the experimental results are reliable and hold for the entire population, the sample needs to be a true representation of the population. That is, the sample needs to be unbiased.

For this purpose, we can use statistical sampling techniques such as Random Sampling, Systematic Sampling, Clustered Sampling, Weighted Sampling, and Stratified Sampling.

The mean, also known as the average, is a central value of a finite set of numbers. Let’s assume a random variable X in the data has the following values:

where N is the number of observations or data points in the sample set or simply the data frequency. Then the sample mean defined by μ , which is very often used to approximate the population mean , can be expressed as follows:

The mean is also referred to as expectation which is often defined by E () or random variable with a bar on the top. For example, the expectation of random variables X and Y, that is E (X) and E (Y), respectively, can be expressed as follows:

Now that we have a solid understanding of the mean as a statistical measure, let's see how we can apply this knowledge practically using Python. Python is a versatile programming language that, with the help of libraries like NumPy, makes it easy to perform complex mathematical operations—including calculating the mean.

In the following code snippet, we demonstrate how to compute the mean of a set of numbers using NumPy. We will start by showing the calculation for a simple array of numbers. Then, we'll address a common scenario encountered in data science: calculating the mean of a dataset that includes undefined or missing values, represented as NaN (Not a Number). NumPy provides a function specifically designed to handle such cases, allowing us to compute the mean while ignoring these NaN values.

Here is how you can perform these operations in Python:

The variance measures how far the data points are spread out from the average value. It's equal to the sum of the squares of the differences between the data values and the average (the mean).

We can express the population variance as follows:

For deriving expectations and variances of different popular probability distribution functions, check out this Github repo .

Standard Deviation

The standard deviation is simply the square root of the variance and measures the extent to which data varies from its mean. The standard deviation defined by sigma can be expressed as follows:

Standard deviation is often preferred over the variance because it has the same units as the data points, which means you can interpret it more easily.

To compute the population variance using Python, we utilize the var function from the NumPy library. By default, this function calculates the population variance by setting the ddof (Delta Degrees of Freedom) parameter to 0. However, when dealing with samples and not the entire population, you would typically set ddof to 1 to get the sample variance.

The code snippet provided shows how to calculate the variance for a set of data. Additionally, it shows how to calculate the variance when there are NaN values in the data. NaN values represent missing or undefined data. When calculating the variance, these NaN values must be handled correctly; otherwise, they can result in a variance that is not a number (NaN), which is uninformative.

Here is how you can calculate the population variance in Python, taking into account the potential presence of NaN values:

The covariance is a measure of the joint variability of two random variables and describes the relationship between these two variables. It is defined as the expected value of the product of the two random variables’ deviations from their means.

The covariance between two random variables X and Z can be described by the following expression, where E (X) and E (Z) represent the means of X and Z, respectively.

Covariance can take negative or positive values as well as a value of 0. A positive value of covariance indicates that two random variables tend to vary in the same direction, whereas a negative value suggests that these variables vary in opposite directions. Finally, the value 0 means that they don’t vary together.

To explore the concept of covariance practically, we will use Python with the NumPy library, which provides powerful numerical operations. The np.cov function can be used to calculate the covariance matrix for two or more datasets. In the matrix, the diagonal elements represent the variance of each dataset, and the off-diagonal elements represent the covariance between each pair of datasets.

Let's look at an example of calculating the covariance between two sets of data:

Correlation

The correlation is also a measure of a relationship. It measures both the strength and the direction of the linear relationship between two variables.

If a correlation is detected, then it means that there is a relationship or a pattern between the values of two target variables. Correlation between two random variables X and Z is equal to the covariance between these two variables divided by the product of the standard deviations of these variables. This can be described by the following expression:

Correlation coefficients’ values range between -1 and 1. Keep in mind that the correlation of a variable with itself is always 1, that is Cor(X, X) = 1 .

Another thing to keep in mind when interpreting correlation is to not confuse it with causation , given that a correlation is not necessarily a causation. Even if there is a correlation between two variables, you cannot conclude that one variable causes a change in the other. This relationship could be coincidental, or a third factor might be causing both variables to change.

Probability Distribution Functions

A function that describes all the possible values, the sample space, and the corresponding probabilities that a random variable can take within a given range, bounded between the minimum and maximum possible values, is called a probability distribution function (pdf) or probability density.

Every pdf needs to satisfy the following two criteria:

where the first criterium states that all probabilities should be numbers in the range of [0,1] and the second criterium states that the sum of all possible probabilities should be equal to 1.

Probability functions are usually classified into two categories: discrete and continuous .

Discrete distribution function describes the random process with countable sample space, like in an example of tossing a coin that has only two possible outcomes. Continuous distribution functions describe the random process with a continuous sample space.

Examples of discrete distribution functions are Bernoulli , Binomial , Poisson , Discrete Uniform . Examples of continuous distribution functions are Normal , Continuous Uniform , Cauchy .

Binomial Distribution

The binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each with the boolean-valued outcome: success (with probability p ) or failure (with probability q = 1 − p).

Let's assume a random variable X follows a Binomial distribution, then the probability of observing k successes in n independent trials can be expressed by the following probability density function:

$$\Pr(X = k) = \binom{n}{k} p^k q^{n-k}$$

The binomial distribution is useful when analyzing the results of repeated independent experiments, especially if you're interested in the probability of meeting a particular threshold given a specific error rate.

Binomial Distribution Mean and Variance

The mean of a binomial distribution, denoted as E ( X )= np , tells you the average number of successes you can expect if you conduct n independent trials of a binary experiment.

A binary experiment is one where there are only two outcomes: success (with probability p ) or failure (with probability q =1− p ).

For example, if you were to flip a coin 100 times and you define a success as the coin landing on heads (let's say the probability of heads is 0.5), the binomial distribution would tell you how likely it is to get any number of heads in those 100 flips. The mean E ( X ) would be 100×0.5=50, indicating that on average, you’d expect to get 50 heads.

The variance Var(X)=npq measures the spread of the distribution, indicating how much the number of successes is likely to deviate from the mean.

Continuing with the coin flip example, the variance would be 100×0.5×0.5=25, which informs you about the variability of the outcomes. A smaller variance would mean the results are more tightly clustered around the mean, whereas a larger variance indicates they’re more spread out.

These concepts are crucial in many fields. For instance:

• Quality Control : Manufacturers might use the binomial distribution to predict the number of defective items in a batch, helping them understand the quality and consistency of their production process.
• Healthcare : In medicine, it could be used to calculate the probability of a certain number of patients responding to a treatment, based on past success rates.
• Finance : In finance, binomial models are used to evaluate the risk of portfolio or investment strategies by predicting the number of times an asset will reach a certain price point.
• Polling and Survey Analysis : When predicting election results or customer preferences, pollsters might use the binomial distribution to estimate how many people will favor a candidate or a product, given the probability drawn from a sample.

Understanding the mean and variance of the binomial distribution is fundamental to interpreting the results and making informed decisions based on the likelihood of different outcomes.

The figure below visualizes an example of Binomial distribution where the number of independent trials is equal to 8 and the probability of success in each trial is equal to 16%.

The Python code below creates a histogram to visualize the distribution of outcomes from 1000 experiments, each consisting of 8 trials with a success probability of 0.16. It uses NumPy to generate the binomial distribution data and Matplotlib to plot the histogram, showing the probability of the number of successes in those trials.

Poisson Distribution

The Poisson distribution is the discrete probability distribution of the number of events occurring in a specified time period, given the average number of times the event occurs over that time period.

Let's assume a random variable X follows a Poisson distribution. Then the probability of observing k events over a time period can be expressed by the following probability function:

where e is Euler’s number and λ lambda, the arrival rate parameter , is the expected value of X. The Poisson distribution function is very popular for its usage in modeling countable events occurring within a given time interval.

Poisson Distribution Mean and  Variance

The Poisson distribution is particularly useful for modeling the number of times an event occurs within a specified time frame. The mean E(X) and variance Var(X)

Var(X)  of a Poisson distribution are both equal to λ, which is the average rate at which events occur (also known as the rate parameter). This makes the Poisson distribution unique, as it is characterized by this single parameter.

The fact that the mean and variance are equal means that as we observe more events, the distribution of the number of occurrences becomes more predictable. It’s used in various fields such as business, engineering, and science for tasks like:

Predicting the number of customer arrivals at a store within an hour. Estimating the number of emails you'd receive in a day.  Understanding the number of defects in a batch of materials.

So, the Poisson distribution helps in making probabilistic forecasts about the occurrence of rare or random events over intervals of time or space.

For example, Poisson distribution can be used to model the number of customers arriving in the shop between 7 and 10 pm, or the number of patients arriving in an emergency room between 11 and 12 pm.

The figure below visualizes an example of Poisson distribution where we count the number of Web visitors arriving at the website where the arrival rate, lambda, is assumed to be equal to 7 minutes.

In practical data analysis, it is often helpful to simulate the distribution of events. Below is a Python code snippet that demonstrates how to generate a series of data points that follow a Poisson distribution using NumPy. We then create a histogram using Matplotlib to visualize the distribution of the number of visitors (as an example) we might expect to see, based on our average rate λ = 7

This histogram helps in understanding the distribution's shape and variability. The most likely number of visitors is around the mean λ, but the distribution shows the probability of seeing fewer or greater numbers as well.

Normal Distribution

The Normal probability distribution is the continuous probability distribution for a real-valued random variable. Normal distribution, also called Gaussian distribution is arguably one of the most popular distribution functions that is commonly used in social and natural sciences for modeling purposes. For example, it is used to model people’s height or test scores.

Let's assume a random variable X follows a Normal distribution. Then its probability density function can be expressed as follows:

where the parameter μ (mu) is the mean of the distribution also referred to as the location parameter , parameter σ (sigma) is the standard deviation of the distribution also referred to as the scale parameter . The number π (pi) is a mathematical constant approximately equal to 3.14.

Normal Distribution Mean and Variance

The figure below visualizes an example of Normal distribution with a mean 0 ( μ = 0 ) and standard deviation of 1 ( σ = 1 ), which is referred to as Standard Normal distribution which is symmetric .

The visualization of the standard normal distribution is crucial because this distribution underpins many statistical methods and probability theory. When data is normally distributed with a mean ( μ ) of 0 and standard deviation (σ) of 1, it is referred to as the standard normal distribution. It's symmetric around the mean, with the shape of the curve often called the "bell curve" due to its bell-like shape.

The standard normal distribution is fundamental for the following reasons:

• Central Limit Theorem: This theorem states that, under certain conditions, the sum of a large number of random variables will be approximately normally distributed. It allows for the use of normal probability theory for sample means and sums, even when the original data is not normally distributed.
• Z-Scores: Values from any normal distribution can be transformed into the standard normal distribution using Z-scores, which indicate how many standard deviations an element is from the mean. This allows for the comparison of scores from different normal distributions.
• Statistical Inference and AB Testing: Many statistical tests, such as t-tests and ANOVAs, assume that the data follows a normal distribution, or they rely on the central limit theorem. Understanding the standard normal distribution helps in the interpretation of these tests' results.
• Confidence Intervals and Hypothesis Testing: The properties of the standard normal distribution are used to construct confidence intervals and to perform hypothesis testing.

All topics which we will cover below!

So, being able to visualize and understand the standard normal distribution is key to applying many statistical techniques accurately.

The Python code below uses NumPy to generate 1000 random samples from a normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1, which are standard parameters for the standard normal distribution. These generated samples are stored in the variable X.

To visualize the distribution of these samples, the code employs Matplotlib to create a histogram. The plt.hist function is used to plot the histogram of the samples with 30 bins, and the density parameter is set to True to normalize the histogram so that the area under it sums to 1. This effectively turns the histogram into a probability density plot.

Additionally, the SciPy library is used to overlay the probability density function (PDF) of the theoretical normal distribution on the histogram. The norm.pdf function generates the y-values for the PDF given an array of x-values. This theoretical curve is plotted in yellow over the histogram to show how closely the random samples fit the expected distribution.

The resulting graph displays the histogram of the generated samples in purple, with the theoretical normal distribution overlaid in yellow. The x-axis represents the range of values that the samples can take, while the y-axis represents the probability density. This visualization is a powerful tool for comparing the empirical distribution of the data with the theoretical model, allowing us to see whether our samples follow the expected pattern of a normal distribution.

Bayes' Theorem

The Bayes' Theorem (often called Bayes' Law ) is arguably the most powerful rule of probability and statistics. It was named after famous English statistician and philosopher, Thomas Bayes.

Bayes' theorem is a powerful probability law that brings the concept of subjectivity into the world of Statistics and Mathematics where everything is about facts. It describes the probability of an event, based on the prior information of conditions that might be related to that event.

For instance, if the risk of getting Coronavirus or Covid-19 is known to increase with age, then Bayes' Theorem allows the risk to an individual of a known age to be determined more accurately. It does this by conditioning it on the age rather than simply assuming that this individual is common to the population as a whole.

The concept of conditional probability , which plays a central role in Bayes' theorem, is a measure of the probability of an event happening, given that another event has already occurred.

Bayes' theorem can be described by the following expression where the X and Y stand for events X and Y, respectively:

• Pr (X|Y): the probability of event X occurring given that event or condition Y has occurred or is true
• Pr (Y|X): the probability of event Y occurring given that event or condition X has occurred or is true
• Pr (X) & Pr (Y): the probabilities of observing events X and Y, respectively

In the case of the earlier example, the probability of getting Coronavirus (event X) conditional on being at a certain age is Pr (X|Y). This is equal to the probability of being at a certain age given that the person got a Coronavirus, Pr (Y|X), multiplied with the probability of getting a Coronavirus, Pr (X), divided by the probability of being at a certain age, Pr (Y).

Linear Regression

Earlier, we introduced the concept of causation between variables, which happens when a variable has a direct impact on another variable.

When the relationship between two variables is linear, then Linear Regression is a statistical method that can help model the impact of a unit change in a variable, the independent variable on the values of another variable, the dependent variable .

Dependent variables are often referred to as response variables or explained variables, whereas independent variables are often referred to as regressors or explanatory variables .

When the Linear Regression model is based on a single independent variable, then the model is called Simple Linear Regression . When the model is based on multiple independent variables, it’s referred to as Multiple Linear Regression.

Simple Linear Regression can be described by the following expression:

where Y is the dependent variable, X is the independent variable which is part of the data, β0 is the intercept which is unknown and constant, β1 is the slope coefficient or a parameter corresponding to the variable X which is unknown and constant as well. Finally, u is the error term that the model makes when estimating the Y values.

The main idea behind linear regression is to find the best-fitting straight line, the regression line, through a set of paired ( X, Y ) data.

One example of the Linear Regression application is modeling the impact of flipper length on penguins’ body mass, which is visualized below:

The R code snippet you've shared is for creating a scatter plot with a linear regression line using the ggplot2 package in R, which is a powerful and widely-used library for creating graphics and visualizations. The code uses a dataset named penguins from the palmerpenguins package, presumably containing data about penguin species, including measurements like flipper length and body mass.

Multiple Linear Regression with three independent variables can be described by the following expression:

Ordinary Least Squares

The ordinary least squares (OLS) is a method for estimating the unknown parameters such as β0 and β1 in a linear regression model. The model is based on the principle of least squares . This minimizes the sum of the squares of the differences between the observed dependent variable and its values that are predicted by the linear function of the independent variable (often referred to as fitted values ).

This difference between the real and predicted values of dependent variable Y is referred to as residual . So OLS minimizes the sum of squared residuals.

This optimization problem results in the following OLS estimates for the unknown parameters β0 and β1 which are also known as coefficient estimates :

Once these parameters of the Simple Linear Regression model are estimated, the fitted values of the response variable can be computed as follows:

Standard Error

The residuals or the estimated error terms can be determined as follows:

It is important to keep in mind the difference between the error terms and residuals. Error terms are never observed, while the residuals are calculated from the data. The OLS estimates the error terms for each observation but not the actual error term. So, the true error variance is still unknown.

Also, these estimates are subject to sampling uncertainty. This means that we will never be able to determine the exact estimate, the true value, of these parameters from sample data in an empirical application. But we can estimate it by calculating the sample residual variance by using the residuals as follows:

This estimate for the variance of sample residuals helps us estimate the variance of the estimated parameters, which is often expressed as follows:

The square root of this variance term is called the standard error of the estimate. This is a key component in assessing the accuracy of the parameter estimates. It is used to calculate test statistics and confidence intervals.

The standard error can be expressed as follows:

It is important to keep in mind the difference between the error terms and residuals. Error terms are never observed, while the residuals are calculated from the data.

OLS Assumptions

The OLS estimation method makes the following assumptions which need to be satisfied to get reliable prediction results:

• The Linearity assumption states that the model is linear in parameters.
• The Random Sample assumption states that all observations in the sample are randomly selected.
• The Exogeneity assumption states that independent variables are uncorrelated with the error terms.
• The Homoskedasticity assumption states that the variance of all error terms is constant.
• The No Perfect Multi-Collinearity assumption states that none of the independent variables is constant and there are no exact linear relationships between the independent variables.

The Python code snippet you've shared performs Ordinary Least Squares (OLS) regression, which is a method used in statistics to estimate the relationship between independent variables and a dependent variable. This process involves calculating the best-fit line through the data points that minimizes the sum of the squared differences between the observed values and the values predicted by the model.

The code defines a function runOLS(Y, X) that takes in a dependent variable Y and an independent variable X and performs the following steps:

• Estimates the OLS coefficients (beta_hat) using the linear algebra solution to the least squares problem.
• Makes predictions ( Y_hat ) using the estimated coefficients and calculates the residuals.
• Computes the residual sum of squares (RSS), total sum of squares (TSS), mean squared error (MSE), root mean squared error (RMSE), and R-squared value, which are common metrics used to assess the fit of the model.
• Calculates the standard error of the coefficient estimates, t-statistics, p-values, and confidence intervals for the estimated coefficients.

These calculations are standard in regression analysis and are used to interpret and understand the strength and significance of the relationship between the variables. The result of this function includes the estimated coefficients and various statistics that help evaluate the model's performance.

Parameter Properties

Under the assumption that the OLS criteria/assumptions we just discussed are satisfied, the OLS estimators of coefficients β0 and β1 are BLUE and Consistent . So what does this mean?

This theorem highlights the properties of OLS estimates where the term BLUE stands for Best Linear Unbiased Estimator . Let's explore what this means in more detail.

The bias of an estimator is the difference between its expected value and the true value of the parameter being estimated. It can be expressed as follows:

When we state that the estimator is unbiased , we mean that the bias is equal to zero. This implies that the expected value of the estimator is equal to the true parameter value, that is:

Unbiasedness does not guarantee that the obtained estimate with any particular sample is equal or close to β. What it means is that, if we repeatedly draw random samples from the population and then computes the estimate each time, then the average of these estimates would be equal or very close to β.

The term Best in the Gauss-Markov theorem relates to the variance of the estimator and is referred to as efficiency . A parameter can have multiple estimators but the one with the lowest variance is called efficient.

Consistency

The term consistency goes hand in hand with the terms sample size and convergence . If the estimator converges to the true parameter as the sample size becomes very large, then this estimator is said to be consistent, that is:

All these properties hold for OLS estimates as summarized in the Gauss-Markov theorem. In other words, OLS estimates have the smallest variance, they are unbiased, linear in parameters, and are consistent. These properties can be mathematically proven by using the OLS assumptions made earlier.

Confidence Intervals

The Confidence Interval is the range that contains the true population parameter with a certain pre-specified probability. This is referred to as the confidence level of the experiment, and it's obtained by using the sample results and the margin of error .

Margin of Error

The margin of error is the difference between the sample results and based on what the result would have been if you had used the entire population.

Confidence Level

The Confidence Level describes the level of certainty in the experimental results. For example, a 95% confidence level means that if you were to perform the same experiment repeatedly 100 times, then 95 of those 100 trials would lead to similar results.

Note that the confidence level is defined before the start of the experiment because it will affect how big the margin of error will be at the end of the experiment.

Confidence Interval for OLS Estimates

As I mentioned earlier, the OLS estimates of the Simple Linear Regression, the estimates for intercept β0 and slope coefficient β1, are subject to sampling uncertainty. But we can construct Confidence Intervals (CIs) for these parameters which will contain the true value of these parameters in 95% of all samples.

That is, 95% confidence interval for β can be interpreted as follows:

• The confidence interval is the set of values for which a hypothesis test cannot be rejected to the level of 5%.
• The confidence interval has a 95% chance to contain the true value of β.

95% confidence interval of OLS estimates can be constructed as follows:

This is based on the parameter estimate, the standard error of that estimate, and the value 1.96 representing the margin of error corresponding to the 5% rejection rule.

This value is determined using the Normal Distribution table , which we'll discuss later on in this handbook.

Meanwhile, the following figure illustrates the idea of 95% CI:

Note that the confidence interval depends on the sample size as well, given that it is calculated using the standard error which is based on sample size.

Statistical Hypothesis Testing

Testing a hypothesis in Statistics is a way to test the results of an experiment or survey to determine how meaningful they the results are.

Basically, you're testing whether the obtained results are valid by figuring out the odds that the results have occurred by chance. If it is the letter, then the results are not reliable and neither is the experiment. Hypothesis Testing is part of the Statistical Inference .

Null and Alternative Hypothesis

Firstly, you need to determine the thesis you wish to test. Then you need to formulate the Null Hypothesis and the Alternative Hypothesis. The test can have two possible outcomes. Based on the statistical results, you can either reject the stated hypothesis or accept it.

As a rule of thumb, statisticians tend to put the version or formulation of the hypothesis under the Null Hypothesis that needs to be rejected , whereas the acceptable and desired version is stated under the Alternative Hypothesis .

Statistical Significance

Let’s look at the earlier mentioned example where we used the Linear Regression model to investigate whether a penguin's Flipper Length, the independent variable, has an impact on Body Mass , the dependent variable.

We can formulate this model with the following statistical expression:

Then, once the OLS estimates of the coefficients are estimated, we can formulate the following Null and Alternative Hypothesis to test whether the Flipper Length has a statistically significant impact on the Body Mass:

where H0 and H1 represent Null Hypothesis and Alternative Hypothesis, respectively.

Rejecting the Null Hypothesis would mean that a one-unit increase in Flipper Length has a direct impact on the Body Mass (given that the parameter estimate of β1 is describing this impact of the independent variable, Flipper Length, on the dependent variable, Body Mass). We can reformulate this hypothesis as follows:

where H0 states that the parameter estimate of β1 is equal to 0, that is Flipper Length effect on Body Mass is statistically insignificant whereas H1 states that the parameter estimate of β1 is not equal to 0, suggesting that Flipper Length effect on Body Mass is statistically significant .

Type I and Type II Errors

When performing Statistical Hypothesis Testing, you need to consider two conceptual types of errors: Type I error and Type II error.

Type I errors occur when the Null is incorrectly rejected, and Type II errors occur when the Null Hypothesis is incorrectly not rejected. A confusion matrix can help you clearly visualize the severity of these two types of errors.

As a rule of thumb, statisticians tend to put the version of the hypothesis under the Null Hypothesis that that needs to be rejected, whereas the acceptable and desired version is stated under the Alternative Hypothesis.

Statistical Tests

Once the you've stataed the Null and the Alternative Hypotheses and defined the test assumptions, the next step is to determine which statistical test is appropriate and to calculate the test statistic .

Whether or not to reject or not reject the Null can be determined by comparing the test statistic with the critical value . This comparison shows whether or not the observed test statistic is more extreme than the defined critical value.

It can have two possible results:

• The test statistic is more extreme than the critical value → the null hypothesis can be rejected
• The test statistic is not as extreme as the critical value → the null hypothesis cannot be rejected

The critical value is based on a pre-specified significance level α (usually chosen to be equal to 5%) and the type of probability distribution the test statistic follows.

The critical value divides the area under this probability distribution curve into the rejection region(s) and non-rejection region . There are numerous statistical tests used to test various hypotheses. Examples of Statistical tests are Student’s t-test , F-test , Chi-squared test , Durbin-Hausman-Wu Endogeneity test , W hite Heteroskedasticity test . In this handbook, we will look at two of these statistical tests: the Student's t-test and the F-test.

Student’s t-test

One of the simplest and most popular statistical tests is the Student’s t-test. You can use it to test various hypotheses, especially when dealing with a hypothesis where the main area of interest is to find evidence for the statistically significant effect of a single variable .

The test statistics of the t-test follows Student’s t distribution and can be determined as follows:

where h0 in the nominator is the value against which the parameter estimate is being tested. So, the t-test statistics are equal to the parameter estimate minus the hypothesized value divided by the standard error of the coefficient estimate.

Let's use this for our earlier hypothesis, where we wanted to test whether Flipper Length has a statistically significant impact on Body Mass or not. This test can be performed using a t-test. The h0 is in that case equal to the 0 since the slope coefficient estimate is tested against a value of 0.

Two-sided vs one-sided t-test

There are two versions of the t-test: a two-sided t-test and a one-sided t-test . Whether you need the former or the latter version of the test depends entirely on the hypothesis that you want to test.

You can use the two-sided or two-tailed t-test when the hypothesis is testing equal versus not equal relationship under the Null and Alternative Hypotheses. It would be similar to the following example:

The two-sided t-test has two rejection regions as visualized in the figure below:

In this version of the t-test, the Null is rejected if the calculated t-statistics is either too small or too large.

Here, the test statistics are compared to the critical values based on the sample size and the chosen significance level. To determine the exact value of the cutoff point, you can use a two-sided t-distribution table .

On the other hand, you can use the one-sided or one-tailed t-test when the hypothesis is testing positive/negative versus negative/positive relationships under the Null and Alternative Hypotheses. It looks like this:

One-sided t-test has a single rejection region . Depending on the hypothesis side, the rejection region is either on the left-hand side or the right-hand side as visualized in the figure below:

In this version of the t-test, the Null is rejected if the calculated t-statistics is smaller/larger than the critical value.

F-test is another very popular statistical test often used to test hypotheses testing a joint statistical significance of multiple variables . This is the case when you want to test whether multiple independent variables have a statistically significant impact on a dependent variable.

Following is an example of a statistical hypothesis that you can test using the F-test:

where the Null states that the three variables corresponding to these coefficients are jointly statistically insignificant, and the Alternative states that these three variables are jointly statistically significant.

The test statistics of the F-test follows F distribution and can be determined as follows:

• the SSRrestricted is the sum of squared residuals of the restricted model , which is the same model excluding from the data the target variables stated as insignificant under the Null
• the SSRunrestricted is the sum of squared residuals of the unrestricted model , which is the model that includes all variables
• the q represents the number of variables that are being jointly tested for the insignificance under the Null
• N is the sample size
• and the k is the total number of variables in the unrestricted model.

SSR values are provided next to the parameter estimates after running the OLS regression, and the same holds for the F-statistics as well.

Following is an example of MLR model output where the SSR and F-statistics values are marked.

F-test has a single rejection region as visualized below:

If the calculated F-statistics is bigger than the critical value, then the Null can be rejected. This suggests that the independent variables are jointly statistically significant. The rejection rule can be expressed as follows:

2-sample T-test

If you want to test whether there is a statistically significant difference between the control and experimental groups’ metrics that are in the form of averages (for example, average purchase amount), metric follows student-t distribution. When the sample size is smaller than 30, you can use 2-sample T-test to test the following hypothesis:

where the sampling distribution of means of Control group follows Student-t distribution with degrees of freedom N_con-1. Also, the sampling distribution of means of the Experimental group also follows the Student-t distribution with degrees of freedom N_exp-1.

Note that the N_con and N_exp are the number of users in the Control and Experimental groups, respectively.

Then you can calculate an estimate for the pooled variance of the two samples as follows:

where σ²_con and σ²_exp are the sample variances of the Control and Experimental groups, respectively. Then the Standard Error is equal to the square root of the estimate of the pooled variance, and can be defined as:

Consequently, the test statistics of the 2-sample T-test with the hypothesis stated earlier can be calculated as follows:

In order to test the statistical significance of the observed difference between sample means, we need to calculate the p-value of our test statistics.

The p-value is the probability of observing values at least as extreme as the common value when this is due to a random chance. Stated differently, the p-value is the probability of obtaining an effect at least as extreme as the one in your sample data, assuming the null hypothesis is true.

Then the p-value of the test statistics can be calculated as follows:

The interpretation of a p -value is dependent on the chosen significance level, alpha, which you choose before running the test during the power analysis .

If the calculated p -value appears to be smaller than equal to alpha (for example, 0.05 for 5% significance level) we can reject the null hypothesis and state that there is a statistically significant difference between the primary metrics of the Control and Experimental groups.

Finally, to determine how accurate the obtained results are and also to comment about the practical significance of the obtained results, you can compute the Confidence Interval of your test by using the following formula:

where the t_(1-alpha/2) is the critical value of the test corresponding to the two-sided t-test with alpha significance level. It can be found using the t-table .

The Python code provided performs a two-sample t-test, which is used in statistics to determine if two sets of data are significantly different from each other. This particular snippet simulates two groups (control and experimental) with data following a t-distribution, calculates the mean and variance for each group, and then performs the following steps:

• It calculates the pooled variance, which is a weighted average of the variances of the two groups.
• It computes the standard error of the difference between the two means.
• It calculates the t-statistic, which is the difference between the two sample means divided by the standard error. This statistic measures how much the groups differ in units of standard error.
• It determines the critical t-value from the t-distribution for the given significance level and degrees of freedom, which is used to decide whether the t-statistic is large enough to indicate a statistically significant difference between the groups.
• It calculates the p-value, which indicates the probability of observing such a difference between means if the null hypothesis (that there is no difference) is true.
• It computes the margin of error and constructs the confidence interval around the difference in means.

Finally, the code prints out the t-statistic, critical t-value, p-value, and confidence interval. These results can be used to infer whether the observed differences in means are statistically significant or likely due to random variation.

2-sample Z-test

There are various situations when you may want to use a 2-sample z-test:

• if you want to test whether there is a statistically significant difference between the control and experimental groups’ metrics that are in the form of averages (for example, average purchase amount) or proportions (for example, Click Through Rate)
• if the metric follows Normal distribution
• when the sample size is larger than 30, such that you can use the Central Limit Theorem (CLT) to state that the sampling distributions of the Control and Experimental groups are asymptotically Normal

Here we will make a distinction between two cases: where the primary metric is in the form of proportions (like Click Through Rate) and where the primary metric is in the form of averages (like average purchase amount).

Case 1: Z-test for comparing proportions (2-sided)

If you want to test whether there is a statistically significant difference between the Control and Experimental groups’ metrics that are in the form of proportions (like CTR) and if the click event occurs independently, you can use a 2-sample Z-test to test the following hypothesis:

where each click event can be described by a random variable that can take two possible values: 1 (success) and 0 (failure). It also follows a Bernoulli distribution (click: success and no click: failure) where p_con and p_exp are the probabilities of clicking (probability of success) of Control and Experimental groups, respectively.

So, after collecting the interaction data of the Control and Experimental users, you can calculate the estimates of these two probabilities as follows:

Since we are testing for the difference in these probabilities, we need to obtain an estimate for the pooled probability of success and an estimate for pooled variance, which can be done as follows:

Then the Standard Error is equal to the square root of the estimate of the pooled variance. It can be defined as:

And so, the test statistics of the 2-sample Z-test for the difference in proportions can be calculated as follows:

Then the p-value of this test statistics can be calculated as follows:

Finally, you can compute the Confidence Interval of the test as follows:

where the z_(1-alpha/2) is the critical value of the test corresponding to the two-sided Z-test with alpha significance level. You can find it using the Z-table .

The rejection region of this two-sided 2-sample Z-test can be visualized by the following graph:

The Python code snippet you’ve provided performs a two-sample Z-test for proportions. This type of test is used to determine whether there is a significant difference between the proportions of two groups. Here’s a brief explanation of the steps the code performs:

• Calculates the sample proportions for both the control and experimental groups.
• Computes the pooled sample proportion, which is an estimate of the proportion assuming the null hypothesis (that there is no difference between the group proportions) is true.
• Calculates the pooled sample variance based on the pooled proportion and the sizes of the two samples.
• Derives the standard error of the difference in sample proportions.
• Calculates the Z-test statistic, which measures the number of standard errors between the sample proportion difference and the null hypothesis.
• Finds the critical Z-value from the standard normal distribution for the given significance level.
• Computes the p-value to assess the evidence against the null hypothesis.
• Calculates the margin of error and the confidence interval for the difference in proportions.
• Outputs the test statistic, critical value, p-value, and confidence interval, and based on the test statistic and critical value, it may print a statement to either reject or not reject the null hypothesis.

The latter part of the code uses Matplotlib to create a visualization of the standard normal distribution and the rejection regions for the two-sided Z-test. This visual aid helps to understand where the test statistic falls in relation to the distribution and the critical values.

Case 2: Z-test for Comparing Means (2-sided)

If you want to test whether there is a statistically significant difference between the Control and Experimental groups’ metrics that are in the form of averages (like average purchase amount) you can use a 2-sample Z-test to test the following hypothesis:

where the sampling distribution of means of the Control group follows Normal distribution with mean mu_con and σ²_con/N_con. Moreover, the sampling distribution of means of the Experimental group also follows the Normal distribution with mean mu_exp and σ²_exp/N_exp.

Then the difference in the means of the control and experimental groups also follows Normal distributions with mean mu_con-mu_exp and variance σ²_con/N_con + σ²_exp/N_exp.

Consequently, the test statistics of the 2-sample Z-test for the difference in means can be calculated as follows:

The Standard Error is equal to the square root of the estimate of the pooled variance and can be defined as:

The Python code provided appears to be set up for conducting a two-sample Z-test, typically used to determine if there is a significant difference between the means of two independent groups. In this context, the code might be comparing two different processes or treatments.

• It generates two arrays of random integers to represent data for a control group ( X_A ) and an experimental group ( X_B ).
• It calculates the sample means ( mu_con , mu_exp ) and variances ( variance_con , variance_exp ) for both groups.
• The pooled variance is computed, which is used in the denominator of the test statistic formula for the Z-test, providing a measure of the data's common variance.
• The Z-test statistic ( T ) is calculated by taking the difference between the two sample means and dividing it by the standard error of the difference.
• The p-value is calculated to test the hypothesis of whether the means of the two groups are statistically different from each other.
• The critical Z-value ( Z_crit ) is determined from the standard normal distribution, which defines the cutoff points for significance.
• A margin of error is computed, and a confidence interval for the difference in means is constructed.
• The test statistic, critical value, p-value, and confidence interval are printed to the console.

Lastly, the code uses Matplotlib to plot the standard normal distribution and highlight the rejection regions for the Z-test. This visualization can help in understanding the result of the Z-test in terms of where the test statistic lies relative to the distribution and the critical values for a two-sided test.

Chi-Squared test

If you want to test whether there is a statistically significant difference between the Control and Experimental groups’ performance metrics (for example their conversions) and you don’t really want to know the nature of this relationship (which one is better) you can use a Chi-Squared test to test the following hypothesis:

Note that the metric should be in the form of a binary variable (for example, conversion or no conversion/click or no click). The data can then be represented in the form of the following table, where O and T correspond to observed and theoretical values, respectively.

Then the test statistics of the Chi-2 test can be expressed as follows:

where the Observed corresponds to the observed data and the Expected corresponds to the theoretical value, and i can take values 0 (no conversion) and 1(conversion). It’s important to see that each of these factors has a separate denominator. The formula for the test statistics when you have two groups only can be represented as follows:

The expected value is simply equal to the number of times each version of the product is viewed multiplied by the probability of it leading to conversion (or to a click in case of CTR).

Note that, since the Chi-2 test is not a parametric test, its Standard Error and Confidence Interval can’t be calculated in a standard way as we did in the parametric Z-test or T-test.

The Python code you've shared is for conducting a Chi-squared test, a statistical hypothesis test that is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories.

In the provided code snippet, it looks like the test is being used to compare two categorical datasets:

• It calculates the Chi-squared test statistic by summing the squared difference between observed ( O ) and expected ( T ) frequencies, divided by the expected frequencies for each category. This is known as the squared relative distance and is used as the test statistic for the Chi-squared test.
• It then calculates the p-value for this test statistic using the degrees of freedom, which in this case is assumed to be 1 (but this would typically depend on the number of categories minus one).
• The Matplotlib library is used to plot the probability density function (pdf) of the Chi-squared distribution with one degree of freedom. It also highlights the rejection region for the test, which corresponds to the critical value of the Chi-squared distribution that the test statistic must exceed for the difference to be considered statistically significant.

The visualization helps to understand the Chi-squared test by showing where the test statistic lies in relation to the Chi-squared distribution and its critical value. If the test statistic is within the rejection region, the null hypothesis of no difference in frequencies can be rejected.

Another quick way to determine whether to reject or to support the Null Hypothesis is by using p-values . The p-value is the probability of the condition under the Null occurring. Stated differently, the p-value is the probability, assuming the null hypothesis is true, of observing a result at least as extreme as the test statistic. The smaller the p-value, the stronger is the evidence against the Null Hypothesis, suggesting that it can be rejected.

The interpretation of a p -value is dependent on the chosen significance level. Most often, 1%, 5%, or 10% significance levels are used to interpret the p-value. So, instead of using the t-test and the F-test, p-values of these test statistics can be used to test the same hypotheses.

The following figure shows a sample output of an OLS regression with two independent variables. In this table, the p-value of the t-test, testing the statistical significance of class_size variable’s parameter estimate, and the p-value of the F-test, testing the joint statistical significance of the class_size, and el_pct variables parameter estimates, are underlined.

The p-value corresponding to the class_size variable is 0.011. When we compare this value to the significance levels 1% or 0.01 , 5% or 0.05, 10% or 0.1, then we can make the following conclusions:

• 0.011 > 0.01 → Null of the t-test can’t be rejected at 1% significance level
• 0.011 < 0.05 → Null of the t-test can be rejected at 5% significance level
• 0.011 < 0.10 → Null of the t-test can be rejected at 10% significance level

So, this p-value suggests that the coefficient of the class_size variable is statistically significant at 5% and 10% significance levels. The p-value corresponding to the F-test is 0.0000. And since 0 is smaller than all three cutoff values (0.01, 0.05, 0.10), we can conclude that the Null of the F-test can be rejected in all three cases.

This suggests that the coefficients of class_size and el_pct variables are jointly statistically significant at 1%, 5%, and 10% significance levels.

Limitation of p-values

Using p-values has many benefits, but it has also limitations. One of the main ones is that the p-value depends on both the magnitude of association and the sample size. If the magnitude of the effect is small and statistically insignificant, the p-value might still show a significant impact because the sample size is large. The opposite can occur as well – an effect can be large, but fail to meet the p<0.01, 0.05, or 0.10 criteria if the sample size is small.

Inferential statistics uses sample data to make reasonable judgments about the population from which the sample data originated. We use it to investigate the relationships between variables within a sample and make predictions about how these variables will relate to a larger population.

Both the Law of Large Numbers (LLN) and the Central Limit Theorem (CLM) have a significant role in Inferential statistics because they show that the experimental results hold regardless of what shape the original population distribution was when the data is large enough.

The more data is gathered, the more accurate the statistical inferences become – hence, the more accurate parameter estimates are generated.

Law of Large Numbers (LLN)

Suppose X1, X2, . . . , Xn are all independent random variables with the same underlying distribution (also called independent identically-distributed or i.i.d), where all X’s have the same mean μ and standard deviation σ . As the sample size grows, the probability that the average of all X’s is equal to the mean μ is equal to 1.

The Law of Large Numbers can be summarized as follows:

Central Limit Theorem (CLM)

Suppose X1, X2, . . . , Xn are all independent random variables with the same underlying distribution (also called independent identically-distributed or i.i.d), where all X’s have the same mean μ and standard deviation σ . As the sample size grows, the probability distribution of X converges in the distribution in Normal distribution with mean μ and variance σ- squared.

The Central Limit Theorem can be summarized as follows:

Stated differently, when you have a population with mean μ and standard deviation σ and you take sufficiently large random samples from that population with replacement, then the distribution of the sample means will be approximately normally distributed.

Dimensionality Reduction Techniques

Dimensionality reduction is the transformation of data from a high-dimensional space into a low-dimensional space such that this low-dimensional representation of the data still contains the meaningful properties of the original data as much as possible.

With the increase in popularity in Big Data, the demand for these dimensionality reduction techniques, reducing the amount of unnecessary data and features, increased as well. Examples of popular dimensionality reduction techniques are Principle Component Analysis , Factor Analysis , Canonical Correlation , Random Forest .

Principle Component Analysis (PCA)

Principal Component Analysis (PCA) is a dimensionality reduction technique that is very often used to reduce the dimensionality of large data sets. It does this by transforming a large set of variables into a smaller set that still contains most of the information or the variation in the original large dataset.

Let’s assume we have a data X with p variables X1, X2, …., Xp with eigenvectors e1, …, ep, and eigenvalues λ1,…, λp. Eigenvalues show the variance explained by a particular data field out of the total variance.

The idea behind PCA is to create new (independent) variables, called Principal Components, that are a linear combination of the existing variable. The i th principal component can be expressed as follows:

Then using the Elbow Rule or Kaiser Rule , you can determine the number of principal components that optimally summarize the data without losing too much information.

It is also important to look at the proportion of total variation (PRTV) that is explained by each principal component to decide whether it is beneficial to include or to exclude it. PRTV for the i th principal component can be calculated using eigenvalues as follows:

The elbow rule or the elbow method is a heuristic approach that we can use to determine the number of optimal principal components from the PCA results.

The idea behind this method is to plot the explained variation as a function of the number of components and pick the elbow of the curve as the number of optimal principal components.

Following is an example of such a scatter plot where the PRTV (Y-axis) is plotted on the number of principal components (X-axis). The elbow corresponds to the X-axis value 2, which suggests that the number of optimal principal components is 2.

Factor Analysis (FA)

Factor analysis or FA is another statistical method for dimensionality reduction. It is one of the most commonly used inter-dependency techniques. We can use it when the relevant set of variables shows a systematic inter-dependence and our objective is to find out the latent factors that create a commonality.

Let’s assume we have a data X with p variables X1, X2, …., Xp. The FA model can be expressed as follows:

• X is a [p x N] matrix of p variables and N observations
• µ is [p x N] population mean matrix
• A is [p x k] common factor loadings matrix
• F [k x N] is the matrix of common factors
• and u [pxN] is the matrix of specific factors.

So, to put it differently, a factor model is as a series of multiple regressions, predicting each of the variables Xi from the values of the unobservable common factors are:

Each variable has k of its own common factors, and these are related to the observations via the factor loading matrix for a single observation as follows:

In factor analysis, the factors are calculated to maximize between-group variance while minimizing in-group varianc e. They are factors because they group the underlying variables. Unlike the PCA, in FA the data needs to be normalized, given that FA assumes that the dataset follows Normal Distribution.

Interview Prep – Top 7 Statistics Questions with Answers

Are you preparing for interviews in statistics, data analysis, or data science? It's crucial to know key statistical concepts and their applications.

Below I've included seven important statistics questions with answers, covering basic statistical tests, probability theory, and the use of statistics in decision-making, like A/B testing.

Question 1: What is the d ifference b etween a t-test and Z-test ?

The question "What is the difference between a t-test and Z-test?" is a common question in data science interviews because it tests the candidate's understanding of basic statistical concepts used in comparing group means.

This knowledge is crucial because choosing the right test affects the validity of conclusions drawn from data, which is a daily task in a data scientist's role when it comes to interpreting experiments, analyzing survey results, or evaluating models.

Both t-tests and Z-tests are statistical methods used to determine if there are significant differences between the means of two groups. But they have key differences:

• Assumptions : You can use a t-test when the sample sizes are small and the population standard deviation is unknown. It doesn't require the sample mean to be normally distributed if the sample size is sufficiently large due to the Central Limit Theorem. The Z-test assumes that both the sample and the population distributions are normally distributed.
• Sample Size : T-tests are typically used for sample sizes smaller than 30, whereas Z-tests are used for larger sample sizes (greater than or equal to 30) when the population standard deviation is known.
• Test Statistic : The t-test uses the t-distribution to calculate the test statistic, taking into account the sample standard deviation. The Z-test uses the standard normal distribution, utilizing the known population standard deviation.
• P-Value : The p-value in a t-test is determined based on the t-distribution, which accounts for the variability in smaller samples. The Z-test uses the standard normal distribution to calculate the p-value, suitable for larger samples or known population parameters.

Question 2: What is a p-value?

The question "What is a p-value?" requires the understanding of a fundamental concept in hypothesis testing that we descussed in this blog in detail with examples. It's not just a number – it's a bridge between the data you collect and the conclusions you draw for data driven decision making.

P-values quantify the evidence against a null hypothesis—how likely it is to observe the collected data if the null hypothesis were true.

For data scientists, p-values are part of everyday language in statistical analysis, model validation, and experimental design. They have to interpret p-values correctly to make informed decisions and often need to explain their implications to stakeholders who might not have deep statistical knowledge.

Thus, understanding p-values helps data scientists to convey the level of certainty or doubt in their findings and to justify subsequent actions or recommendations.

So here you need to show your understanding of what p-value measures and connect it to statistical significance and hypothesis testing.

The p-value measures the probability of observing a test statistic at least as extreme as the one observed, under the assumption that the null hypothesis is true. It helps in deciding whether the observed data significantly deviate from what would be expected under the null hypothesis.

If the p-value is lower than a predetermined threshold (alpha level, usually set at 0.05), the null hypothesis is rejected, indicating that the observed result is statistically significant.

Question 3: What are limitations of p-values?

P-values are a staple of inferential statistics, providing a metric for evaluating evidence against a null hypothesis. In these question you need to name couple of them.

• Dependence on Sample Size : The p-value is sensitive to the sample size. Large samples might yield significant p-values even for trivial effects, while small samples may not detect significant effects even if they exist.
• Not a Measure of Effect Size or Importance : A small p-value does not necessarily mean the effect is practically significant – it simply indicates it's unlikely to have occurred by chance.
• Misinterpretation : P-values can be misinterpreted as the probability that the null hypothesis is true, which is incorrect. They only measure the evidence against the null hypothesis.

Question 4: What is a Confidence Level?

A confidence level represents the frequency with which an estimated confidence interval would contain the true population parameter if the same process were repeated multiple times.

For example, a 95% confidence level means that if the study were repeated 100 times, approximately 95 of the confidence intervals calculated from those studies would be expected to contain the true population parameter.

Question 5: What is the Probability of Picking 5 Red and 5 Blue Balls Without Replacement?

What is the probability of picking exactly 5 red balls and 5 blue balls in 10 picks without replacement from a set of 100 balls, where there are 70 red balls and 30 blue balls? The text describes how to calculate this probability using combinatorial mathematics and the hypergeometric distribution.

In this question, you're dealing with a classic probability problem that involves combinatorial principles and the concept of probability without replacement. The context is a finite set of balls, each draw affecting the subsequent ones because the composition of the set changes with each draw.

To approach this problem, you need to consider:

• The total number of balls : If the question doesn't specify this, you need to ask or make a reasonable assumption based on the context.
• Initial proportion of balls : Know the initial count of red and blue balls in the set.
• Sequential probability : Remember that each time you draw a ball, you don't put it back, so the probability of drawing a ball of a certain color changes with each draw.
• Combinations : Calculate the number of ways to choose 5 red balls from the total red balls and 5 blue balls from the total blue balls, then divide by the number of ways to choose any 10 balls from the total.

Thinking through these points will guide you in formulating the solution based on the hypergeometric distribution, which describes the probability of a given number of successes in draws without replacement from a finite population.

This question tests your ability to apply probability theory to a dynamic scenario, a skill that's invaluable in data-driven decision-making and statistical modeling.

To find the probability of picking exactly 5 red balls and 5 blue balls in 10 picks without replacement, we calculate the probability of picking 5 red balls out of 70 and 5 blue balls out of 30, and then divide by the total ways to pick 10 balls out of 100:

Let's calculate this probability:

Question 6: Explain Bayes' Theorem and its importance in calculating posterior probabilities.

Provide an example of how it might be used in genetic testing to determine the likelihood of an individual carrying a certain gene.

Bayes' Theorem is a cornerstone of probability theory that enables the updating of initial beliefs (prior probabilities) with new evidence to obtain updated beliefs (posterior probabilities). This question wants to test candidates ability to explain the concept, mathematical framework for incorporating new evidence into existing predictions or models.

Bayes' Theorem is a fundamental theorem in probability theory and statistics that describes the probability of an event, based on prior knowledge of conditions that might be related to the event. It's crucial for calculating posterior probabilities, which are the probabilities of hypotheses given observed evidence.

• P ( A ∣ B ) is the posterior probability: the probability of hypothesis A given the evidence B .
• P(B∣A) is the likelihood: the probability of observing evidence B given that hypothesis A is true.
• P(A) is the prior probability: the initial probability of hypothesis A , before observing evidence B .
• P(B) is the marginal probability: the total probability of observing evidence B B under all possible hypotheses.

Question 7: Describe how you would statistically determine if the results of an A/B test are significant - walk me through AB Testing process.

In this question, the interviewer is assessing your comprehensive knowledge of the A/B testing framework. They are looking for evidence that you can navigate the full spectrum of A/B testing procedures, which is essential for data scientists and AI professionals tasked with optimizing features, making data-informed decisions, and testing software products.

The interviewer wants to confirm that you understand each step in the process, beginning with formulating statistical hypotheses derived from business objectives. They are interested in your ability to conduct a power analysis and discuss its components, including determining effect size, significance level, and power, all critical in calculating the minimum sample size needed to detect a true effect and prevent p-hacking.

The discussion on randomization, data collection, and monitoring checks whether you grasp how to maintain the integrity of the test conditions. You should also be prepared to explain the selection of appropriate statistical tests, calculation of test statistics, p-values, and interpretation of results for both statistical and practical significance.

Ultimately, the interviewer is testing whether you can act as a data advocate: someone who can meticulously run A/B tests, interpret the results, and communicate findings and recommendations effectively to stakeholders, thereby driving data-driven decision-making within the organization.

To Learn AB Testing check my AB Testing Crash Course on YouTube .

In an A/B test, my first step is to establish clear business and statistical hypotheses. For example, if we’re testing a new webpage layout, the business hypothesis might be that the new layout increases user engagement. Statistically, this translates to expecting a higher mean engagement score for the new layout compared to the old.

Next, I’d conduct a power analysis. This involves deciding on an effect size that's practically significant for our business context—say, a 10% increase in engagement. I'd choose a significance level, commonly 0.05, and aim for a power of 80%, reducing the likelihood of Type II errors.

The power analysis, which takes into account the effect size, significance level, and power, helps determine the minimum sample size needed. This is crucial for ensuring that our test is adequately powered to detect the effect we care about and for avoiding p-hacking by committing to a sample size upfront.

With our sample size determined, I’d ensure proper randomization in assigning users to the control and test groups, to eliminate selection bias. During the test, I’d closely monitor data collection for any anomalies or necessary adjustments.

Upon completion of the data collection, I’d choose an appropriate statistical test based on the data distribution and variance homogeneity—typically a t-test if the sample size is small or a normal distribution can’t be assumed, or a Z-test for larger samples with a known variance.

Calculating the test statistic and the corresponding p-value allows us to test the null hypothesis. If the p-value is less than our chosen alpha level, we reject the null hypothesis, suggesting that the new layout has a statistically significant impact on engagement.

In addition to statistical significance, I’d evaluate the practical significance by looking at the confidence interval for the effect size and considering the business impact.

Finally, I’d document the entire process and results, then communicate them to stakeholders in a clear, non-technical language. This includes not just the statistical significance, but also how the results translate to business outcomes. As a data advocate, my goal is to support data-driven decisions that align with our business objectives and user experience strategy

For getting more interview questions from Stats to Deep Learning - with over 400 Q&A as well as personalized interview preparation check out our Free Resource Hub and our Data Science Bootcamp with Free Trial .

Thank you for choosing this guide as your learning companion. As you continue to explore the vast field of machine learning, I hope you do so with confidence, precision, and an innovative spirit. Best wishes in all your future endeavors!

About the Author

I am Tatev Aslanyan , Senior Machine Learning and AI Researcher, and Co-Founder of LunarTech where we are making Data Science and AI accessible to everyone. I have had the privilege of working in Data Science across numerous countries, including the US, UK, Canada, and the Netherlands.

With an MSc and BSc in Econometrics under my belt, my journey in Machine and AI has been nothing short of incredible. Drawing from my technical studies during my Bachelors & Masters, along with over 5 years of hands-on experience in the Data Science Industry, in Machine Learning and AI, I've gathered this high-level summary of ML topics to share with you.

After studying this guide, if you're keen to dive even deeper and structured learning is your style, consider joining us at LunarTech , we offer individual courses and Bootcamp in Data Science, Machine Learning and AI.

We provide a comprehensive program that offers an in-depth understanding of the theory, hands-on practical implementation, extensive practice material, and tailored interview preparation to set you up for success at your own phase.

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