research meaning in statistics

Statistics Made Easy

The Importance of Statistics in Research (With Examples)

The field of statistics is concerned with collecting, analyzing, interpreting, and presenting data.

In the field of research, statistics is important for the following reasons:

Reason 1 : Statistics allows researchers to design studies such that the findings from the studies can be extrapolated to a larger population.

Reason 2 : Statistics allows researchers to perform hypothesis tests to determine if some claim about a new drug, new procedure, new manufacturing method, etc. is true.

Reason 3 : Statistics allows researchers to create confidence intervals to capture uncertainty around population estimates.

In the rest of this article, we elaborate on each of these reasons.

Reason 1: Statistics Allows Researchers to Design Studies

Researchers are often interested in answering questions about populations like:

What is the average weight of a certain species of bird?
What is the average height of a certain species of plant?
What percentage of citizens in a certain city support a certain law?

One way to answer these questions is to go around and collect data on every single individual in the population of interest.

However, this is typically too costly and time-consuming which is why researchers instead take a sample of the population and use the data from the sample to draw conclusions about the population as a whole.

Example of taking a sample from a population

There are many different methods researchers can potentially use to obtain individuals to be in a sample. These are known as sampling methods .

There are two classes of sampling methods:

Probability sampling methods : Every member in a population has an equal probability of being selected to be in the sample.
Non-probability sampling methods : Not every member in a population has an equal probability of being selected to be in the sample.

By using probability sampling methods, researchers can maximize the chances that they obtain a sample that is representative of the overall population.

This allows researchers to extrapolate the findings from the sample to the overall population.

Read more about the two classes of sampling methods here .

Reason 2: Statistics Allows Researchers to Perform Hypothesis Tests

Another way that statistics is used in research is in the form of hypothesis tests .

These are tests that researchers can use to determine if there is a statistical significance between different medical procedures or treatments.

For example, suppose a scientist believes that a new drug is able to reduce blood pressure in obese patients. To test this, he measures the blood pressure of 30 patients before and after using the new drug for one month.

He then performs a paired samples t- test using the following hypotheses:

H 0 : μ after = μ before (the mean blood pressure is the same before and after using the drug)
H A : μ after < μ before (the mean blood pressure is less after using the drug)

If the p-value of the test is less than some significance level (e.g. α = .05), then he can reject the null hypothesis and conclude that the new drug leads to reduced blood pressure.

Note : This is just one example of a hypothesis test that is used in research. Other common tests include a one sample t-test , two sample t-test , one-way ANOVA , and two-way ANOVA .

Reason 3: Statistics Allows Researchers to Create Confidence Intervals

Another way that statistics is used in research is in the form of confidence intervals .

A confidence interval is a range of values that is likely to contain a population parameter with a certain level of confidence.

For example, suppose researchers are interested in estimating the mean weight of a certain species of turtle.

Instead of going around and weighing every single turtle in the population, researchers may instead take a simple random sample of turtles with the following information:

Sample size n = 25
Sample mean weight x = 300
Sample standard deviation s = 18.5

Using the confidence interval for a mean formula , researchers may then construct the following 95% confidence interval:

95% Confidence Interval: 300 +/- 1.96*(18.5/√ 25 ) = [292.75, 307.25]

The researchers would then claim that they’re 95% confident that the true mean weight for this population of turtles is between 292.75 pounds and 307.25 pounds.

Additional Resources

The following articles explain the importance of statistics in other fields:

The Importance of Statistics in Healthcare The Importance of Statistics in Nursing The Importance of Statistics in Business The Importance of Statistics in Economics The Importance of Statistics in Education

Published by Zach

Department of Statistics - Donald Bren School of Information & Computer Sciences

What is statistics.

Statistics is the science concerned with developing and studying methods for collecting, analyzing, interpreting and presenting empirical data. Statistics is a highly interdisciplinary field; research in statistics finds applicability in virtually all scientific fields and research questions in the various scientific fields motivate the development of new statistical methods and theory. In developing methods and studying the theory that underlies the methods statisticians draw on a variety of mathematical and computational tools.

Two fundamental ideas in the field of statistics are uncertainty and variation. There are many situations that we encounter in science (or more generally in life) in which the outcome is uncertain. In some cases the uncertainty is because the outcome in question is not determined yet (e.g., we may not know whether it will rain tomorrow) while in other cases the uncertainty is because although the outcome has been determined already we are not aware of it (e.g., we may not know whether we passed a particular exam).

Probability is a mathematical language used to discuss uncertain events and probability plays a key role in statistics. Any measurement or data collection effort is subject to a number of sources of variation. By this we mean that if the same measurement were repeated, then the answer would likely change. Statisticians attempt to understand and control (where possible) the sources of variation in any situation.

We encourage you to continue exploring our website to learn more about statistics, our academic programs, our students and faculty, as well as the cutting-edge research we are doing in the field.

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1.1: Basic Definitions and Concepts

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Learning Objectives

To learn the basic definitions used in statistics and some of its key concepts.

We begin with a simple example. There are millions of passenger automobiles in the United States. What is their average value? It is obviously impractical to attempt to solve this problem directly by assessing the value of every single car in the country, add up all those values, then divide by the number of values, one for each car. In practice the best we can do would be to estimate the average value. A natural way to do so would be to randomly select some of the cars, say $200$ of them, ascertain the value of each of those cars, and find the average of those $200$ values. The set of all those millions of vehicles is called the population of interest, and the number attached to each one, its value, is a measurement . The average value is a parameter : a number that describes a characteristic of the population, in this case monetary worth. The set of $200$ cars selected from the population is called a sample , and the $200$ numbers, the monetary values of the cars we selected, are the sample data . The average of the data is called a statistic : a number calculated from the sample data. This example illustrates the meaning of the following definitions.

Definitions: populations and samples

A population is any specific collection of objects of interest. A sample is any subset or subcollection of the population, including the case that the sample consists of the whole population, in which case it is termed a census.

Definitions: measurements and Sample Data

A measurement is a number or attribute computed for each member of a population or of a sample. The measurements of sample elements are collectively called the sample data .

Definition: parameters

A parameter is a number that summarizes some aspect of the population as a whole. A statistic is a number computed from the sample data.

Continuing with our example, if the average value of the cars in our sample was $$8,357$, then it seems reasonable to conclude that the average value of all cars is about $$8,357$. In reasoning this way we have drawn an inference about the population based on information obtained from the sample . In general, statistics is a study of data: describing properties of the data, which is called descriptive statistics , and drawing conclusions about a population of interest from information extracted from a sample, which is called inferential statistics . Computing the single number $$8,357$ to summarize the data was an operation of descriptive statistics; using it to make a statement about the population was an operation of inferential statistics.

Definition: Statistics

Statistics is a collection of methods for collecting, displaying, analyzing, and drawing conclusions from data.

Definition: Descriptive statistics

Descriptive statistics is the branch of statistics that involves organizing, displaying, and describing data.

Definition: Inferential statistics

Inferential statistics is the branch of statistics that involves drawing conclusions about a population based on information contained in a sample taken from that population.

Definition: Qualitative data

Qualitative data are measurements for which there is no natural numerical scale, but which consist of attributes, labels, or other non-numerical characteristics.

Definition: Quantitative data

Quantitative data are numerical measurements that arise from a natural numerical scale.

Qualitative data can generate numerical sample statistics. In the automobile example, for instance, we might be interested in the proportion of all cars that are less than six years old. In our same sample of $200$ cars we could note for each car whether it is less than six years old or not, which is a qualitative measurement. If $172$ cars in the sample are less than six years old, which is $0.86$ or $86\% $, then we would estimate the parameter of interest, the population proportion, to be about the same as the sample statistic, the sample proportion, that is, about $0.86$.

The relationship between a population of interest and a sample drawn from that population is perhaps the most important concept in statistics, since everything else rests on it. This relationship is illustrated graphically in Figure $\PageIndex{1}$. The circles in the large box represent elements of the population. In the figure there was room for only a small number of them but in actual situations, like our automobile example, they could very well number in the millions. The solid black circles represent the elements of the population that are selected at random and that together form the sample. For each element of the sample there is a measurement of interest, denoted by a lower case $x$ (which we have indexed as $x_1 , \ldots, x_n$ to tell them apart); these measurements collectively form the sample data set. From the data we may calculate various statistics. To anticipate the notation that will be used later, we might compute the sample mean $\bar{x}$ and the sample proportion $\hat{p}$, and take them as approximations to the population mean $\mu$ (this is the lower case Greek letter mu, the traditional symbol for this parameter) and the population proportion $p$, respectively. The other symbols in the figure stand for other parameters and statistics that we will encounter.

Key Takeaway

Statistics is a study of data: describing properties of data (descriptive statistics) and drawing conclusions about a population based on information in a sample (inferential statistics).
The distinction between a population together with its parameters and a sample together with its statistics is a fundamental concept in inferential statistics.
Information in a sample is used to make inferences about the population from which the sample was drawn.

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Statistical Analysis in Research: Meaning, Methods and Types

Home » Videos » Statistical Analysis in Research: Meaning, Methods and Types

The scientific method is an empirical approach to acquiring new knowledge by making skeptical observations and analyses to develop a meaningful interpretation. It is the basis of research and the primary pillar of modern science. Researchers seek to understand the relationships between factors associated with the phenomena of interest. In some cases, research works with vast chunks of data, making it difficult to observe or manipulate each data point. As a result, statistical analysis in research becomes a means of evaluating relationships and interconnections between variables with tools and analytical techniques for working with large data. Since researchers use statistical power analysis to assess the probability of finding an effect in such an investigation, the method is relatively accurate. Hence, statistical analysis in research eases analytical methods by focusing on the quantifiable aspects of phenomena.

What is Statistical Analysis in Research? A Simplified Definition

Statistical analysis uses quantitative data to investigate patterns, relationships, and patterns to understand real-life and simulated phenomena. The approach is a key analytical tool in various fields, including academia, business, government, and science in general. This statistical analysis in research definition implies that the primary focus of the scientific method is quantitative research. Notably, the investigator targets the constructs developed from general concepts as the researchers can quantify their hypotheses and present their findings in simple statistics.

When a business needs to learn how to improve its product, they collect statistical data about the production line and customer satisfaction. Qualitative data is valuable and often identifies the most common themes in the stakeholders’ responses. On the other hand, the quantitative data creates a level of importance, comparing the themes based on their criticality to the affected persons. For instance, descriptive statistics highlight tendency, frequency, variation, and position information. While the mean shows the average number of respondents who value a certain aspect, the variance indicates the accuracy of the data. In any case, statistical analysis creates simplified concepts used to understand the phenomenon under investigation. It is also a key component in academia as the primary approach to data representation, especially in research projects, term papers and dissertations.

Most Useful Statistical Analysis Methods in Research

Using statistical analysis methods in research is inevitable, especially in academic assignments, projects, and term papers. It’s always advisable to seek assistance from your professor or you can try research paper writing by CustomWritings before you start your academic project or write statistical analysis in research paper. Consulting an expert when developing a topic for your thesis or short mid-term assignment increases your chances of getting a better grade. Most importantly, it improves your understanding of research methods with insights on how to enhance the originality and quality of personalized essays. Professional writers can also help select the most suitable statistical analysis method for your thesis, influencing the choice of data and type of study.

Descriptive Statistics

Descriptive statistics is a statistical method summarizing quantitative figures to understand critical details about the sample and population. A description statistic is a figure that quantifies a specific aspect of the data. For instance, instead of analyzing the behavior of a thousand students, research can identify the most common actions among them. By doing this, the person utilizes statistical analysis in research, particularly descriptive statistics.

Measures of central tendency . Central tendency measures are the mean, mode, and media or the averages denoting specific data points. They assess the centrality of the probability distribution, hence the name. These measures describe the data in relation to the center.
Measures of frequency . These statistics document the number of times an event happens. They include frequency, count, ratios, rates, and proportions. Measures of frequency can also show how often a score occurs.
Measures of dispersion/variation . These descriptive statistics assess the intervals between the data points. The objective is to view the spread or disparity between the specific inputs. Measures of variation include the standard deviation, variance, and range. They indicate how the spread may affect other statistics, such as the mean.
Measures of position . Sometimes researchers can investigate relationships between scores. Measures of position, such as percentiles, quartiles, and ranks, demonstrate this association. They are often useful when comparing the data to normalized information.

Inferential Statistics

Inferential statistics is critical in statistical analysis in quantitative research. This approach uses statistical tests to draw conclusions about the population. Examples of inferential statistics include t-tests, F-tests, ANOVA, p-value, Mann-Whitney U test, and Wilcoxon W test. This

Common Statistical Analysis in Research Types

Although inferential and descriptive statistics can be classified as types of statistical analysis in research, they are mostly considered analytical methods. Types of research are distinguishable by the differences in the methodology employed in analyzing, assembling, classifying, manipulating, and interpreting data. The categories may also depend on the type of data used.

Predictive Analysis

Predictive research analyzes past and present data to assess trends and predict future events. An excellent example of predictive analysis is a market survey that seeks to understand customers’ spending habits to weigh the possibility of a repeat or future purchase. Such studies assess the likelihood of an action based on trends.

Prescriptive Analysis

On the other hand, a prescriptive analysis targets likely courses of action. It’s decision-making research designed to identify optimal solutions to a problem. Its primary objective is to test or assess alternative measures.

Causal Analysis

Causal research investigates the explanation behind the events. It explores the relationship between factors for causation. Thus, researchers use causal analyses to analyze root causes, possible problems, and unknown outcomes.

Mechanistic Analysis

This type of research investigates the mechanism of action. Instead of focusing only on the causes or possible outcomes, researchers may seek an understanding of the processes involved. In such cases, they use mechanistic analyses to document, observe, or learn the mechanisms involved.

Exploratory Data Analysis

Similarly, an exploratory study is extensive with a wider scope and minimal limitations. This type of research seeks insight into the topic of interest. An exploratory researcher does not try to generalize or predict relationships. Instead, they look for information about the subject before conducting an in-depth analysis.

The Importance of Statistical Analysis in Research

As a matter of fact, statistical analysis provides critical information for decision-making. Decision-makers require past trends and predictive assumptions to inform their actions. In most cases, the data is too complex or lacks meaningful inferences. Statistical tools for analyzing such details help save time and money, deriving only valuable information for assessment. An excellent statistical analysis in research example is a randomized control trial (RCT) for the Covid-19 vaccine. You can download a sample of such a document online to understand the significance such analyses have to the stakeholders. A vaccine RCT assesses the effectiveness, side effects, duration of protection, and other benefits. Hence, statistical analysis in research is a helpful tool for understanding data.

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Effective Use of Statistics in Research – Methods and Tools for Data Analysis

Remember that impending feeling you get when you are asked to analyze your data! Now that you have all the required raw data, you need to statistically prove your hypothesis. Representing your numerical data as part of statistics in research will also help in breaking the stereotype of being a biology student who can’t do math.

Statistical methods are essential for scientific research. In fact, statistical methods dominate the scientific research as they include planning, designing, collecting data, analyzing, drawing meaningful interpretation and reporting of research findings. Furthermore, the results acquired from research project are meaningless raw data unless analyzed with statistical tools. Therefore, determining statistics in research is of utmost necessity to justify research findings. In this article, we will discuss how using statistical methods for biology could help draw meaningful conclusion to analyze biological studies.

Table of Contents

Role of Statistics in Biological Research

Statistics is a branch of science that deals with collection, organization and analysis of data from the sample to the whole population. Moreover, it aids in designing a study more meticulously and also give a logical reasoning in concluding the hypothesis. Furthermore, biology study focuses on study of living organisms and their complex living pathways, which are very dynamic and cannot be explained with logical reasoning. However, statistics is more complex a field of study that defines and explains study patterns based on the sample sizes used. To be precise, statistics provides a trend in the conducted study.

Biological researchers often disregard the use of statistics in their research planning, and mainly use statistical tools at the end of their experiment. Therefore, giving rise to a complicated set of results which are not easily analyzed from statistical tools in research. Statistics in research can help a researcher approach the study in a stepwise manner, wherein the statistical analysis in research follows –

1. Establishing a Sample Size

Usually, a biological experiment starts with choosing samples and selecting the right number of repetitive experiments. Statistics in research deals with basics in statistics that provides statistical randomness and law of using large samples. Statistics teaches how choosing a sample size from a random large pool of sample helps extrapolate statistical findings and reduce experimental bias and errors.

2. Testing of Hypothesis

When conducting a statistical study with large sample pool, biological researchers must make sure that a conclusion is statistically significant. To achieve this, a researcher must create a hypothesis before examining the distribution of data. Furthermore, statistics in research helps interpret the data clustered near the mean of distributed data or spread across the distribution. These trends help analyze the sample and signify the hypothesis.

3. Data Interpretation Through Analysis

When dealing with large data, statistics in research assist in data analysis. This helps researchers to draw an effective conclusion from their experiment and observations. Concluding the study manually or from visual observation may give erroneous results; therefore, thorough statistical analysis will take into consideration all the other statistical measures and variance in the sample to provide a detailed interpretation of the data. Therefore, researchers produce a detailed and important data to support the conclusion.

Types of Statistical Research Methods That Aid in Data Analysis

Statistical analysis is the process of analyzing samples of data into patterns or trends that help researchers anticipate situations and make appropriate research conclusions. Based on the type of data, statistical analyses are of the following type:

1. Descriptive Analysis

The descriptive statistical analysis allows organizing and summarizing the large data into graphs and tables . Descriptive analysis involves various processes such as tabulation, measure of central tendency, measure of dispersion or variance, skewness measurements etc.

2. Inferential Analysis

The inferential statistical analysis allows to extrapolate the data acquired from a small sample size to the complete population. This analysis helps draw conclusions and make decisions about the whole population on the basis of sample data. It is a highly recommended statistical method for research projects that work with smaller sample size and meaning to extrapolate conclusion for large population.

3. Predictive Analysis

Predictive analysis is used to make a prediction of future events. This analysis is approached by marketing companies, insurance organizations, online service providers, data-driven marketing, and financial corporations.

4. Prescriptive Analysis

Prescriptive analysis examines data to find out what can be done next. It is widely used in business analysis for finding out the best possible outcome for a situation. It is nearly related to descriptive and predictive analysis. However, prescriptive analysis deals with giving appropriate suggestions among the available preferences.

5. Exploratory Data Analysis

EDA is generally the first step of the data analysis process that is conducted before performing any other statistical analysis technique. It completely focuses on analyzing patterns in the data to recognize potential relationships. EDA is used to discover unknown associations within data, inspect missing data from collected data and obtain maximum insights.

6. Causal Analysis

Causal analysis assists in understanding and determining the reasons behind “why” things happen in a certain way, as they appear. This analysis helps identify root cause of failures or simply find the basic reason why something could happen. For example, causal analysis is used to understand what will happen to the provided variable if another variable changes.

7. Mechanistic Analysis

This is a least common type of statistical analysis. The mechanistic analysis is used in the process of big data analytics and biological science. It uses the concept of understanding individual changes in variables that cause changes in other variables correspondingly while excluding external influences.

Important Statistical Tools In Research

Researchers in the biological field find statistical analysis in research as the scariest aspect of completing research. However, statistical tools in research can help researchers understand what to do with data and how to interpret the results, making this process as easy as possible.

1. Statistical Package for Social Science (SPSS)

It is a widely used software package for human behavior research. SPSS can compile descriptive statistics, as well as graphical depictions of result. Moreover, it includes the option to create scripts that automate analysis or carry out more advanced statistical processing.

2. R Foundation for Statistical Computing

This software package is used among human behavior research and other fields. R is a powerful tool and has a steep learning curve. However, it requires a certain level of coding. Furthermore, it comes with an active community that is engaged in building and enhancing the software and the associated plugins.

3. MATLAB (The Mathworks)

It is an analytical platform and a programming language. Researchers and engineers use this software and create their own code and help answer their research question. While MatLab can be a difficult tool to use for novices, it offers flexibility in terms of what the researcher needs.

4. Microsoft Excel

Not the best solution for statistical analysis in research, but MS Excel offers wide variety of tools for data visualization and simple statistics. It is easy to generate summary and customizable graphs and figures. MS Excel is the most accessible option for those wanting to start with statistics.

5. Statistical Analysis Software (SAS)

It is a statistical platform used in business, healthcare, and human behavior research alike. It can carry out advanced analyzes and produce publication-worthy figures, tables and charts .

6. GraphPad Prism

It is a premium software that is primarily used among biology researchers. But, it offers a range of variety to be used in various other fields. Similar to SPSS, GraphPad gives scripting option to automate analyses to carry out complex statistical calculations.

This software offers basic as well as advanced statistical tools for data analysis. However, similar to GraphPad and SPSS, minitab needs command over coding and can offer automated analyses.

Use of Statistical Tools In Research and Data Analysis

Statistical tools manage the large data. Many biological studies use large data to analyze the trends and patterns in studies. Therefore, using statistical tools becomes essential, as they manage the large data sets, making data processing more convenient.

Following these steps will help biological researchers to showcase the statistics in research in detail, and develop accurate hypothesis and use correct tools for it.

There are a range of statistical tools in research which can help researchers manage their research data and improve the outcome of their research by better interpretation of data. You could use statistics in research by understanding the research question, knowledge of statistics and your personal experience in coding.

Have you faced challenges while using statistics in research? How did you manage it? Did you use any of the statistical tools to help you with your research data? Do write to us or comment below!

Frequently Asked Questions

Statistics in research can help a researcher approach the study in a stepwise manner: 1. Establishing a sample size 2. Testing of hypothesis 3. Data interpretation through analysis

Statistical tools in research can help researchers understand what to do with data and how to interpret the results, making this process as easy as possible. They can manage large data sets, making data processing more convenient. A great number of tools are available to carry out statistical analysis of data like SPSS, SAS (Statistical Analysis Software), and Minitab.

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Introduction to Statistics

(15 reviews)

David Lane, Rice University

Publisher: David Lane

Language: English

Formats Available

Conditions of use.

Learn more about reviews.

Reviewed by Terri Torres, professor, Oregon Institute of Technology on 8/17/23

This author covers all the topics that would be covered in an introductory statistics course plus some. I could imagine using it for two courses at my university, which is on the quarter system. I would rather have the problem of too many topics... read more

Comprehensiveness rating: 5 see less

Content Accuracy rating: 5

Yes, Lane is both thorough and accurate.

Relevance/Longevity rating: 5

What is covered is what is usually covered in an introductory statistics book. The only topic I may, given sufficient time, cover is bootstrapping.

Clarity rating: 5

The book is clear and well-written. For the trickier topics, simulations are included to help with understanding.

Consistency rating: 5

All is organized in a way that is consistent with the previous topic.

Modularity rating: 5

The text is organized in a way that easily enables navigation.

Organization/Structure/Flow rating: 5

The text is organized like most statistics texts.

Interface rating: 5

Easy navigation.

Grammatical Errors rating: 5

I didn't see any grammatical errors.

Cultural Relevance rating: 5

Nothing is included that is culturally insensitive.

The videos that accompany this text are short and easy to watch and understand. Videos should be short enough to teach, but not so long that they are tiresome. This text includes almost everything: videos, simulations, case studies---all nicely organized in one spot. In addition, Lane has promised to send an instructor's manual and slide deck.

Reviewed by Professor Sandberg, Professor, Framingham State University on 6/29/21

This text covers all the usual topics in an Introduction to Statistics for college students. In addition, it has some additional topics that are useful. read more

This text covers all the usual topics in an Introduction to Statistics for college students. In addition, it has some additional topics that are useful.

I did not find any errors.

Some of the examples are dated. And the frequent use of male/female examples need updating in terms of current gender splits.

I found it was easy to read and understand and I expect that students would also find the writing clear and the explanations accessible.

Even with different authors of chapter, the writing is consistent.

The text is well organized into sections making it easy to assign individual topics and sections.

The topics are presented in the usual order. Regression comes later in the text but there is a difference of opinions about whether to present it early with descriptive statistics for bivariate data or later with inferential statistics.

I had no problem navigating the text online.

The writing is grammatical correct.

I saw no issues that would be offensive.

I did like this text. It seems like it would be a good choice for most introductory statistics courses. I liked that the Monty Hall problem was included in the probability section. The author offers to provide an instructor's manual, PowerPoint slides and additional questions. These additional resources are very helpful and not always available with online OER texts.

Reviewed by Emilio Vazquez, Associate Professor, Trine University on 4/23/21

This appears to be an excellent textbook for an Introductory Course in Statistics. It covers subjects in enough depth to fulfill the needs of a beginner in Statistics work yet is not so complex as to be overwhelming. read more

I found no errors in their discussions. Did not work out all of the questions and answers but my sampling did not reveal any errors.

Some of the examples may need updating depending on the times but the examples are still relevant at this time.

This is a Statistics text so a little dry. I found that the derivation of some of the formulas was not explained. However the background is there to allow the instructor to derive these in class if desired.

The text is consistent throughout using the same verbiage in various sections.

The text dose lend itself to reasonable reading assignments. For example the chapter (Chapter 3) on Summarizing Distributions covers Central Tendency and its associated components in an easy 20 pages with Measures of Variability making up most of the rest of the chapter and covering approximately another 20 pages. Exercises are available at the end of each chapter making it easy for the instructor to assign reading and exercises to be discussed in class.

The textbook flows easily from Descriptive to Inferential Statistics with chapters on Sampling and Estimation preceding chapters on hypothesis testing

I had no problems with navigation

All textbooks have a few errors but certainly nothing glaring or making text difficult

I saw no issues and I am part of a cultural minority in the US

Overall I found this to be a excellent in-depth overview of Statistical Theory, Concepts and Analysis. The length of the textbook appears to be more than adequate for a one-semester course in Introduction to Statistics. As I no longer teach a full statistics course but simply a few lectures as part of our Research Curriculum, I am recommending this book to my students as a good reference. Especially as it is available on-line and in Open Access.

Reviewed by Audrey Hickert, Assistant Professor, Southern Illinois University Carbondale on 3/29/21

All of the major topics of an introductory level statistics course for social science are covered. Background areas include levels of measurement and research design basics. Descriptive statistics include all major measures of central tendency and dispersion/variation. Building blocks for inferential statistics include sampling distributions, the standard normal curve (z scores), and hypothesis testing sections. Inferential statistics include how to calculate confidence intervals, as well as conduct tests of one-sample tests of the population mean (Z- and t-tests), two-sample tests of the difference in population means (Z- and t-tests), chi square test of independence, correlation, and regression. Doesn’t include full probability distribution tables (e.g., t or Z), but those can be easily found online in many places.

I did not find any errors or issues of inaccuracy. When a particular method or practice is debated in the field, the authors acknowledge it (and provide citations in some circumstances).

Relevance/Longevity rating: 4

Basic statistics are standard, so the core information will remain relevant in perpetuity. Some of the examples are dated (e.g., salaries from 1999), but not problematic.

Clarity rating: 4

All of the key terms, formulas, and logic for statistical tests are clearly explained. The book sometimes uses different notation than other entry-level books. For example, the variance formula uses "M" for mean, rather than x-bar.

The explanations are consistent and build from and relate to corresponding sections that are listed in each unit.

Modularity is a strength of this text in both the PDF and interactive online format. Students can easily navigate to the necessary sections and each starts with a “Prerequisites” list of other sections in the book for those who need the additional background material. Instructors could easily compile concise sub-sections of the book for readings.

The presentation of topics differs somewhat from the standard introductory social science statistics textbooks I have used before. However, the modularity allows the instructor and student to work through the discrete sections in the desired order.

Interface rating: 4

For the most part the display of all images/charts is good and navigation is straightforward. One concern is that the organization of the Table of Contents does not exactly match the organizational outline at the start of each chapter in the PDF version. For example, sometimes there are more detailed sub-headings at the start of chapter and occasionally slightly different section headings/titles. There are also inconsistencies in section listings at start of chapters vs. start of sub-sections.

The text is easy to read and free from any obvious grammatical errors.

Although some of the examples are outdated, I did not review any that were offensive. One example of an outdated reference is using descriptive data on “Men per 100 Women” in U.S. cities as “useful if we are looking for an opposite-sex partner”.

This is a good introduction level statistics text book if you have a course with students who may be intimated by longer texts with more detailed information. Just the core basics are provided here and it is easy to select the sections you need. It is a good text if you plan to supplement with an array of your own materials (lectures, practice, etc.) that are specifically tailored to your discipline (e.g., criminal justice and criminology). Be advised that some formulas use different notation than other standard texts, so you will need to point that out to students if they differ from your lectures or assessment materials.

Reviewed by Shahar Boneh, Professor, Metropolitan State University of Denver on 3/26/21, updated 4/22/21

The textbook is indeed quite comprehensive. It can accommodate any style of introductory statistics course. read more

The textbook is indeed quite comprehensive. It can accommodate any style of introductory statistics course.

The text seems to be statistically accurate.

It is a little too extensive, which requires instructors to cover it selectively, and has a potential to confuse the students.

It is written clearly.

Consistency rating: 4

The terminology is fairly consistent. There is room for some improvement.

By the nature of the subject, the topics have to be presented in a sequential and coherent order. However, the book breaks things down quite effectively.

Organization/Structure/Flow rating: 3

Some of the topics are interleaved and not presented in the order I would like to cover them.

Good interface.

The grammar is ok.

The book seems to be culturally neutral, and not offensive in any way.

I really liked the simulations that go with the book. Parts of the book are a little too advanced for students who are learning statistics for the first time.

Reviewed by Julie Gray, Adjunct Assistant Professor, University of Texas at Arlington on 2/26/21

The textbook is for beginner-level students. The concept development is appropriate--there is always room to grow to high higher level, but for an introduction, the basics are what is needed. This is a well-thought-through OER textbook project by... read more

I found all the material accurate.

Essentially, statistical concepts at the introductory level are accepted as universal. This suggests that the relevance of this textbook will continue for a long time.

The book is well written for introducing beginners to statistical concepts. The figures, tables, and animated examples reinforce the clarity of the written text.

Yes, the information is consistent; when it is introduced in early chapters it ties in well in later chapters that build on and add more understanding for the topic.

Modularity rating: 4

The book is well-written with attention to modularity where possible. Due to the nature of statistics, that is not always possible. The content is presented in the order that I usually teach these concepts.

The organization of the book is good, I particularly like the sample lecture slide presentations and the problem set with solutions for use in quizzes and exams. These are available by writing to the author. It is wonderful to have access to these helpful resources for instructors to use in preparation.

I did not find any interface issues.

The book is well written. In my reading I did not notice grammatical errors.

For this subject and in the examples given, I did not notice any cultural issues.

For the field of social work where qualitative data is as common as quantitative, the importance of giving students the rationale or the motivation to learn the quantitative side is understated. To use this text as an introductory statistics OER textbook in a social work curriculum, the instructor will want to bring in field-relevant examples to engage and motivate students. The field needs data-driven decision making and evidence-based practices to become more ubiquitous than not. Preparing future social workers by teaching introductory statistics is essential to meet that goal.

Reviewed by Mamata Marme, Assistant Professor, Augustana College on 6/25/19

This textbook offers a fairly comprehensive summary of what should be discussed in an introductory course in Statistics. The statistical literacy exercises are particularly interesting. It would be helpful to have the statistical tables... read more

Comprehensiveness rating: 4 see less

The terminology and notation used in the textbook is pretty standard. The content is accurate.

The statistical literacy example are up to date but will need to be updated fairly regularly to keep the textbook fresh. The applications within the chapter are accessible and can be used fairly easily over a couple of editions.

The textbook does not necessarily explain the derivation of some of the formulae and this will need to be augmented by the instructor in class discussion. What is beneficial is that there are multiple ways that a topic is discussed using graphs, calculations and explanations of the results. Statistics textbooks have to cover a wide variety of topics with a fair amount of depth. To do this concisely is difficult. There is a fine line between being concise and clear, which this textbook does well, and being somewhat dry. It may be up to the instructor to bring case studies into the readings we are going through the topics rather than wait until the end of the chapter.

The textbook uses standard notation and terminology. The heading section of each chapter is closely tied to topics that are covered. The end of chapter problems and the statistical literacy applications are closely tied to the material covered.

The authors have done a good job treating each chapter as if they stand alone. The lack of connection to a past reference may create a sense of disconnect between the topics discussed

The text's "modularity" does make the flow of the material a little disconnected. If would be better if there was accountability of what a student should already have learnt in a different section. The earlier material is easy to find but not consistently referred to in the text.

I had no problem with the interface. The online version is more visually interesting than the pdf version.

I did not see any grammatical errors.

Cultural Relevance rating: 4

I am not sure how to evaluate this. The examples are mostly based on the American experience and the data alluded to mostly domestic. However, I am not sure if that creates a problem in understanding the methodology.

Overall, this textbook will cover most of the topics in a survey of statistics course.

Reviewed by Alexandra Verkhovtseva, Professor, Anoka-Ramsey Community College on 6/3/19

This is a comprehensive enough text, considering that it is not easy to create a comprehensive statistics textbook. It is suitable for an introductory statistics course for non-math majors. It contains twenty-one chapters, covering the wide range... read more

The content is pretty accurate, I did not find any biases or errors.

The book contains fairly recent data presented in the form of exercises, examples and applications. The topics are up-to-date, and appropriate technology is used for examples, applications, and case studies.

The language is simple and clear, which is a good thing, since students are usually scared of this class, and instructors are looking for something to put them at ease. I would, however, try to make it a little more interesting, exciting, or may be even funny.

Consistency is good, the book has a great structure. I like how each chapter has prerequisites and learner outcomes, this gives students a good idea of what to expect. Material in this book is covered in good detail.

The text can be easily divided into sub-sections, some of which can be omitted if needed. The chapter on regression is covered towards the end (chapter 14), but part of it can be covered sooner in the course.

The book contains well organized chapters that makes reading through easy and understandable. The order of chapters and sections is clear and logical.

The online version has many functions and is easy to navigate. This book also comes with a PDF version. There is no distortion of images or charts. The text is clean and clear, the examples provided contain appropriate format of data presentation.

No grammatical errors found.

The text uses simple and clear language, which is helpful for non-native speakers. I would include more culturally-relevant examples and case studies. Overall, good text.

In all, this book is a good learning experience. It contains tools and techniques that free and easy to use and also easy to modify for both, students and instructors. I very much appreciate this opportunity to use this textbook at no cost for our students.

Reviewed by Dabrina Dutcher, Assistant Professor, Bucknell University on 3/4/19

This is a reasonably thorough first-semester statistics book for most classes. It would have worked well for the general statistics courses I have taught in the past but is not as suitable for specialized introductory statistics courses for engineers or business applications. That is OK, they have separate texts for that! The only sections that feel somewhat light in terms of content are the confidence intervals and ANOVA sections. Given that these topics are often sort of crammed in at the end of many introductory classes, that might not be problematic for many instructors. It should also be pointed out that while there are a couple of chapters on probability, this book spends presents most formulas as "black boxes" rather than worry about the derivation or origin of the formulas. The probability sections do not include any significant combinatorics work, which is sometimes included at this level.

I did not find any errors in the formulas presented but I did not work many end-of-chapter problems to gauge the accuracy of their answers.

There isn't much changing in the introductory stats world, so I have no concerns about the book becoming outdated rapidly. The examples and problems still feel relevant and reasonably modern. My only concern is that the statistical tool most often referenced in the book are TI-83/84 type calculators. As students increasingly buy TI-89s or Inspires, these sections of the book may lose relevance faster than other parts.

Solid. The book gives a list of key terms and their definitions at the end of each chapter which is a nice feature. It also has a formula review at the end of each chapter. I can imagine that these are heavily used by students when studying! Formulas are easy to find and read and are well defined. There are a few areas that I might have found frustrating as a student. For example, the explanation for the difference in formulas for a population vs sample standard deviation is quite weak. Again, this is a book that focuses on sort of a "black-box" approach but you may have to supplement such sections for some students.

I did not detect any problems with inconsistent symbol use or switches in terminology.

Modularity rating: 3

This low rating should not be taken as an indicator of an issue with this book but would be true of virtually any statistics book. Different books still use different variable symbols even for basic calculated statistics. So trying to use a chapter of this book without some sort of symbol/variable cheat-sheet would likely be frustrating to the students.

However, I think it would be possible to skip some chapters or use the chapters in a different order without any loss of functionality.

This book uses a very standard order for the material. The chapter on regressions comes later than it does in some texts but it doesn't really matter since that chapter never seems to fit smoothly anywhere.

There are numerous end of chapter problems, some with answers, available in this book. I'm vacillating on whether these problems would be more useful if they were distributed after each relevant section or are better clumped at the end of the whole chapter. That might be a matter of individual preference.

I did not detect any problems.

I found no errors. However, there were several sections where the punctuation seemed non-ideal. This did not affect the over-all useability of the book though

I'm not sure how well this book would work internationally as many of the examples contain domestic (American) references. However, I did not see anything offensive or biased in the book.

Reviewed by Ilgin Sager, Assistant Professor, University of Missouri - St. Louis on 1/14/19

As the title implies, this is a brief introduction textbook. It covers the fundamental of the introductory statistics, however not a comprehensive text on the subject. A teacher can use this book as the sole text of an introductory statistics. The prose format of definitions and theorems make theoretical concepts accessible to non-math major students. The textbook covers all chapters required in this level course.

It is accurate; the subject matter in the examples to be up to date, is timeless and wouldn't need to be revised in future editions; there is no error except a few typographical errors. There are no logic errors or incorrect explanations.

This text will remain up to date for a long time since it has timeless examples and exercises, it wouldn't be outdated. The information is presented clearly with a simple way and the exercises are beneficial to follow the information.

The material is presented in a clear, concise manner. The text is easy readable for the first time statistics student.

The structure of the text is very consistent. Topics are presented with examples, followed by exercises. Problem sets are appropriate for the level of learner.

When the earlier matters need to be referenced, it is easy to find; no trouble reading the book and finding results, it has a consistent scheme. This book is set very well in sections.

The text presents the information in a logical order.

The learner can easily follow up the material; there is no interface problem.

There is no logic errors and incorrect explanations, a few typographical errors is just to be ignored.

Not applicable for this textbook.

Reviewed by Suhwon Lee, Associate Teaching Professor, University of Missouri on 6/19/18

This book is pretty comprehensive for being a brief introductory book. This book covers all necessary content areas for an introduction to Statistics course for non-math majors. The text book provides an effective index, plenty of exercises, review questions, and practice tests. It provides references and case studies. The glossary and index section is very helpful for students and can be used as a great resource.

Content appears to be accurate throughout. Being an introductory book, the book is unbiased and straight to the point. The terminology is standard.

The content in textbook is up to date. It will be very easy to update it or make changes at any point in time because of the well-structured contents in the textbook.

The author does a great job of explaining nearly every new term or concept. The book is easy to follow, clear and concise. The graphics are good to follow. The language in the book is easily understandable. I found most instructions in the book to be very detailed and clear for students to follow.

Overall consistency is good. It is consistent in terms of terminology and framework. The writing is straightforward and standardized throughout the text and it makes reading easier.

The authors do a great job of partitioning the text and labeling sections with appropriate headings. The table of contents is well organized and easily divisible into reading sections and it can be assigned at different points within the course.

Organization/Structure/Flow rating: 4

Overall, the topics are arranged in an order that follows natural progression in a statistics course with some exception. They are addressed logically and given adequate coverage.

The text is free of any issues. There are no navigation problems nor any display issues.

The text contains no grammatical errors.

The text is not culturally insensitive or offensive in any way most of time. Some examples might need to consider citing the sources or use differently to reflect current inclusive teaching strategies.

Overall, it's well-written and good recourse to be an introduction to statistical methods. Some materials may not need to be covered in an one-semester course. Various examples and quizzes can be a great recourse for instructor.

Reviewed by Jenna Kowalski, Mathematics Instructor, Anoka-Ramsey Community College on 3/27/18

The text includes the introductory statistics topics covered in a college-level semester course. An effective index and glossary are included, with functional hyperlinks. read more

The text includes the introductory statistics topics covered in a college-level semester course. An effective index and glossary are included, with functional hyperlinks.

Content Accuracy rating: 3

The content of this text is accurate and error-free, based on a random sampling of various pages throughout the text. Several examples included information without formal citation, leading the reader to potential bias and discrimination. These examples should be corrected to reflect current values of inclusive teaching.

The text contains relevant information that is current and will not become outdated in the near future. The statistical formulas and calculations have been used for centuries. The examples are direct applications of the formulas and accurately assess the conceptual knowledge of the reader.

The text is very clear and direct with the language used. The jargon does require a basic mathematical and/or statistical foundation to interpret, but this foundational requirement should be met with course prerequisites and placement testing. Graphs, tables, and visual displays are clearly labeled.

The terminology and framework of the text is consistent. The hyperlinks are working effectively, and the glossary is valuable. Each chapter contains modules that begin with prerequisite information and upcoming learning objectives for mastery.

The modules are clearly defined and can be used in conjunction with other modules, or individually to exemplify a choice topic. With the prerequisite information stated, the reader understands what prior mathematical understanding is required to successfully use the module.

The topics are presented well, but I recommend placing Sampling Distributions, Advanced Graphs, and Research Design ahead of Probability in the text. I think this rearranged version of the index would better align with current Introductory Statistics texts. The structure is very organized with the prerequisite information stated and upcoming learner outcomes highlighted. Each module is well-defined.

Adding an option of returning to the previous page would be of great value to the reader. While progressing through the text systematically, this is not an issue, but when the reader chooses to skip modules and read select pages then returning to the previous state of information is not easily accessible.

No grammatical errors were found while reviewing select pages of this text at random.

Cultural Relevance rating: 3

Several examples contained data that were not formally cited. These examples need to be corrected to reflect current inclusive teaching strategies. For example, one question stated that “while men are XX times more likely to commit murder than women, …” This data should be cited, otherwise the information can be interpreted as biased and offensive.

An included solutions manual for the exercises would be valuable to educators who choose to use this text.

Reviewed by Zaki Kuruppalil, Associate Professor, Ohio University on 2/1/18

This is a comprehensive book on statistical methods, its settings and most importantly the interpretation of the results. With the advent of computers and software’s, complex statistical analysis can be done very easily. But the challenge is the knowledge of how to set the case, setting parameters (for example confidence intervals) and knowing its implication on the interpretation of the results. If not done properly this could lead to deceptive inferences, inadvertently or purposely. This book does a great job in explaining the above using many examples and real world case studies. If you are looking for a book to learn and apply statistical methods, this is a great one. I think the author could consider revising the title of the book to reflect the above, as it is more than just an introduction to statistics, may be include the word such as practical guide.

The contents of the book seems accurate. Some plots and calculations were randomly selected and checked for accuracy.

The book topics are up to date and in my opinion, will not be obsolete in the near future. I think the smartest thing the author has done is, not tied the book with any particular software such as minitab or spss . No matter what the software is, standard deviation is calculated the same way as it is always. The only noticeable exception in this case was using the Java Applet for calculating Z values in page 261 and in page 416 an excerpt of SPSS analysis is provided for ANOVA calculations.

The contents and examples cited are clear and explained in simple language. Data analysis and presentation of the results including mathematical calculations, graphical explanation using charts, tables, figures etc are presented with clarity.

Terminology is consistant. Framework for each chapter seems consistent with each chapter beginning with a set of defined topics, and each of the topic divided into modules with each module having a set of learning objectives and prerequisite chapters.

The text book is divided into chapters with each chapter further divided into modules. Each of the modules have detailed learning objectives and prerequisite required. So you can extract a portion of the book and use it as a standalone to teach certain topics or as a learning guide to apply a relevant topic.

Presentation of the topics are well thought and are presented in a logical fashion as if it would be introduced to someone who is learning the contents. However, there are some issues with table of contents and page numbers, for example chapter 17 starts in page 597 not 598. Also some tables and figures does not have a number, for instance the graph shown in page 114 does not have a number. Also it would have been better if the chapter number was included in table and figure identification, for example Figure 4-5 . Also in some cases, for instance page 109, the figures and titles are in two different pages.

No major issues. Only suggestion would be, since each chapter has several modules, any means such as a header to trace back where you are currently, would certainly help.

Grammatical Errors rating: 4

Easy to read and phrased correctly in most cases. Minor grammatical errors such as missing prepositions etc. In some cases the author seems to have the habbit of using a period after the decimal. For instance page 464, 467 etc. For X = 1, Y' = (0.425)(1) + 0.785 = 1.21. For X = 2, Y' = (0.425)(2) + 0.785 = 1.64.

However it contains some statements (even though given as examples) that could be perceived as subjective, which the author could consider citing the sources. For example from page 11: Statistics include numerical facts and figures. For instance: • The largest earthquake measured 9.2 on the Richter scale. • Men are at least 10 times more likely than women to commit murder. • One in every 8 South Africans is HIV positive. • By the year 2020, there will be 15 people aged 65 and over for every new baby born.

Solutions for the exercises would be a great teaching resource to have

Reviewed by Randy Vander Wal, Professor, The Pennsylvania State University on 2/1/18

As a text for an introductory course, standard topics are covered. It was nice to see some topics such as power, sampling, research design and distribution free methods covered, as these are often omitted in abbreviated texts. Each module introduces the topic, has appropriate graphics, illustration or worked example(s) as appropriate and concluding with many exercises. An instructor’s manual is available by contacting the author. A comprehensive glossary provides definitions for all the major terms and concepts. The case studies give examples of practical applications of statistical analyses. Many of the case studies contain the actual raw data. To note is that the on-line e-book provides several calculators for the essential distributions and tests. These are provided in lieu of printed tables which are not included in the pdf. (Such tables are readily available on the web.)

The content is accurate and error free. Notation is standard and terminology is used accurately, as are the videos and verbal explanations therein. Online links work properly as do all the calculators. The text appears neutral and unbiased in subject and content.

The text achieves contemporary relevance by ending each section with a Statistical Literacy example, drawn from contemporary headlines and issues. Of course, the core topics are time proven. There is no obvious material that may become “dated”.

The text is very readable. While the pdf text may appear “sparse” by absence varied colored and inset boxes, pictures etc., the essential illustrations and descriptions are provided. Meanwhile for this same content the on-line version appears streamlined, uncluttered, enhancing the value of the active links. Moreover, the videos provide nice short segments of “active” instruction that are clear and concise. Despite being a mathematical text, the text is not overly burdened by formulas and numbers but rather has “readable feel”.

This terminology and symbol use are consistent throughout the text and with common use in the field. The pdf text and online version are also consistent by content, but with the online e-book offering much greater functionality.

The chapters and topics may be used in a selective manner. Certain chapters have no pre-requisite chapter and in all cases, those required are listed at the beginning of each module. It would be straightforward to select portions of the text and reorganize as needed. The online version is highly modular offering students both ease of navigation and selection of topics.

Chapter topics are arranged appropriately. In an introductory statistics course, there is a logical flow given the buildup to the normal distribution, concept of sampling distributions, confidence intervals, hypothesis testing, regression and additional parametric and non-parametric tests. The normal distribution is central to an introductory course. Necessary precursor topics are covered in this text, while its use in significance and hypothesis testing follow, and thereafter more advanced topics, including multi-factor ANOVA.

Each chapter is structured with several modules, each beginning with pre-requisite chapter(s), learning objectives and concluding with Statistical Literacy sections providing a self-check question addressing the core concept, along with answer, followed by an extensive problem set. The clear and concise learning objectives will be of benefit to students and the course instructor. No solutions or answer key is provided to students. An instructor’s manual is available by request.

The on-line interface works well. In fact, I was pleasantly surprised by its options and functionality. The pdf appears somewhat sparse by comparison to publisher texts, lacking pictures, colored boxes, etc. But the on-line version has many active links providing definitions and graphic illustrations for key terms and topics. This can really facilitate learning as making such “refreshers” integral to the new material. Most sections also have short videos that are professionally done, with narration and smooth graphics. In this way, the text is interactive and flexible, offering varied tools for students. To note is that the interactive e-book works for both IOS and OS X.

The text in pdf form appeared to free of grammatical errors, as did the on-line version, text, graphics and videos.

This text contains no culturally insensitive or offensive content. The focus of the text is on concepts and explanation.

The text would be a great resource for students. The full content would be ambitious for a 1-semester course, such use would be unlikely. The text is clearly geared towards students with no statistics background nor calculus. The text could be used in two styles of course. For 1st year students early chapters on graphs and distributions would be the starting point, omitting later chapters on Chi-square, transformations, distribution-free and size effect chapters. Alternatively, for upper level students the introductory chapters could be bypassed with the latter chapters then covered to completion.

This text adopts a descriptive style of presentation with topics well and fully explained, much like the “Dummy series”. For this, it may seem a bit “wordy”, but this can well serve students and notably it complements powerpoint slides that are generally sparse on written content. This text could be used as the primary text, for regular lectures, or as reference for a “flipped” class. The e-book videos are an enabling tool if this approach is adopted.

Reviewed by David jabon, Associate Professor, DePaul University on 8/15/17

This text covers all the standard topics in a semester long introductory course in statistics. It is particularly well indexed and very easy to navigate. There is comprehensive hyperlinked glossary. read more

This text covers all the standard topics in a semester long introductory course in statistics. It is particularly well indexed and very easy to navigate. There is comprehensive hyperlinked glossary.

The material is completely accurate. There are no errors. The terminology is standard with one exception: the book calls what most people call the interquartile range, the H-spread in a number of places. Ideally, the term "interquartile range" would be used in place of every reference to "H-spread." "Interquartile range" is simply a better, more descriptive term of the concept that it describes. It is also more commonly used nowadays.

This book came out a number of years ago, but the material is still up to date. Some more recent case studies have been added.

The writing is very clear. There are also videos for almost every section. The section on boxplots uses a lot of technical terms that I don't find are very helpful for my students (hinge, H-spread, upper adjacent value).

The text is internally consistent with one exception that I noted (the use of the synonymous words "H-spread" and "interquartile range").

The text book is brokenly into very short sections, almost to a fault. Each section is at most two pages long. However at the end of each of these sections there are a few multiple choice questions to test yourself. These questions are a very appealing feature of the text.

The organization, in particular the ordering of the topics, is rather standard with a few exceptions. Boxplots are introduced in Chapter II before the discussion of measures of center and dispersion. Most books introduce them as part of discussion of summaries of data using measure of center and dispersion. Some statistics instructors may not like the way the text lumps all of the sampling distributions in a single chapter (sampling distribution of mean, sampling distribution for the difference of means, sampling distribution of a proportion, sampling distribution of r). I have tried this approach, and I now like this approach. But it is a very challenging chapter for students.

The book's interface has no features that distracted me. Overall the text is very clean and spare, with no additional distracting visual elements.

The book contains no grammatical errors.

The book's cultural relevance comes out in the case studies. As of this writing there are 33 such case studies, and they cover a wide range of issues from health to racial, ethnic, and gender disparity.

Each chapter as a nice set of exercises with selected answers. The thirty three case studies are excellent and can be supplement with some other online case studies. An instructor's manual and PowerPoint slides can be obtained by emailing the author. There are direct links to online simulations within the text. This text is very high quality textbook in every way.

1. Introduction
2. Graphing Distributions
3. Summarizing Distributions
4. Describing Bivariate Data
5. Probability
6. Research Design
7. Normal Distributions
8. Advanced Graphs
9. Sampling Distributions
10. Estimation
11. Logic of Hypothesis Testing
12. Testing Means
14. Regression
15. Analysis of Variance
16. Transformations
17. Chi Square
18. Distribution-Free Tests
19. Effect Size
20. Case Studies
21. Glossary

Ancillary Material

Ancillary materials are available by contacting the author or publisher .

About the Book

Introduction to Statistics is a resource for learning and teaching introductory statistics. This work is in the public domain. Therefore, it can be copied and reproduced without limitation. However, we would appreciate a citation where possible. Please cite as: Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University. Instructor's manual, PowerPoint Slides, and additional questions are available.

About the Contributors

David Lane is an Associate Professor in the Departments of Psychology, Statistics, and Management at the Rice University. Lane is the principal developer of this resource although many others have made substantial contributions. This site was developed at Rice University, University of Houston-Clear Lake, and Tufts University.

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What Is Statistics?

Understanding statistics.

Descriptive & Inferential Statistics

Mean, Median, and Mode

Understanding statistical data.

Levels of Measurement
Sampling Techniques

Uses of Statistics

The bottom line.

Corporate Finance
Financial Analysis

Statistics in Math: Definition, Types, and Importance

Katrina Ávila Munichiello is an experienced editor, writer, fact-checker, and proofreader with more than fourteen years of experience working with print and online publications.

Statistics is a branch of applied mathematics that involves the collection, description, analysis, and inference of conclusions from quantitative data. The mathematical theories behind statistics rely heavily on differential and integral calculus, linear algebra, and probability theory.

People who do statistics are referred to as statisticians. They're particularly concerned with determining how to draw reliable conclusions about large groups and general events from the behavior and other observable characteristics of small samples. These small samples represent a portion of the large group or a limited number of instances of a general phenomenon.

Key Takeaways

Statistics is the study and manipulation of data, including ways to gather, review, analyze, and draw conclusions from data.
The two major areas of statistics are descriptive and inferential statistics.
Statistics can be communicated at different levels ranging from non-numerical descriptor (nominal-level) to numerical in reference to a zero-point (ratio-level).
Several sampling techniques can be used to compile statistical data, including simple random, systematic, stratified, or cluster sampling.
Statistics are present in almost every department of every company and are an integral part of investing as well.

Investopedia / Dennis Madamba

Statistics are used in virtually all scientific disciplines, such as the physical and social sciences as well as in business, the humanities, government, and manufacturing. Statistics is fundamentally a branch of applied mathematics that developed from the application of mathematical tools, including calculus and linear algebra to probability theory.

In practice, statistics is the idea that we can learn about the properties of large sets of objects or events (a population ) by studying the characteristics of a smaller number of similar objects or events (a sample ). Gathering comprehensive data about an entire population is too costly, difficult, or impossible in many cases so statistics start with a sample that can be conveniently or affordably observed.

Statisticians measure and gather data about the individuals or elements of a sample, then they analyze this data to generate descriptive statistics. They can then use these observed characteristics of the sample data, which are properly called "statistics," to make inferences or educated guesses about the unmeasured (or unmeasured) characteristics of the broader population, known as the parameters.

Statistics informally dates back centuries. An early record of correspondence between French mathematicians Pierre de Fermat and Blaise Pascal in 1654 is often cited as an early example of statistical probability analysis.

Descriptive and Inferential Statistics

The two major areas of statistics are known as descriptive statistics , which describes the properties of sample and population data, and inferential statistics, which uses those properties to test hypotheses and draw conclusions. Descriptive statistics include mean (average), variance, skewness , and kurtosis . Inferential statistics include linear regression analysis, analysis of variance (ANOVA), logit/Probit models, and null hypothesis testing.

Descriptive Statistics

Descriptive statistics mostly focus on the central tendency, variability, and distribution of sample data. Central tendency means the estimate of the characteristics, a typical element of a sample or population. It includes descriptive statistics such as mean , median , and mode .

Variability refers to a set of statistics that show how much difference there is among the elements of a sample or population along the characteristics measured. It includes metrics such as range, variance , and standard deviation .

The distribution refers to the overall "shape" of the data, which can be depicted on a chart such as a histogram or a dot plot, and includes properties such as the probability distribution function, skewness, and kurtosis. Descriptive statistics can also describe differences between observed characteristics of the elements of a data set. They can help us understand the collective properties of the elements of a data sample and form the basis for testing hypotheses and making predictions using inferential statistics.

Inferential Statistics

Inferential statistics are tools that statisticians use to draw conclusions about the characteristics of a population, drawn from the characteristics of a sample, and to determine how certain they can be of the reliability of those conclusions. Based on the sample size and distribution, statisticians can calculate the probability that statistics, which measure the central tendency, variability, distribution, and relationships between characteristics within a data sample, provide an accurate picture of the corresponding parameters of the whole population from which the sample is drawn.

Inferential statistics are used to make generalizations about large groups, such as estimating average demand for a product by surveying a sample of consumers' buying habits or attempting to predict future events. This might mean projecting the future return of a security or asset class based on returns in a sample period.

Regression analysis is a widely used technique of statistical inference used to determine the strength and nature of the relationship (the correlation) between a dependent variable and one or more explanatory (independent) variables. The output of a regression model is often analyzed for statistical significance, which refers to the claim that a result from findings generated by testing or experimentation is not likely to have occurred randomly or by chance. It's likely to be attributable to a specific cause elucidated by the data.

Having statistical significance is important for academic disciplines or practitioners that rely heavily on analyzing data and research.

The terms mean, median and mode fall under the umbrella of central tendency. They describe an element that’s typical in a given sample group. You can find the mean descriptor by adding the numbers in the group and dividing the result by the number of data set observations.

The middle number in the set is the median. Half of all included numbers are higher than the median and half are lesser. The median home value in a neighborhood would be $350,000 if five homes were located there and valued at $500,000, $400,000, $350,000, $325,000, and $300,000. Two values are higher and two are lower.

Mode identifies the number that falls between the highest and lowest values. It appears most frequently in the data set.

The root of statistics is driven by variables. A variable is a data set that can be counted that marks a characteristic or attribute of an item. For example, a car can have variables such as make, model, year, mileage, color, or condition. By combining the variables across a set of data, such as the colors of all cars in a given parking lot, statistics allows us to better understand trends and outcomes.

There are two main types of variables. First, qualitative variables are specific attributes that are often non-numeric. Many of the examples given in the car example are qualitative. Other examples of qualitative variables in statistics are gender, eye color, or city of birth. Qualitative data is most often used to determine what percentage of an outcome occurs for any given qualitative variable. Qualitative analysis often does not rely on numbers. For example, trying to determine what percentage of women own a business analyzes qualitative data.

The second type of variable in statistics is quantitative variables. Quantitative variables are studied numerically and only have weight when they're about a non-numerical descriptor. Similar to quantitative analysis, this information is rooted in numbers. In the car example above, the mileage driven is a quantitative variable, but the number 60,000 holds no value unless it is understood that is the total number of miles driven.

Quantitative variables can be further broken into two categories. First, discrete variables have limitations in statistics and infer that there are gaps between potential discrete variable values. The number of points scored in a football game is a discrete variable because :

There can be no decimals, and
It is impossible for a team to score only one point

Statistics also makes use of continuous quantitative variables. These values run along a scale. Discrete values have limitations, but continuous variables are often measured into decimals. Any value within possible limits can be obtained when measuring the height of the football players, and the heights can be measured down to 1/16th of an inch, if not further.

Statisticians can hold various titles and positions within a company. The average total compensation for a statistician with one to three years experience as of December 2021 was $84,958 as of May 2022. This increased to $111,846 with 15 years experience.

Statistical Levels of Measurement

There are several resulting levels of measurement after analyzing variables and outcomes. Statistics can quantify outcomes in four ways.

Nominal-level Measurement

There's no numerical or quantitative value, and qualities are not ranked. Nominal-level measurements are instead simply labels or categories assigned to other variables. It's easiest to think of nominal-level measurements as non-numerical facts about a variable.

Example: The name of the President elected in 2020 was Joseph Robinette Biden, Jr.

Ordinal-level Measurement

Outcomes can be arranged in an order, but all data values have the same value or weight. Although numerical, ordinal-level measurements can't be subtracted against each other in statistics because only the position of the data point matters. Ordinal levels are often incorporated into nonparametric statistics and compared against the total variable group.

Example: American Fred Kerley was the 2nd fastest man at the 2020 Tokyo Olympics based on 100-meter sprint times.

Interval-level Measurement

Outcomes can be arranged in order but differences between data values may now have meaning. Two data points are often used to compare the passing of time or changing conditions within a data set. There is often no "starting point" for the range of data values, and calendar dates or temperatures may not have a meaningful intrinsic zero value.

Example: Inflation hit 8.6% in May 2022. The last time inflation was this high was in December 1981 .

Ratio-level Measurement

Outcomes can be arranged in order and differences between data values now have meaning. But there's a starting point or "zero value" that can be used to further provide value to a statistical value. The ratio between data values has meaning, including its distance away from zero.

Example: The lowest meteorological temperature recorded was -128.6 degrees Fahrenheit in Antarctica.

Statistics Sampling Techniques

It would often not be possible to gather data from every data point within a population to gather statistical information. Statistics relies instead on different sampling techniques to create a representative subset of the population that's easier to analyze. In statistics, there are several primary types of sampling in statistics.

Simple Random Sampling

Simple random sampling calls for every member within the population to have an equal chance of being selected for analysis. The entire population is used as the basis for sampling, and any random generator based on chance can select the sample items. For example, 100 individuals are lined up and 10 are chosen at random.

Systemic Sampling

Systematic sampling calls for a random sample as well, but its technique is slightly modified to make it easier to conduct. A single random number is generated and individuals are then selected at a specified regular interval until the sample size is complete. For example, 100 individuals are lined up and numbered. The 7th individual is selected for the sample followed by every subsequent 9th individual until 10 sample items have been selected.

Stratified Sampling

Stratified sampling calls for more control over your sample. The population is divided into subgroups based on similar characteristics. Then you calculate how many people from each subgroup would represent the entire population. For example, 100 individuals are grouped by gender and race. Then a sample from each subgroup is taken in proportion to how representative that subgroup is of the population.

Cluster Sampling

Cluster sampling calls for subgroups as well, but each subgroup should be representative of the population. The entire subgroup is randomly selected instead of randomly selecting individuals within a subgroup.

Not sure which Major League Baseball player should have won Most Valuable Player last year? Statistics, often used to determine value, is often cited when the award for best player is awarded. Statistics can include batting average, number of home runs hit, and stolen bases.

Statistics is prominent in finance, investing, business, and in the world. Much of the information you see and the data you're given is derived from statistics, which are used in all facets of a business.

Statistics in investing include average trading volume, 52-week low, 52-week high, beta, and correlation between asset classes or securities.
Statistics in economics include GDP, unemployment, consumer pricing, inflation, and other economic growth metrics.
Statistics in marketing include conversion rates, click-through rates, search quantities, and social media metrics.
Statistics in accounting include liquidity, solvency, and profitability metrics across time.
Statistics in information technology include bandwidth, network capabilities, and hardware logistics.
Statistics in human resources include employee turnover, employee satisfaction, and average compensation relative to the market.

Why Is Statistics Important?

Statistics provide the information to educate how things work. They're used to conduct research, evaluate outcomes, develop critical thinking, and make informed decisions. Statistics can be used to inquire about almost any field of study to investigate why things happen, when they occur, and whether reoccurrence is predictable.

What's the Difference Between Descriptive and Inferential Statistics?

Descriptive statistics are used to describe or summarize the characteristics of a sample or data set, such as a variable's mean, standard deviation, or frequency. Inferential statistics employs any number of techniques to relate variables in a data set to one another. An example would be using correlation or regression analysis. These can then be used to estimate forecasts or infer causality.

Who Uses Statistics?

Statistics are used widely across an array of applications and professions. Statistics are done whenever data are collected and analyzed. This can range from government agencies to academic research to analyzing investments.

How Are Statistics Used in Economics and Finance?

Economists collect and look at all sorts of data ranging from consumer spending to housing starts to inflation to GDP growth. In finance, analysts and investors collect data about companies, industries, sentiment, and market data on price and volume. The use of inferential statistics in these fields is known as econometrics. Several important financial models, from CAPM to Modern Portfolio Theory (MPT) and the Black-Scholes options pricing model, rely on statistical inference.

Statistics is the practice of analyzing pieces of information that might seem conflicting or unrelated at first glance and on the surface. It can lead to a solid career as a statistician, but it can also be a handy metric in everyday life, perhaps when you’re analyzing the odds that your favorite team will win the Super Bowl before you place a bet, gauging the viability of an investment, or determining whether you’re being comparatively overcharged for a product or service.

Britannica. " Probability and Statistics ."

Coursera. " How Much Do Statisticians Make? 2023 Statistician Salary Guide ."

Olympics. " Tokyo 2020, Athletics, Men's 100m ."

Bureau of Labor Statistics. " Consumer Price Index ."

Arizona State University. " World Meteorological Organization's World Weather & Climate Extremes Archive ."

Baseball Reference. " MLB Most Valuable Player MVP Award Winners ."

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Types of statistical analysis, importance of statistical analysis, benefits of statistical analysis, statistical analysis process, statistical analysis methods, statistical analysis software, statistical analysis examples, career in statistical analysis, choose the right program, become proficient in statistics today, what is statistical analysis types, methods and examples.

Statistical analysis is the process of collecting and analyzing data in order to discern patterns and trends. It is a method for removing bias from evaluating data by employing numerical analysis. This technique is useful for collecting the interpretations of research, developing statistical models, and planning surveys and studies.

Statistical analysis is a scientific tool in AI and ML that helps collect and analyze large amounts of data to identify common patterns and trends to convert them into meaningful information. In simple words, statistical analysis is a data analysis tool that helps draw meaningful conclusions from raw and unstructured data.

The conclusions are drawn using statistical analysis facilitating decision-making and helping businesses make future predictions on the basis of past trends. It can be defined as a science of collecting and analyzing data to identify trends and patterns and presenting them. Statistical analysis involves working with numbers and is used by businesses and other institutions to make use of data to derive meaningful information.

Given below are the 6 types of statistical analysis:

Descriptive Analysis

Descriptive statistical analysis involves collecting, interpreting, analyzing, and summarizing data to present them in the form of charts, graphs, and tables. Rather than drawing conclusions, it simply makes the complex data easy to read and understand.

Inferential Analysis

The inferential statistical analysis focuses on drawing meaningful conclusions on the basis of the data analyzed. It studies the relationship between different variables or makes predictions for the whole population.

Predictive Analysis

Predictive statistical analysis is a type of statistical analysis that analyzes data to derive past trends and predict future events on the basis of them. It uses machine learning algorithms, data mining , data modelling , and artificial intelligence to conduct the statistical analysis of data.

Prescriptive Analysis

The prescriptive analysis conducts the analysis of data and prescribes the best course of action based on the results. It is a type of statistical analysis that helps you make an informed decision.

Exploratory Data Analysis

Exploratory analysis is similar to inferential analysis, but the difference is that it involves exploring the unknown data associations. It analyzes the potential relationships within the data.

Causal Analysis

The causal statistical analysis focuses on determining the cause and effect relationship between different variables within the raw data. In simple words, it determines why something happens and its effect on other variables. This methodology can be used by businesses to determine the reason for failure.

Statistical analysis eliminates unnecessary information and catalogs important data in an uncomplicated manner, making the monumental work of organizing inputs appear so serene. Once the data has been collected, statistical analysis may be utilized for a variety of purposes. Some of them are listed below:

The statistical analysis aids in summarizing enormous amounts of data into clearly digestible chunks.
The statistical analysis aids in the effective design of laboratory, field, and survey investigations.
Statistical analysis may help with solid and efficient planning in any subject of study.
Statistical analysis aid in establishing broad generalizations and forecasting how much of something will occur under particular conditions.
Statistical methods, which are effective tools for interpreting numerical data, are applied in practically every field of study. Statistical approaches have been created and are increasingly applied in physical and biological sciences, such as genetics.
Statistical approaches are used in the job of a businessman, a manufacturer, and a researcher. Statistics departments can be found in banks, insurance businesses, and government agencies.
A modern administrator, whether in the public or commercial sector, relies on statistical data to make correct decisions.
Politicians can utilize statistics to support and validate their claims while also explaining the issues they address.

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Statistical analysis can be called a boon to mankind and has many benefits for both individuals and organizations. Given below are some of the reasons why you should consider investing in statistical analysis:

It can help you determine the monthly, quarterly, yearly figures of sales profits, and costs making it easier to make your decisions.
It can help you make informed and correct decisions.
It can help you identify the problem or cause of the failure and make corrections. For example, it can identify the reason for an increase in total costs and help you cut the wasteful expenses.
It can help you conduct market analysis and make an effective marketing and sales strategy.
It helps improve the efficiency of different processes.

Given below are the 5 steps to conduct a statistical analysis that you should follow:

Step 1: Identify and describe the nature of the data that you are supposed to analyze.
Step 2: The next step is to establish a relation between the data analyzed and the sample population to which the data belongs.
Step 3: The third step is to create a model that clearly presents and summarizes the relationship between the population and the data.
Step 4: Prove if the model is valid or not.
Step 5: Use predictive analysis to predict future trends and events likely to happen.

Although there are various methods used to perform data analysis, given below are the 5 most used and popular methods of statistical analysis:

Mean or average mean is one of the most popular methods of statistical analysis. Mean determines the overall trend of the data and is very simple to calculate. Mean is calculated by summing the numbers in the data set together and then dividing it by the number of data points. Despite the ease of calculation and its benefits, it is not advisable to resort to mean as the only statistical indicator as it can result in inaccurate decision making.

Standard Deviation

Standard deviation is another very widely used statistical tool or method. It analyzes the deviation of different data points from the mean of the entire data set. It determines how data of the data set is spread around the mean. You can use it to decide whether the research outcomes can be generalized or not.

Regression is a statistical tool that helps determine the cause and effect relationship between the variables. It determines the relationship between a dependent and an independent variable. It is generally used to predict future trends and events.

Hypothesis Testing

Hypothesis testing can be used to test the validity or trueness of a conclusion or argument against a data set. The hypothesis is an assumption made at the beginning of the research and can hold or be false based on the analysis results.

Sample Size Determination

Sample size determination or data sampling is a technique used to derive a sample from the entire population, which is representative of the population. This method is used when the size of the population is very large. You can choose from among the various data sampling techniques such as snowball sampling, convenience sampling, and random sampling.

Everyone can't perform very complex statistical calculations with accuracy making statistical analysis a time-consuming and costly process. Statistical software has become a very important tool for companies to perform their data analysis. The software uses Artificial Intelligence and Machine Learning to perform complex calculations, identify trends and patterns, and create charts, graphs, and tables accurately within minutes.

Look at the standard deviation sample calculation given below to understand more about statistical analysis.

The weights of 5 pizza bases in cms are as follows:

Calculation of Mean = (9+2+5+4+12)/5 = 32/5 = 6.4

Calculation of mean of squared mean deviation = (6.76+19.36+1.96+5.76+31.36)/5 = 13.04

Sample Variance = 13.04

Standard deviation = √13.04 = 3.611

A Statistical Analyst's career path is determined by the industry in which they work. Anyone interested in becoming a Data Analyst may usually enter the profession and qualify for entry-level Data Analyst positions right out of high school or a certificate program — potentially with a Bachelor's degree in statistics, computer science, or mathematics. Some people go into data analysis from a similar sector such as business, economics, or even the social sciences, usually by updating their skills mid-career with a statistical analytics course.

Statistical Analyst is also a great way to get started in the normally more complex area of data science. A Data Scientist is generally a more senior role than a Data Analyst since it is more strategic in nature and necessitates a more highly developed set of technical abilities, such as knowledge of multiple statistical tools, programming languages, and predictive analytics models.

Aspiring Data Scientists and Statistical Analysts generally begin their careers by learning a programming language such as R or SQL. Following that, they must learn how to create databases, do basic analysis, and make visuals using applications such as Tableau. However, not every Statistical Analyst will need to know how to do all of these things, but if you want to advance in your profession, you should be able to do them all.

Based on your industry and the sort of work you do, you may opt to study Python or R, become an expert at data cleaning, or focus on developing complicated statistical models.

You could also learn a little bit of everything, which might help you take on a leadership role and advance to the position of Senior Data Analyst. A Senior Statistical Analyst with vast and deep knowledge might take on a leadership role leading a team of other Statistical Analysts. Statistical Analysts with extra skill training may be able to advance to Data Scientists or other more senior data analytics positions.

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Hope this article assisted you in understanding the importance of statistical analysis in every sphere of life. Artificial Intelligence (AI) can help you perform statistical analysis and data analysis very effectively and efficiently.

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Indian J Crit Care Med
v.25(Suppl 2); 2021 May

An Introduction to Statistics: Choosing the Correct Statistical Test

Priya ranganathan.

1 Department of Anaesthesiology, Critical Care and Pain, Tata Memorial Centre, Homi Bhabha National Institute, Mumbai, Maharashtra, India

The choice of statistical test used for analysis of data from a research study is crucial in interpreting the results of the study. This article gives an overview of the various factors that determine the selection of a statistical test and lists some statistical testsused in common practice.

How to cite this article: Ranganathan P. An Introduction to Statistics: Choosing the Correct Statistical Test. Indian J Crit Care Med 2021;25(Suppl 2):S184–S186.

In a previous article in this series, we looked at different types of data and ways to summarise them. 1 At the end of the research study, statistical analyses are performed to test the hypothesis and either prove or disprove it. The choice of statistical test needs to be carefully performed since the use of incorrect tests could lead to misleading conclusions. Some key questions help us to decide the type of statistical test to be used for analysis of study data. 2

W hat is the R esearch H ypothesis ?

Sometimes, a study may just describe the characteristics of the sample, e.g., a prevalence study. Here, the statistical analysis involves only descriptive statistics . For example, Sridharan et al. aimed to analyze the clinical profile, species distribution, and susceptibility pattern of patients with invasive candidiasis. 3 They used descriptive statistics to express the characteristics of their study sample, including mean (and standard deviation) for normally distributed data, median (with interquartile range) for skewed data, and percentages for categorical data.

Studies may be conducted to test a hypothesis and derive inferences from the sample results to the population. This is known as inferential statistics . The goal of inferential statistics may be to assess differences between groups (comparison), establish an association between two variables (correlation), predict one variable from another (regression), or look for agreement between measurements (agreement). Studies may also look at time to a particular event, analyzed using survival analysis.

A re the C omparisons M atched (P aired ) or U nmatched (U npaired )?

Observations made on the same individual (before–after or comparing two sides of the body) are usually matched or paired . Comparisons made between individuals are usually unpaired or unmatched . Data are considered paired if the values in one set of data are likely to be influenced by the other set (as can happen in before and after readings from the same individual). Examples of paired data include serial measurements of procalcitonin in critically ill patients or comparison of pain relief during sequential administration of different analgesics in a patient with osteoarthritis.

W hat are the T ype of D ata B eing M easured ?

The test chosen to analyze data will depend on whether the data are categorical (and whether nominal or ordinal) or numerical (and whether skewed or normally distributed). Tests used to analyze normally distributed data are known as parametric tests and have a nonparametric counterpart that is used for data, which is distribution-free. 4 Parametric tests assume that the sample data are normally distributed and have the same characteristics as the population; nonparametric tests make no such assumptions. Parametric tests are more powerful and have a greater ability to pick up differences between groups (where they exist); in contrast, nonparametric tests are less efficient at identifying significant differences. Time-to-event data requires a special type of analysis, known as survival analysis.

H ow M any M easurements are B eing C ompared ?

The choice of the test differs depending on whether two or more than two measurements are being compared. This includes more than two groups (unmatched data) or more than two measurements in a group (matched data).

T ests for C omparison

( Table 1 lists the tests commonly used for comparing unpaired data, depending on the number of groups and type of data. As an example, Megahed and colleagues evaluated the role of early bronchoscopy in mechanically ventilated patients with aspiration pneumonitis. 5 Patients were randomized to receive either early bronchoscopy or conventional treatment. Between groups, comparisons were made using the unpaired t test for normally distributed continuous variables, the Mann–Whitney U -test for non-normal continuous variables, and the chi-square test for categorical variables. Chowhan et al. compared the efficacy of left ventricular outflow tract velocity time integral (LVOTVTI) and carotid artery velocity time integral (CAVTI) as predictors of fluid responsiveness in patients with sepsis and septic shock. 6 Patients were divided into three groups— sepsis, septic shock, and controls. Since there were three groups, comparisons of numerical variables were done using analysis of variance (for normally distributed data) or Kruskal–Wallis test (for skewed data).

Tests for comparison of unpaired data

A common error is to use multiple unpaired t -tests for comparing more than two groups; i.e., for a study with three treatment groups A, B, and C, it would be incorrect to run unpaired t -tests for group A vs B, B vs C, and C vs A. The correct technique of analysis is to run ANOVA and use post hoc tests (if ANOVA yields a significant result) to determine which group is different from the others.

( Table 2 lists the tests commonly used for comparing paired data, depending on the number of groups and type of data. As discussed above, it would be incorrect to use multiple paired t -tests to compare more than two measurements within a group. In the study by Chowhan, each parameter (LVOTVTI and CAVTI) was measured in the supine position and following passive leg raise. These represented paired readings from the same individual and comparison of prereading and postreading was performed using the paired t -test. 6 Verma et al. evaluated the role of physiotherapy on oxygen requirements and physiological parameters in patients with COVID-19. 7 Each patient had pretreatment and post-treatment data for heart rate and oxygen supplementation recorded on day 1 and day 14. Since data did not follow a normal distribution, they used Wilcoxon's matched pair test to compare the prevalues and postvalues of heart rate (numerical variable). McNemar's test was used to compare the presupplemental and postsupplemental oxygen status expressed as dichotomous data in terms of yes/no. In the study by Megahed, patients had various parameters such as sepsis-related organ failure assessment score, lung injury score, and clinical pulmonary infection score (CPIS) measured at baseline, on day 3 and day 7. 5 Within groups, comparisons were made using repeated measures ANOVA for normally distributed data and Friedman's test for skewed data.

Tests for comparison of paired data

T ests for A ssociation between V ariables

( Table 3 lists the tests used to determine the association between variables. Correlation determines the strength of the relationship between two variables; regression allows the prediction of one variable from another. Tyagi examined the correlation between ETCO 2 and PaCO 2 in patients with chronic obstructive pulmonary disease with acute exacerbation, who were mechanically ventilated. 8 Since these were normally distributed variables, the linear correlation between ETCO 2 and PaCO 2 was determined by Pearson's correlation coefficient. Parajuli et al. compared the acute physiology and chronic health evaluation II (APACHE II) and acute physiology and chronic health evaluation IV (APACHE IV) scores to predict intensive care unit mortality, both of which were ordinal data. Correlation between APACHE II and APACHE IV score was tested using Spearman's coefficient. 9 A study by Roshan et al. identified risk factors for the development of aspiration pneumonia following rapid sequence intubation. 10 Since the outcome was categorical binary data (aspiration pneumonia— yes/no), they performed a bivariate analysis to derive unadjusted odds ratios, followed by a multivariable logistic regression analysis to calculate adjusted odds ratios for risk factors associated with aspiration pneumonia.

Tests for assessing the association between variables

T ests for A greement between M easurements

( Table 4 outlines the tests used for assessing agreement between measurements. Gunalan evaluated concordance between the National Healthcare Safety Network surveillance criteria and CPIS for the diagnosis of ventilator-associated pneumonia. 11 Since both the scores are examples of ordinal data, Kappa statistics were calculated to assess the concordance between the two methods. In the previously quoted study by Tyagi, the agreement between ETCO 2 and PaCO 2 (both numerical variables) was represented using the Bland–Altman method. 8

Tests for assessing agreement between measurements

T ests for T ime-to -E vent D ata (S urvival A nalysis )

Time-to-event data represent a unique type of data where some participants have not experienced the outcome of interest at the time of analysis. Such participants are considered to be “censored” but are allowed to contribute to the analysis for the period of their follow-up. A detailed discussion on the analysis of time-to-event data is beyond the scope of this article. For analyzing time-to-event data, we use survival analysis (with the Kaplan–Meier method) and compare groups using the log-rank test. The risk of experiencing the event is expressed as a hazard ratio. Cox proportional hazards regression model is used to identify risk factors that are significantly associated with the event.

Hasanzadeh evaluated the impact of zinc supplementation on the development of ventilator-associated pneumonia (VAP) in adult mechanically ventilated trauma patients. 12 Survival analysis (Kaplan–Meier technique) was used to calculate the median time to development of VAP after ICU admission. The Cox proportional hazards regression model was used to calculate hazard ratios to identify factors significantly associated with the development of VAP.

The choice of statistical test used to analyze research data depends on the study hypothesis, the type of data, the number of measurements, and whether the data are paired or unpaired. Reviews of articles published in medical specialties such as family medicine, cytopathology, and pain have found several errors related to the use of descriptive and inferential statistics. 12 – 15 The statistical technique needs to be carefully chosen and specified in the protocol prior to commencement of the study, to ensure that the conclusions of the study are valid. This article has outlined the principles for selecting a statistical test, along with a list of tests used commonly. Researchers should seek help from statisticians while writing the research study protocol, to formulate the plan for statistical analysis.

Priya Ranganathan https://orcid.org/0000-0003-1004-5264

Source of support: Nil

Conflict of interest: None

R eferences

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How to Find the Mean | Definition, Examples & Calculator

How to Find the Mean | Definition, Examples & Calculator

Published on October 9, 2020 by Pritha Bhandari . Revised on June 21, 2023.

The mean (aka the arithmetic mean, different from the geometric mean ) of a dataset is the sum of all values divided by the total number of values. It’s the most commonly used measure of central tendency and is often referred to as the “average.”

Arithmetic mean calculator, mean formulas for populations and samples, steps for calculating the mean by hand, outlier effect on the mean, when can you use the mean, median or mode, other interesting articles, frequently asked questions about the mean, here's why students love scribbr's proofreading services.

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In research, you often collect data from samples and perform inferential statistics to understand the population they came from.

The formulas for the sample mean and the population mean only differ in mathematical notation. Population attributes use capital letters while sample attributes use lowercase letters.

Population mean

The population mean can also be denoted as μ.

Sample mean

The sample mean is also referred to as M .

There are two steps for calculating the mean:

Add up all the values in the data set.
Divide this number by the number of values.

We’ll walk through these steps with a sample data set.

Let’s say you want to find the average amount people spend on a restaurant meal in your neighborhood. You ask a sample of 8 neighbors how much they spent the last time they went out for dinner, and find the mean cost.

Step 1: Find the sum of the values by adding them all up

Because we’re working with a sample, we use the sample formula.

Step 2: Divide the sum by the number of values

In the formula, n is the number of values in your data set. Our data set has 8 values.

The mean tells us that in our sample, participants spent an average of 50 USD on their restaurant bill.

Outliers are extreme values that differ from most values in the data set. Because all values are used in the calculation of the mean, an outlier can have a dramatic effect on the mean by pulling the mean away from the majority of the values.

Let’s see what happens to the mean when we add an outlier to our data set.

As we can see, adding just one outlier to our data set raised the mean by 20 USD. In this case, a different measure of central tendency, like the median , would be more appropriate.

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The mean is the most widely used measure of central tendency because it uses all values in its calculation. The best measure of central tendency depends on your type of variable and the shape of your distribution.

Type of variable

The mean can only be calculated for quantitative variables (e.g., height), and it can’t be found for categorical variables (e.g., gender).

In categorical variables, data is placed into groupings without exact numerical values, so the mean cannot be calculated. For categorical variables, the mode is the best measure of central tendency because it tells you the most common characteristic or popular choice for your sample.

But for continuous or discrete variables, you have exact numerical values. With these, you can easily calculate the mean or median.

Distribution shape

The mean is best for data sets with normal distributions . In a normal distribution, data is symmetrically distributed with no skew . Most values cluster around a central region, with values tapering off as they go further away from the center.

The mean, mode and median are exactly the same in a normal distribution.

The mean, median and mode in a normal distribution

In skewed distributions, more values fall on one side of the center than the other, and the mean, median and mode all differ from each other. One side has a more spread out and longer tail with fewer scores at one end than the other.

The mean, median, and mode in a positively skewed distribution

For skewed distributions and distributions with outliers, the mean is easily influenced by extreme values and may not accurately represent the central tendency. The median is a better measure for these distributions as it takes a value from the middle of the distribution.

Alternatively, you can systematically review and remove outliers from your dataset in the data cleansing process.

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

Statistical power
Pearson correlation
Degrees of freedom
Statistical significance

Methodology

Cluster sampling
Stratified sampling
Focus group
Systematic review
Ethnography
Double-Barreled Question

Research bias

Implicit bias
Publication bias
Cognitive bias
Placebo effect
Pygmalion effect
Hindsight bias
Overconfidence bias

You can find the mean , or average, of a data set in two simple steps:

Find the sum of the values by adding them all up.
Divide the sum by the number of values in the data set.

This method is the same whether you are dealing with sample or population data or positive or negative numbers.

The arithmetic mean is the most commonly used mean. It’s often simply called the mean or the average. But there are some other types of means you can calculate depending on your research purposes:

Weighted mean: some values contribute more to the mean than others.
Geometric mean : values are multiplied rather than summed up.
Harmonic mean: reciprocals of values are used instead of the values themselves.

Measures of central tendency help you find the middle, or the average, of a data set.

The 3 most common measures of central tendency are the mean, median and mode.

The mode is the most frequent value.
The median is the middle number in an ordered data set.
The mean is the sum of all values divided by the total number of values.

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.

For data from skewed distributions, the median is better than the mean because it isn’t influenced by extremely large values.

The mode is the only measure you can use for nominal or categorical data that can’t be ordered.

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