z score for hypothesis testing calculator

Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

• By IIT Bombay Learn HTML Java Tutorial Django Tutorial PHP Tutorial
• On-Demand (Videos) Core Python Certification Complete Python Certification Course Online Create Own Cryptocurrency C Programming Online
• Live Courses (1:1 Live Sesions) Coding For Kids Online C Programming Complete Python For Kids Online Essential Python For Kids Online Complete C++ Programming For Kids
• For Kids Coding For Kids Online C Programming Essential Python For Kids Online Complete Python For Kids Online C Programming For Kids Complete C++ Programming For Kids

EVM - Cryptocurrency

Cryptocurrency

Z-test Calculator

Streamline your statistical calculations with the z-test calculator by newtum.

Discover the power of our Z-test Calculator, expertly developed by Newtum. This essential tool simplifies statistical testing, making it accessible for professionals and students alike. Unveil the mysteries of z-scores and enhance your data analysis skills.

Understanding the Statistical Significance Tool

The Z-test Calculator is a statistical tool designed to determine if there is a significant difference between sample and population means. It's ideal for researchers and students engaged in hypothesis testing and data analysis.

Z-test Calculation Formula Explained

Learn the critical formula used in the Z-test Calculator and its significance in statistical analysis. Understanding this formula is vital for accurate hypothesis testing and research conclusions.

• Define the null and alternative hypotheses.
• Calculate the sample mean (x̄) and population mean (μ).
• Determine the standard deviation (σ) and the sample size (n).
• Compute the standard error of the mean (σ/√n).
• Use the formula Z = (x̄ - μ) / (σ/√n) to calculate the Z-score.

Step-by-Step Guide to Using the Z-test Calculator

Our Z-test Calculator is incredibly user-friendly. Just follow the simple instructions below, and you'll be on your way to obtaining quick and accurate z-score results.

• Enter the sample mean into the designated field.
• Input the population mean.
• Provide the standard deviation of the population.
• Specify the sample size.
• Click 'Calculate' to get your Z-score and p-value.

Discover the Superior Features of Our Z-test Calculator

• User-Friendly Interface: Navigate with ease.
• Instant Results: Get your answers without delays.
• Data Security: Your data remains on your device.
• Accessibility Across Devices: Use on any modern device.
• No Installation Needed: Access directly from your browser.
• Examples for Clarity: Understand with practical scenarios.
• Transparent Process: No hidden steps or calculations.
• Educational Resource: Enhance your statistical knowledge.
• Responsive Customer Support: We're here to assist you.
• Regular Updates: Benefit from the latest features.
• Privacy Assurance: Your data is safe with us.
• Efficient Age Retrieval: Quick and accurate.
• Language Accessibility: Use in your preferred language.
• Engaging and Informative Content: Learn while you use.
• Fun and Interactive Learning: Enjoy the process.
• Shareable Results: Easily export your findings.
• Responsive Design: Works flawlessly on any screen size.
• Educational Platform Integration: Perfect for e-learning environments.
• Comprehensive Documentation: All the information you need.

Applications and Use Cases for the Z-test Calculator

• Analyze the difference between sample means and population means.
• Validate research findings with statistical significance.
• Enhance academic projects with precise hypothesis testing.
• Apply in various scientific and market research studies.
• Utilize in quality control processes for product consistency.

Illustrating the Z-test Calculator with Practical Examples

Example 1: Suppose your sample mean (x) is 105, the population mean (y) is 100, the population standard deviation is 15, and the sample size is 30. Plugging these into the Z-test formula, we get a Z-score, which we then compare against the standard normal distribution.

Example 2: If your sample mean is 130, the population mean is 120, the standard deviation is 20, and the sample size is 50, the Z-test Calculator will give you a Z-score indicating the probability of this difference occurring by chance.

Ensuring Data Security with the Z-test Calculator

Our Z-test Calculator not only provides precise statistical analysis but also guarantees the utmost data security. As the calculations are performed entirely within your browser, your data never leaves your computer, ensuring complete confidentiality. This tool is a crucial asset for users who prioritize privacy while seeking reliable statistical solutions. Rest assured that with our Z-test Calculator, your data is processed securely without any risk of server-side exposure.

Frequently Asked Questions About Z-test Calculator

• What is a Z-test Calculator used for?
• How accurate is the Z-test Calculator?
• Can I use the Z-test Calculator for any sample size?
• Is there a cost associated with using the Z-test Calculator?
• How does the Z-test Calculator ensure the privacy of my data?

People also viewed

104, Building No. 5, Sector 3, Millennium Business Park, Mahape, Navi Mumbai - 400710

• Core Python Certification
• Create Own Cryptocurrency
• Python for Kids
• Learn HTML (IIT)
• Learn PHP(IIT)
• Java Tutorial (IIT)
• Django Tutorial (IIT)
• C Prog. for Kids
• Python For Kids Online
• C++ for Kids
• Verify Certificate
• Book Free Demo
• Online Compiler
• Generate Genesis Block

Copyright © 2024 Newtum. All Right Reserved.

• Privacy policy
• Terms & Conditions

Z Score Calculator

Use this Z table calculator to easily calculate the Z-score from a given raw score. Also computes areas under the normal curve ( p -values) cut off by a given score. A table of Z scores and corresponding p-values is included, as well as the z score formula. Also calculates Z from p.

Related calculators

• Using the Z score calculator
• What is a Z Distribution?
• What is a "Z score"?
• Z score formula
• Example calculations

    Using the Z score calculator

The z score calculator can be used to derive a z statistic from a raw score and known or estimated distribution mean and standard deviation . If the variance is known instead, then the standard deviation is simply its square root. The output also contains probabilities calculated for different areas under the standard normal curve which correspond to a one-tailed or two-tailed test of significance. The cumulative probabilities are calculated using the standard normal cumulative distribution function (CDF) .

The z statistic calculator can also be used in inverse - to obtain a Z critical value corresponding to a given probability. Simply select "Z score from P" and enter the p-value threshold in the field to obtain the standard score defining the critical region.

    What is a Z Distribution?

The Z distribution is simply the standard normal distribution of the random variable Z meaning it is a normal distribution with mean 0 and variance and standard deviation equal to 1 [1,2,3] . The Z distribution with key quantiles is shown on the graph below:

The fact that the distribution is standardized means that the quantiles are known, and that area between any two Z scores is also known. For example, 68.27% of values would fall between -1 and 1 standard deviations of a Z distribution. Similarly, just over 95% of its probability density falls between -2 and +2 standard deviations. The entire distribution density sums to 1 and just like other normal distributions it is fully defined by its first two moments. These and other qualities make it a useful tool in statistics and probability calculation of various sorts.

Our z score calculator uses the CDF of the Z distribution to find the area under the standard normal curve above, below, between, or outside regions defined by given scores. It uses the inverse CDF to calculate Z scores from p-values.

    What is a "Z score"?

The z-score, also referred to as standard score and z-value is a signed real valued dimensionless quantity which indicates the number of standard deviations by which a given observed data point is distanced from the mean or expected value of a distribution. Standard refers to the fact that they are computed against the standard normal distribution (a.k.a. Z distribution) which is fully defined by its mean and standard deviation of zero and one, respectively.

Z values have numerous applications in statistical inference and estimation. Most commonly they are used in a Z-test of significance as well as confidence interval calculations [4] . They are also used in process control and quality assurance applications (e.g. six sigma ). Due to Z values being standardized scores they are useful in comparing measurements across different scales which is often needed in both scientific and applied disciplines.

    Z score formula

The formula for calculating a z score from a raw score from is given by the simple equation:

Using the above formula one can easily convert a raw score from a normal distribution with known or estimated mean and standard deviation to a standard score. An online Z calculator can perform the arithmetic for you quickly and easily.

    Z table

A Z table contains tabulated values of the Z distribution and their corresponding quantiles, or percentages. Since a table of Z scores can be used to judge either a point null hypothesis (e.g. the effect is exactly zero), as well as the more commonly needed composite nulls such as μ 1 ≤ 0, a proper Z table should contain both types of cumulative probabilities. In the table below, a one-tailed p-value / percentile refers to the area under the standard normal curve to the right of the Z score ( P(X > z) ), whereas a two-tailed p-value / percentile refers to the cumulative probability contained in the union of the areas to the right of -Z and to the left of Z ( P(X < -|z| ∪ X > |z|) ).

Table of commonly used Z-score cut-offs for defining critical regions for normally distributed random variables:

Note that since the standard normal distribution is symmetrical, a two-tailed p-value is exactly twice that of a one-tailed one for the same value of Z. Obviously, such a table has limited utility nowadays when it is much easier to use a free online z table calculator like ours. Refer to the documentation and graphs in our critical value calculator page for more on critical values and regions.

    Example calculations

It is easy to find the Z score corresponding to a given raw score, given that one knows the mean and standard deviation of the normal distribution to which the raw score belongs. Using the z statistic formula above we can easily compute that a raw score from a standard normal distribution is equivalent to the Z score since z = (x - μ) / σ = x for μ = 0 and σ = 1.

In another example, a raw score of 1600 from a distribution with mean 1000 and variance 90,000 is given. How do we find the Z score? First, convert the variance (σ 2 ) to standard deviation (σ) by taking its square root : σ = √90000 = 300. Then the solution is simply:

z = (1600 - 1000) / 300 = 600 / 300 = 2

This tells us that the raw score of 1600 is 2 standard deviations away from the mean. The normal CDF can then be used to arrive at whatever area under the standard normal curve is of interest. For example, to reject the hypothesis that the true value related to the observation is not lower than or equal to zero, one needs to compute

p = P(X > z) = P(X > 2) = 0.0228

which is the 2.28% percentile. For the last part one can look up the Z table for the row where Z = 2, or use a Z score probability calculator if the value is not tabulated or a greater precision is needed.

    References

1 Gauss, C.F. (1809) "Theoria motvs corporvm coelestivm in sectionibvs conicis Solem ambientivm" [Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections]

2 Laplace, P-S (1774). "Mémoire sur la probabilité des causes par les événements". Mémoires de l'Académie Royale des Sciences de Paris (Savants étrangers), Tome 6: 621–656. Translated by Stephen M. Stigler in Statistical Science 1(3), 1986.

3 Laplace, P-S (1812). "Théorie analytique des probabilités" [Analytical theory of probabilities]

4 Mayo D.G., Spanos A. (2010) – "Error Statistics", in P. S. Bandyopadhyay & M. R. Forster (Eds.), Philosophy of Statistics, (7, 152–198). Handbook of the Philosophy of Science . The Netherlands: Elsevier.

Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Z Score Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/z-score-calculator.php URL [Accessed Date: 12 Apr, 2024].

Our statistical calculators have been featured in scientific papers and articles published in high-profile science journals by:

The author of this tool

     Statistical calculators

Find Z Score Calculator

Statscalculator.com, calculating z-score, related tools.

• Descriptive Statistics
• Confidence Interval
• Correlation Coefficient
• Outlier Detection
• t-test - 1 Sample
• t-test - 2 Sample

Population Mean

Standard Deviation

Calculation Detail

Z = (raw score - population mean) / standard deviation, z = (7 - 2.5) / 3, tool overview: z score calculator.

This Find Z Score calculator is used to convert your raw score into a standardized z score. This uses the Z score formula to automate the process of looking up this information in a z table . Enter the three factors (population mean, standard deviation, and raw score) into the calculator and it will generate the appropriate Z-Score. This standard score identifies where your raw score would fall in a standard normal distribution.

The find z score calculator will also convert your Z-score into an appropriate p-value, given in this case as percentile. This identifies the probability of seeing that raw score within a standard normal distribution, expressed as the percentage of the normal distribution under that mark. If you already have a Z-statistic and just need to calculate the p value, we have a simple p value calculator. We have another version that works for calculating the p value from a t statistic if you are working with small sample sizes and need to use Student's t-distribution.

For those of you using the find z score calculator to check statistics homework, we present a breakout of the relevant calculations. This includes both the formula and a version populated with numbers from the calculator. This is intended to help you check your calculations. It replaces a z table . If you're using R, check out the rnorm function .

FAQ - Interpreting Z Score Results

How do you calculate z score.

You can use a z score calculator (like this one) or a z score table. THe table works well with practice. You calculate your test statistic and compare it to the normal distribution curve (aka the bell curve). You may also have reference values from class.

What is Hypothesis Testing?

A rigorous way of looking at a sample and comparing it to the shape of a distribution associated with the assumption nothing is going on. We refer to this as the null hypothesis. If the data in your sample lies outside the expected range of the distribution, we will accept the alternate hypothesis - which suggests the source of your data is from a different distribution. Remember all distributions converge on the normal distribution (the normal curve) with a larger sample size due to the central limit theorem.

How Many Standard Deviation Is Significant?

How is this used.

This is used both in industry and in education. A test score is intended as a representative measurement of your progress towards a binary goal: you don't know the material (null hypothesis) or do (an alternate hypothesis). We assume the results of the default state are a random variable with a visible population mean and population standard deviation. (Deviation is a measure of how much volatility to expect). The mean value is compared the curve.

Positive z score / negative z score?For a standard normal distribution, you can use the sign of the z score to assess where someone stands. A positive z score is above the mean (average) result. A negative z score is below average (median result). [for a true standard normal, the mean and median are identical).TransformationsIn some cases, you can apply another function to transform the raw value.

Speaking as a practicing analyst, concepts such as a Z-Score can be very useful when comparing the performance of similar processes measured under different conditions. For example, a larger machine might have a different level of output than a smaller machine but swings and variation in that output could certainly vary based on similar factors. This type of analysis allows you to translate these differences into a common basis and make intelligent comparisons.

Other Tools: P Value From Z Score , P Value From T Score , Confidence Interval (proportion) , t critical value calculator , z critical value calculator

• Calculators
• Descriptive Statistics
• Merchandise
• Which Statistics Test?

Single Sample Z Score Calculator

This tool calculates the z -score of the mean of a single sample. It can be used to make a judgement about whether the sample differs significantly on some axis from the population from which it was originally drawn.

By default, this tool works on the assumption that you already know the mean value of your sample scores and the number of individuals in your sample. If you want the tool to calculate these for you from raw values, please select the checkbox below.

To use this calculator, just input your population mean, population variance, sample mean and the number of individuals in the sample into the text boxes below. You also need to select a significance level and whether your hypothesis is one or two-tailed. Hit the calculate button (below) when you're ready.

Calculation not performed yet.

Z-Test for Statistical Hypothesis Testing Explained

The Z-test is a statistical hypothesis test used to determine where the distribution of the test statistic we are measuring, like the mean , is part of the normal distribution .

There are multiple types of Z-tests, however, we’ll focus on the easiest and most well known one, the one sample mean test. This is used to determine if the difference between the mean of a sample and the mean of a population is statistically significant.

What Is a Z-Test?

A Z-test is a type of statistical hypothesis test where the test-statistic follows a normal distribution.  

The name Z-test comes from the Z-score of the normal distribution. This is a measure of how many standard deviations away a raw score or sample statistics is from the populations’ mean.

Z-tests are the most common statistical tests conducted in fields such as healthcare and data science . Therefore, it’s an essential concept to understand.

Requirements for a Z-Test

In order to conduct a Z-test, your statistics need to meet a few requirements, including:

• A Sample size that’s greater than 30. This is because we want to ensure our sample mean comes from a distribution that is normal. As stated by the c entral limit theorem , any distribution can be approximated as normally distributed if it contains more than 30 data points.
• The standard deviation and mean of the population is known .
• The sample data is collected/acquired randomly .

More on Data Science:   What Is Bootstrapping Statistics?

Z-Test Steps

There are four steps to complete a Z-test. Let’s examine each one.

4 Steps to a Z-Test

• State the null hypothesis.
• State the alternate hypothesis.
• Choose your critical value.
• Calculate your Z-test statistics. 

1. State the Null Hypothesis

The first step in a Z-test is to state the null hypothesis, H_0 . This what you believe to be true from the population, which could be the mean of the population, μ_0 :

2. State the Alternate Hypothesis

Next, state the alternate hypothesis, H_1 . This is what you observe from your sample. If the sample mean is different from the population’s mean, then we say the mean is not equal to μ_0:

3. Choose Your Critical Value

Then, choose your critical value, α , which determines whether you accept or reject the null hypothesis. Typically for a Z-test we would use a statistical significance of 5 percent which is z = +/- 1.96 standard deviations from the population’s mean in the normal distribution:

This critical value is based on confidence intervals.

4. Calculate Your Z-Test Statistic

Compute the Z-test Statistic using the sample mean, μ_1 , the population mean, μ_0 , the number of data points in the sample, n and the population’s standard deviation, σ :

If the test statistic is greater (or lower depending on the test we are conducting) than the critical value, then the alternate hypothesis is true because the sample’s mean is statistically significant enough from the population mean.

Another way to think about this is if the sample mean is so far away from the population mean, the alternate hypothesis has to be true or the sample is a complete anomaly.

More on Data Science: Basic Probability Theory and Statistics Terms to Know

Z-Test Example

Let’s go through an example to fully understand the one-sample mean Z-test.

A school says that its pupils are, on average, smarter than other schools. It takes a sample of 50 students whose average IQ measures to be 110. The population, or the rest of the schools, has an average IQ of 100 and standard deviation of 20. Is the school’s claim correct?

The null and alternate hypotheses are:

Where we are saying that our sample, the school, has a higher mean IQ than the population mean.

Now, this is what’s called a right-sided, one-tailed test as our sample mean is greater than the population’s mean. So, choosing a critical value of 5 percent, which equals a Z-score of 1.96 , we can only reject the null hypothesis if our Z-test statistic is greater than 1.96.

If the school claimed its students’ IQs were an average of 90, then we would use a left-tailed test, as shown in the figure above. We would then only reject the null hypothesis if our Z-test statistic is less than -1.96.

Computing our Z-test statistic, we see:

Therefore, we have sufficient evidence to reject the null hypothesis, and the school’s claim is right.

Hope you enjoyed this article on Z-tests. In this post, we only addressed the most simple case, the one-sample mean test. However, there are other types of tests, but they all follow the same process just with some small nuances.  

Built In’s expert contributor network publishes thoughtful, solutions-oriented stories written by innovative tech professionals. It is the tech industry’s definitive destination for sharing compelling, first-person accounts of problem-solving on the road to innovation.

Great Companies Need Great People. That's Where We Come In.

Z-score Calculator

Use this calculator to compute the z-score of a normal distribution.

Z-score and Probability Converter

Please provide any one value to convert between z-score and probability. This is the equivalent of referencing a z-table.

Probability between Two Z-scores

Use this calculator to find the probability (area P in the diagram) between two z-scores.

Related Standard Deviation Calculator

What is z-score?

The z-score, also referred to as standard score, z-value, and normal score, among other things, is a dimensionless quantity that is used to indicate the signed, fractional, number of standard deviations by which an event is above the mean value being measured. Values above the mean have positive z-scores, while values below the mean have negative z-scores.

The z-score can be calculated by subtracting the population mean from the raw score, or data point in question (a test score, height, age, etc.), then dividing the difference by the population standard deviation:

where x is the raw score, μ is the population mean, and σ is the population standard deviation. For a sample, the formula is similar, except that the sample mean and population standard deviation are used instead of the population mean and population standard deviation.

The z-score has numerous applications and can be used to perform a z-test, calculate prediction intervals, process control applications, comparison of scores on different scales, and more.

A z-table, also known as a standard normal table or unit normal table, is a table that consists of standardized values that are used to determine the probability that a given statistic is below, above, or between the standard normal distribution. A z-score of 0 indicates that the given point is identical to the mean. On the graph of the standard normal distribution, z = 0 is therefore the center of the curve. A positive z-value indicates that the point lies to the right of the mean, and a negative z-value indicates that the point lies left of the mean. There are a few different types of z-tables.

The values in the table below represent the area between z = 0 and the given z-score.

How to read the z-table

In the table above,

• the column headings define the z-score to the hundredth's place.
• the row headings define the z-score to the tenth's place.
• each value in the table is the area between z = 0 and the z-score of the given value, which represents the probability that a data point will lie within the referenced region in the standard normal distribution.

For example, referencing the right-tail z-table above, a data point with a z-score of 1.12 corresponds to an area of 0.36864 (row 13, column 4). This means that for a normally distributed population, there is a 36.864% chance, a data point will have a z-score between 0 and 1.12.

Because there are various z-tables, it is important to pay attention to the given z-table to know what area is being referenced.

Input 2 Proportions

Formula and output, when do you use z-test two independent sample means.

When performing a Z-test on two independent samples you want to sum the variances of the means and then take the square root to find the standard deviation of variance sum. In this example you are given the standard deviations for each sample thus you need to take the square of the standar deviations to find the variances: - Variance sample 1 = 0.26^2 = 0.0676 - Variance sample 2 = 0.22^2 = 0.0484 Now when you have the variances you use the formula for Z-test two independent samples or you can use the calculator provided. The Z-value is 3.648 which is above the critical value of 2.5758 (two tailed test), thus there is a significant difference between the soils.

Z Score Calculator

• Enter the Raw Score, Mean (μ), and Standard Deviation (σ) for your data.
• Click "Calculate Z-Score" to calculate the Z-Score and related values.
• Results, including the Z-Score, p-values, and confidence level, will be displayed below.
• Calculation steps will also be shown to explain how the Z-Score was computed.
• A chart visualizes the Z-Score in the context of the normal distribution.
• You can clear the entries, copy the results, and view calculation history.

What is Z Score

A z-score, also known as a standard score, is a statistical measure that indicates how many standard deviations a specific data point is away from the mean of the entire dataset . In simpler terms, it tells you how unusual or typical a particular value is compared to the rest of the data.

Key Aspects of Z Score

1. Standardization:

• Z-scores transform raw data points into a standardized scale with a mean of 0 and a standard deviation of 1.
• This allows for meaningful comparisons of data points from different datasets, even if they have different units or scales.

2. Relative Position:

• The z-score directly indicates how far a data point is from the mean in terms of standard deviations.
• A positive z-score means the point is above the mean, while a negative z-score means it’s below the mean.
• The further the z-score is from zero, the more unusual the value is within the dataset.

3. Normal Distribution:

• Z-scores are particularly useful when working with normally distributed data.
• In a normal distribution, approximately 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
• By knowing the z-score, you can quickly determine the percentile rank of a data point within a normal distribution.

4. Outlier Detection:

• Z-scores can be used to identify outliers, which are significantly different values from the rest of the data.
• A common rule of thumb is that data points with z-scores greater than 3 or less than -3 are considered potential outliers.

5. Hypothesis Testing:

• Z-scores play a crucial role in hypothesis testing, where they are used to assess the likelihood of observed results occurring by chance.
• They are used to calculate p-values, which indicate the statistical significance of findings.

6. Data Transformation:

• Z-scores can be used to transform non-normal distributions into a normal distribution, which is required for statistical analyses that assume normality.

7. Standard Normal Distribution:

• The standard normal distribution, also known as the “Z-distribution,” is a special normal distribution with a mean of 0 and a standard deviation of 1.
• It’s a valuable reference for interpreting z-scores and understanding probabilities associated with different z-score values.

All Formulae Related to Z Score

Here are the key formulae related to z-scores:

1. Calculating the z-score:

• z = the z-score
• x = the raw data point
• μ = the mean of the dataset
• σ = the standard deviation of the dataset

2. Finding the raw data point from a z-score:

3. Calculating percentile rank in a normal distribution:

where the cumulative area under the z-curve can be found using a z-table or statistical software.

4. Converting a z-score to a probability:

5. Converting a probability to a z-score:

This requires using a z-table or statistical software to find the inverse of the cumulative normal distribution function.

Additional formulae for specific applications:

• Hypothesis testing:  Z-tests involve calculating a test statistic ( z-score ) and comparing it to a critical value to determine statistical significance.
• Confidence intervals:  The margin of error for a confidence interval can be calculated using z-scores and the standard error of the estimate.
• Correlation analysis:  The correlation coefficient (r) can be converted to a z-score to test its statistical significance.

Practical Uses of Z-Score Calculator

Here are some practical uses of z-score calculators:

1. Comparing Scores from Different Tests:

• Scenario:  You took two exams in different subjects, each with different scales and means. A z-score calculator can help you compare your performance on both exams on a standardized scale.
• How it works:  Calculate the z-scores for your scores on each exam. The exam with the higher z-score indicates a better relative performance, regardless of the original scores or scales.

2. Evaluating Individual Performance in a Group:

• Scenario:  You received a score of 85 on a class exam where the mean was 75 and the standard deviation was 5. A z-score calculator can tell you how well you performed compared to your classmates.
• How it works:  Calculate your z-score (z = (85 – 75) / 5 = 2). A z-score of 2 means you scored 2 standard deviations above the average, indicating excellent performance relative to the group.

3. Determining Normality of Data Distribution:

• Scenario:  You have a dataset of customer purchase amounts and want to check if it follows a normal distribution. A z-score calculator can help you assess normality.
• How it works:  Calculate the z-scores for all data points. If the distribution of z-scores is approximately bell-shaped and symmetric, it suggests a normal distribution.

4. Detecting Outliers:

• Scenario:  You’re analyzing website traffic data and need to identify unusual spikes or drops. A z-score calculator can help you detect potential outliers.
• How it works:  Calculate the z-scores for daily traffic numbers. Data points with z-scores significantly higher or lower than the rest might be considered outliers, warranting further investigation.

5. Calculating Percentile Ranks:

• Scenario:  You scored 1500 on the SAT, and you want to know what percentage of test-takers scored lower than you. A z-score calculator can estimate your percentile rank.
• How it works:  Assuming SAT scores are normally distributed, use a z-table or calculator to find the area under the standard normal curve to the left of your z-score. This area represents the percentile rank.

6. Making Predictions in Finance:

• Scenario:  You’re analyzing stock prices and want to assess the probability of a certain stock price movement. Z-scores can help you make predictions based on historical volatility.
• How it works:  Calculate the z-score for a potential price change, and use a z-table to find the corresponding probability. This can inform investment decisions based on risk assessment.

Applications of Z Score in Various Fields

Here are some applications of z-scores in various fields:

1. Statistics:

• Comparing data from different datasets:  Z-scores allow for standardized comparisons even when scales and means differ, enabling meaningful analysis across diverse studies or populations.
• Hypothesis testing:  Z-tests are foundational for determining whether observed results are statistically significant or likely due to chance.
• Outlier detection:  Z-scores help identify unusual values that might warrant further investigation or exclusion from analysis to ensure data integrity.

2. Finance:

• Risk assessment and investment analysis:  Z-scores measure volatility and potential returns of investments, aiding in portfolio optimization and risk management strategies.
• Statistical arbitrage:  Z-scores form the basis of statistical arbitrage models, which seek to exploit pricing inefficiencies in financial markets.
• Credit scoring:  Z-scores are used in credit scoring models to assess the creditworthiness of individuals and businesses, influencing loan approval decisions and interest rates.

3. Education:

• Standardized testing:  Z-scores enable comparisons of student performance across different tests and grade levels, facilitating fair assessments and tracking progress.
• Identifying gifted and talented students:  Z-scores help identify students with exceptional abilities or those requiring additional support, promoting tailored educational approaches.
• Evaluating educational programs and interventions:  Z-scores can measure the effectiveness of instructional methods or interventions by comparing student outcomes before and after implementation.

4. Psychology:

• Psychological assessment:  Z-scores are used to standardize scores on psychological tests and questionnaires, enabling comparisons with norms and identifying potential clinical concerns.
• Research studies:  Z-scores are used to analyze experimental data, assess the effectiveness of interventions, and draw meaningful conclusions from research findings.

5. Healthcare:

• Monitoring patient health:  Z-scores track vital signs, lab results, and other health indicators over time, aiding in early detection of health issues and evaluating treatment effectiveness.
• Epidemiology:  Z-scores are used to identify risk factors for diseases and track disease outbreaks, informing public health policies and interventions.

6. Manufacturing and Quality Control:

• Process control:  Z-scores monitor manufacturing processes to detect deviations from expected quality standards, ensuring consistent product quality and reducing defects.
• Statistical process control (SPC):  Z-scores are central to SPC techniques, which aim to prevent defects and improve process efficiency.

7. Science and Engineering:

• Experiment design and analysis:  Z-scores guide sample size calculations, ensure experimental validity, and assess the significance of findings in various scientific fields.
• Signal processing:  Z-scores are used in signal processing techniques to detect anomalies and extract meaningful information from noisy data.

Benefits of Using the Z-Score Calculator

• Efficiency : Quickly computes z-scores without manual calculations.
• Accuracy : Reduces the risk of human error in computation.
• Ease of Use : Simplifies the process for those unfamiliar with statistical formulas.
• Practical Learning Tool : Helps students and professionals understand and apply z-scores.

The Z-Score Calculator is an invaluable tool for statisticians, researchers, students, and professionals across various fields. It simplifies the process of calculating and understanding z-scores, facilitating data normalization, comparison, and interpretation.

By providing an easy and accurate means to calculate z-scores, this tool aids in the analysis of data sets, ensuring that the nuances and insights embedded in statistical data are accessible to a broader audience. Whether used for academic purposes, professional analysis, or personal interest, the Z-Score Calculator is a testament to the importance of statistical tools in the age of data-driven decision-making.

• NIST/SEMATECH e-Handbook of Statistical Methods – https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm
• Z-score – Wikipedia – https://en.wikipedia.org/wiki/Standard_score

Last Updated : 27 February, 2024

I’ve put so much effort writing this blog post to provide value to you. It’ll be very helpful for me, if you consider sharing it on social media or with your friends/family. SHARING IS ♥️

Sandeep Bhandari holds a Bachelor of Engineering in Computers from Thapar University (2006). He has 20 years of experience in the technology field. He has a keen interest in various technical fields, including database systems, computer networks, and programming. You can read more about him on his bio page.

Similar Reads

• GPA Calculator
• Inverse Trigonometric Functions Calculator
• Parallelogram Calculator
• Pascal’s Triangle Calculator
• Trigonometric Functions Calculator ƒ(π)

Share this post!

20 thoughts on “z score calculator”.

This article provides detailed and comprehensive information about z-scores. It’s an excellent resource for anyone studying statistics or data analysis.

This article serves as a comprehensive guide to understanding z-scores and their relevance in statistical analysis. A must-read for anyone delving into this field.

This article is an excellent reference for anyone needing to understand z-scores and their practical applications. It’s comprehensive and well-articulated.

While informative, the length of the article may deter some readers from fully engaging with its content. A more compact version might be beneficial.

I find the length to be justified considering the complexity of the topic. It’s a tradeoff for thoroughness and comprehensiveness.

The article delves into the intricacies of z-scores, providing valuable insights. It’s a commendable effort to elucidate this statistical concept.

The definition and practical uses sections are particularly insightful. This article provides a solid foundation for understanding z-scores in various contexts.

I totally agree! The explanation of the key aspects of z-scores is particularly well done and easy to understand.

Absolutely, the depth of coverage on z-scores and their practical implications is commendable. The article stands out as an educational aid.

I couldn’t agree more. The detailed explanation of formulae and practical examples make this article a standout resource.

I understand your point, but the depth of information provided is valuable for those seeking an in-depth understanding of z-scores.

Absolutely! I found the practical uses of z-score calculators to be especially valuable. They provide tangible examples of how z-scores are applied in real-world scenarios.

While the information is good, I find the article to be quite verbose and overwhelming. It might benefit from a more concise and structured presentation.

I appreciate the inclusion of practical use cases, as it helps contextualize the concept of z-scores. This article is a valuable learning tool.

Agreed. The real-world applications provided here are essential for grasping the significance of z-scores in practice.

The content is dense, but it offers a wealth of information for those willing to engage with it. A great resource for statistical analysis enthusiasts.

I see your point, but I appreciate the level of detail. It’s a complex concept, and the thorough explanation is helpful.

While the article is thorough, it might benefit from more interactive elements to help visualize the concepts. Visual aids could enhance the learning experience.

The practical use cases indeed enhance the article’s educational value. Understanding how z-scores are applied in real scenarios is crucial.

I agree. Incorporating visualizations could further elucidate the concepts and engage readers more effectively.

Leave a Comment

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

Z-test for One Population Mean

Instructions: This calculator conducts a Z-test for one population mean (\(\mu\)), with known population standard deviation (\(\sigma\)). Please select the null and alternative hypotheses, type the hypothesized mean, the significance level, the sample mean, the population standard deviation, and the sample size, and the results of the z-test will be displayed for you:

How to Conduct a Z-Test for One Population Mean?

More about the z-test for one mean so you can better interpret the results obtained by this solver: A z-test for one mean is a hypothesis test that attempts to make a claim about the population mean (\(\mu\)).

The test has two non-overlapping hypotheses, the null and the alternative hypothesis. The null hypothesis is a statement about the population mean, under the assumption of no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of a one sample z-test for one population mean are:

• Depending on our knowledge about the "no effect" situation, the z-test can be two-tailed, left-tailed or right-tailed
• The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true
• The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true
• In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis

Uses of this z-test calculator

What can you do with this z-test statistic calculator for hypothesis testing? The formula for a z-statistic is

The null hypothesis is rejected when the z-statistic lies on the rejection region, which is determined by the significance level (\(\alpha\)) and the type of tail (two-tailed, left-tailed or right-tailed).

What if the population standard deviation is not known?

It frequently happens that you don't actually know the population standard deviation, in which case you need to use a t-test for one mean calculator instead, which adjusts for that by using the sample standard deviation, by using a slightly different distribution (the t-distribution)

How to calculate p-value in the context of a z-test?

The answer depends on whether you are using a two-tailed, a left-tailed or a right-tailed test. Say you have the calculated z-statistic, \(Z_{obs}\).

• For a two-tailed test, the p-value is computed as: \(p = \Pr( Z > |Z_{obs}|) \)
• For a left-tailed test, the p-value is computed as: \(p = \Pr( Z < Z_{obs}) \)
• For a right-tailed test, the p-value is computed as: \(p = \Pr( Z > Z_{obs}) \)

where \(Z\) has a standard normal distribution.

Other types of Z-calculators

In case that you need to compare two population means, when you know the corresponding population standard deviations, you need to use this z-test for two means with known population standard deviations instead.

Outlier Detection

Don't forget to detect outliers before running a z-test for one mean. It is important that outliers are detected and removed before conducting the test, but the results of the test statistics may be slanted.

Example: Application of the Z-test calculator

Question : Assume that you want to test whether or not the population mean is 12.3. You collect a representative random sample of size n = 16, and you find that the sample mean is 11.3. Also, you know that the population is 2.3. Do the sample data provide enough evidence to reject the claim that the population mean is 12.3? Use a two-tailed test, with a significance level of 0.01.

The following information has been provided:

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is \(\alpha = 0.01\), and the critical value for a two-tailed test is \(z_c = 2.58\).

The rejection region for this two-tailed test is \(R = \{z: |z| > 2.576\}\)

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that \(|z| = 1.739 \le z_c = 2.576\), it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is \(p = 0.082\), and since \(p = 0.082 \ge 0.01\), it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean \(\mu\) is different than 12.3, at the \(\alpha = 0.01\) significance level.

Confidence Interval

The 99% confidence interval is \(9.819 < \mu < 12.781\).

Related Calculators

log in to your account

Reset password.

One Sample Z-Test Calculator

About one sample z-test calculator (formula).

A One Sample Z-Test is a statistical test used to determine whether the mean of a single sample differs significantly from a known population mean or a hypothesized mean. This test is commonly used in hypothesis testing when you have a single set of data points and want to determine if it’s representative of a larger population or if there’s a significant difference between the sample and the population.

Here’s the formula for a One Sample Z-Test:

Z = (X̄ – μ) / (σ / √(n))

• Z is the Z-statistic.
• X̄ (pronounced as “X-bar”) is the sample mean.
• μ (pronounced as “mu”) is the population mean (the known mean or the hypothesized mean).
• σ (pronounced as “sigma”) is the population standard deviation (if known).
• n is the sample size.

The steps to perform a One Sample Z-Test are as follows:

• H0: The sample mean is equal to the population mean (μ).
• Ha: The sample mean is not equal to the population mean (μ), indicating a two-tailed test. Alternatively, you can use a one-tailed test if you have a specific direction in mind (greater than or less than).
• Collect your sample data and calculate the sample mean (X̄) and, if possible, the population standard deviation (σ).
• Determine the significance level (α), which represents the probability of making a Type I error (rejecting the null hypothesis when it is true). Common choices for α include 0.05 and 0.01.
• Calculate the Z-statistic using the formula mentioned above.
• Compare the calculated Z-statistic to the critical Z-value(s) from the standard normal distribution table or use a statistical calculator. The critical value(s) correspond to your chosen significance level (α) and the type of test (two-tailed or one-tailed).
• If |Z| > critical value: Reject the null hypothesis (H0) in favor of the alternative hypothesis (Ha).
• If |Z| ≤ critical value: Fail to reject the null hypothesis (H0).
• Draw a conclusion based on your decision and report the results.

This test helps you determine whether the observed difference between your sample mean and the population mean is statistically significant or if it could have occurred due to random sampling variation.

Keep in mind that for practical purposes, it’s often recommended to use statistical software or calculators to perform One Sample Z-Tests because they can handle the calculations and critical value lookup efficiently.

Leave a Comment Cancel reply

Save my name, email, and website in this browser for the next time I comment.

• [email protected]

Z-Test Calculator

Z-Score: –

Conclusion: –

Note : This calculator assumes a two-tailed test, so if you’re interested in a one-tailed test, you’ll need to interpret the Z-score manually.

Table of Contents

How to use this z-test calculator.

• Select the Confidence Level : Choose the confidence level suitable for your test.
• Enter Sample Mean (x̄) : Type the average of the sample data in the corresponding field.
• Enter Population Mean (μ) : Type the average of the entire population in the corresponding field.
• Enter Sample Size (n) : Type the number of observations in your sample.
• Enter Population Standard Deviation (σ) : Type the standard deviation of the population.
• Click the Calculate Button : After entering all the data, click the “Calculate” button.
• View the Results : Look at the “Z-Score” and “Conclusion” sections to interpret the results.

Interpretations

• If the conclusion is “Reject null hypothesis,” it means that the sample mean is significantly different from the population mean at the selected confidence level.
• If the conclusion is “Fail to reject null hypothesis,” it means that there’s not enough evidence to suggest that the sample mean is different from the population mean at the selected confidence level.

Demystifying the Z-Test: A Simple Guide to Understanding When and How to Use It

Let’s delve into the fascinating world of statistical hypothesis testing, particularly focusing on one of its most fundamental tools: the Z-test. Now, if the mere mention of ‘statistics’ or ‘Z-test’ sends you into a spiral of confusion, worry not! By the end of this article, you’ll not only understand what a Z-test is but also when and how to use it

What is a Z-Test, Anyway?

So, let’s start with the basics. A Z-test is a type of statistical hypothesis test that helps you determine whether the mean of a sample you’ve collected is statistically significantly different from the known mean of a whole population. In simpler terms, it’s like a fact-checker that tells you if the average value you’ve calculated from a small group is a reliable representation of a much larger group. Imagine polling a hundred people to predict the outcome of a presidential election. A Z-test can tell you whether that sample of 100 people is a good indicator of how the entire country will vote.

Why Would You Need a Z-Test?

The answer is quite straightforward: you often can’t survey an entire population. Whether you’re a researcher studying the effect of a new drug, a marketer trying to gauge the success of a recent campaign, or a student collecting data for a project, you’ll likely be working with samples. And here’s where the Z-test comes in handy—it helps you make accurate inferences about a whole population based on the sample data you have.

The Ingredients for a Z-Test

Armed with these numbers, you’re all set to perform a Z-test.

• Sample Mean (x̄) : The average of the sample data you’ve collected.
• Population Mean (μ) : The known average of the entire population.
• Sample Size (n) : The number of observations in your sample.
• Population Standard Deviation (σ) : The spread of the entire population data.

The Nuts and Bolts: How to Perform a Z-Test

The crux of a Z-test lies in calculating the Z-score, a simple formula that looks something like this:

Don’t be intimidated by the formula; it’s simpler than it looks. You’re basically taking the difference between the sample mean and the population mean, and then dividing that by the standard error (which is the population standard deviation divided by the square root of the sample size). This Z-score will be your guide in determining whether to reject or accept the null hypothesis.

Interpreting the Z-Score

Once you’ve got the Z-score, the next step is to compare it against a critical Z-value that corresponds to a confidence level you’ve chosen (usually 95% or 99%). If the absolute value of the Z-score is greater than the critical Z-value, you can reject the null hypothesis. This means that the sample mean is significantly different from the population mean. On the flip side, if the Z-score is less than the critical Z-value, you fail to reject the null hypothesis, implying there’s not enough evidence to say the sample is different from the population.

When to Use Which? (One-Tailed vs Two-Tailed)

The terms ‘one-tailed’ and ‘two-tailed’ refer to the number of directions in which we are interested in testing our sample mean against the population mean. Allow me to clarify:

One-Tailed Test

In a one-tailed test, you’re essentially asking a directional question. You want to know if the sample mean is either significantly greater or significantly less than the population mean, but not both. For example, let’s say you’re a marketer who has tweaked an online ad campaign, and you want to know if the changes have increased the click-through rate. You’re not interested in whether the rate decreased; you only want to know if it went up. This is a perfect situation for a one-tailed test.

The critical Z-value in a one-tailed test is generally going to be larger in absolute terms if you’re looking at the tail at the extreme end of the distribution. This makes it ‘easier’ to reject the null hypothesis, but be cautious—this also increases the likelihood of a Type I error (falsely rejecting a true null hypothesis).

Two-Tailed Test

On the other hand, a two-tailed test doesn’t assume a direction. You’re asking if the sample mean is different from the population mean—either greater or less. Imagine you’re a quality control manager at a factory, and you want to know if a new manufacturing process has changed the average weight of a product. In this case, you’re interested in knowing if the weight has either increased or decreased, making a two-tailed test the appropriate choice.

The critical Z-value in a two-tailed test is generally smaller in absolute terms, making it ‘harder’ to reject the null hypothesis, which can be a safer approach to avoid Type I errors.

When to Use Which?

• Use a One-Tailed Test When : You have a specific direction in mind (greater or less), and you are okay with the increased risk of a Type I error for the benefit of a more sensitive test.
• Use a Two-Tailed Test When : You are interested in any kind of difference between the sample mean and population mean, irrespective of the direction. This is often considered a more conservative approach.

Confidence Levels and Two-Tailed Tests

You may have heard of terms like ‘confidence levels’ or ‘two-tailed tests.’ These are just ways to refine your Z-test. The confidence level, usually set at 95%, gives you the probability that your sample mean will fall within a certain range of the population mean. A two-tailed test, on the other hand, checks for deviations in both directions—whether the sample mean is either significantly higher or lower than the population mean.

Putting It All Together

So there you have it- a complete guide to understanding and using the Z-test. While it might seem daunting at first, remember that the Z-test is essentially a tool that helps you make sense of the data you have, allowing you to draw meaningful conclusions about a larger group based on a sample. Whether you’re in academia, business, or just satisfying your own curiosity, mastering the Z-test is a valuable skill that will help you navigate the complex, yet exciting, world of data and statistics.

Loved it? How about checking our All-in-One Descriptive Statistical Calculator or may be other Statistical Tools .

Other Useful Tools

Variance to Standard Deviation

Molarity Calculator

Destiny Number Calculator

Word Counter

Rolling Die

Png to WebP Converter

Character Counter

Projectile Motion Calculator

Leave a reply cancel reply.

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

Critical Value Calculator

How to use critical value calculator, what is a critical value, critical value definition, how to calculate critical values, z critical values, t critical values, chi-square critical values (χ²), f critical values, behind the scenes of the critical value calculator.

Welcome to the critical value calculator! Here you can quickly determine the critical value(s) for two-tailed tests, as well as for one-tailed tests. It works for most common distributions in statistical testing: the standard normal distribution N(0,1) (that is when you have a Z-score), t-Student, chi-square, and F-distribution .

What is a critical value? And what is the critical value formula? Scroll down – we provide you with the critical value definition and explain how to calculate critical values in order to use them to construct rejection regions (also known as critical regions).

The critical value calculator is your go-to tool for swiftly determining critical values in statistical tests, be it one-tailed or two-tailed. To effectively use the calculator, follow these steps:

In the first field, input the distribution of your test statistic under the null hypothesis: is it a standard normal N (0,1), t-Student, chi-squared, or Snedecor's F? If you are not sure, check the sections below devoted to those distributions, and try to localize the test you need to perform.

In the field What type of test? choose the alternative hypothesis : two-tailed, right-tailed, or left-tailed.

If needed, specify the degrees of freedom of the test statistic's distribution. If you need more clarification, check the description of the test you are performing. You can learn more about the meaning of this quantity in statistics from the degrees of freedom calculator .

Set the significance level, α \alpha α . By default, we pre-set it to the most common value, 0.05, but you can adjust it to your needs.

The critical value calculator will display your critical value(s) and the rejection region(s).

Click the advanced mode if you need to increase the precision with which the critical values are computed.

For example, let's envision a scenario where you are conducting a one-tailed hypothesis test using a t-Student distribution with 15 degrees of freedom. You have opted for a right-tailed test and set a significance level (α) of 0.05. The results indicate that the critical value is 1.7531, and the critical region is (1.7531, ∞). This implies that if your test statistic exceeds 1.7531, you will reject the null hypothesis at the 0.05 significance level.

👩‍🏫 Want to learn more about critical values? Keep reading!

In hypothesis testing, critical values are one of the two approaches which allow you to decide whether to retain or reject the null hypothesis. The other approach is to calculate the p-value (for example, using the p-value calculator ).

The critical value approach consists of checking if the value of the test statistic generated by your sample belongs to the so-called rejection region , or critical region , which is the region where the test statistic is highly improbable to lie . A critical value is a cut-off value (or two cut-off values in the case of a two-tailed test) that constitutes the boundary of the rejection region(s). In other words, critical values divide the scale of your test statistic into the rejection region and the non-rejection region.

Once you have found the rejection region, check if the value of the test statistic generated by your sample belongs to it :

• If so, it means that you can reject the null hypothesis and accept the alternative hypothesis; and
• If not, then there is not enough evidence to reject H 0 .

But how to calculate critical values? First of all, you need to set a significance level , α \alpha α , which quantifies the probability of rejecting the null hypothesis when it is actually correct. The choice of α is arbitrary; in practice, we most often use a value of 0.05 or 0.01. Critical values also depend on the alternative hypothesis you choose for your test , elucidated in the next section .

To determine critical values, you need to know the distribution of your test statistic under the assumption that the null hypothesis holds. Critical values are then points with the property that the probability of your test statistic assuming values at least as extreme at those critical values is equal to the significance level α . Wow, quite a definition, isn't it? Don't worry, we'll explain what it all means.

First, let us point out it is the alternative hypothesis that determines what "extreme" means. In particular, if the test is one-sided, then there will be just one critical value; if it is two-sided, then there will be two of them: one to the left and the other to the right of the median value of the distribution.

Critical values can be conveniently depicted as the points with the property that the area under the density curve of the test statistic from those points to the tails is equal to α \alpha α :

Left-tailed test: the area under the density curve from the critical value to the left is equal to α \alpha α ;

Right-tailed test: the area under the density curve from the critical value to the right is equal to α \alpha α ; and

Two-tailed test: the area under the density curve from the left critical value to the left is equal to α / 2 \alpha/2 α /2 , and the area under the curve from the right critical value to the right is equal to α / 2 \alpha/2 α /2 as well; thus, total area equals α \alpha α .

As you can see, finding the critical values for a two-tailed test with significance α \alpha α boils down to finding both one-tailed critical values with a significance level of α / 2 \alpha/2 α /2 .

The formulae for the critical values involve the quantile function , Q Q Q , which is the inverse of the cumulative distribution function ( c d f \mathrm{cdf} cdf ) for the test statistic distribution (calculated under the assumption that H 0 holds!): Q = c d f − 1 Q = \mathrm{cdf}^{-1} Q = cdf − 1 .

Once we have agreed upon the value of α \alpha α , the critical value formulae are the following:

• Left-tailed test :
• Right-tailed test :
• Two-tailed test :

In the case of a distribution symmetric about 0 , the critical values for the two-tailed test are symmetric as well:

Unfortunately, the probability distributions that are the most widespread in hypothesis testing have somewhat complicated c d f \mathrm{cdf} cdf formulae. To find critical values by hand, you would need to use specialized software or statistical tables. In these cases, the best option is, of course, our critical value calculator! 😁

Use the Z (standard normal) option if your test statistic follows (at least approximately) the standard normal distribution N(0,1) .

In the formulae below, u u u denotes the quantile function of the standard normal distribution N(0,1):

Left-tailed Z critical value: u ( α ) u(\alpha) u ( α )

Right-tailed Z critical value: u ( 1 − α ) u(1-\alpha) u ( 1 − α )

Two-tailed Z critical value: ± u ( 1 − α / 2 ) \pm u(1- \alpha/2) ± u ( 1 − α /2 )

Check out Z-test calculator to learn more about the most common Z-test used on the population mean. There are also Z-tests for the difference between two population means, in particular, one between two proportions.

Use the t-Student option if your test statistic follows the t-Student distribution . This distribution is similar to N(0,1) , but its tails are fatter – the exact shape depends on the number of degrees of freedom . If this number is large (>30), which generically happens for large samples, then the t-Student distribution is practically indistinguishable from N(0,1). Check our t-statistic calculator to compute the related test statistic.

In the formulae below, Q t , d Q_{\text{t}, d} Q t , d ​ is the quantile function of the t-Student distribution with d d d degrees of freedom:

Left-tailed t critical value: Q t , d ( α ) Q_{\text{t}, d}(\alpha) Q t , d ​ ( α )

Right-tailed t critical value: Q t , d ( 1 − α ) Q_{\text{t}, d}(1 - \alpha) Q t , d ​ ( 1 − α )

Two-tailed t critical values: ± Q t , d ( 1 − α / 2 ) \pm Q_{\text{t}, d}(1 - \alpha/2) ± Q t , d ​ ( 1 − α /2 )

Visit the t-test calculator to learn more about various t-tests: the one for a population mean with an unknown population standard deviation , those for the difference between the means of two populations (with either equal or unequal population standard deviations), as well as about the t-test for paired samples .

Use the χ² (chi-square) option when performing a test in which the test statistic follows the χ²-distribution .

You need to determine the number of degrees of freedom of the χ²-distribution of your test statistic – below, we list them for the most commonly used χ²-tests.

Here we give the formulae for chi square critical values; Q χ 2 , d Q_{\chi^2, d} Q χ 2 , d ​ is the quantile function of the χ²-distribution with d d d degrees of freedom:

Left-tailed χ² critical value: Q χ 2 , d ( α ) Q_{\chi^2, d}(\alpha) Q χ 2 , d ​ ( α )

Right-tailed χ² critical value: Q χ 2 , d ( 1 − α ) Q_{\chi^2, d}(1 - \alpha) Q χ 2 , d ​ ( 1 − α )

Two-tailed χ² critical values: Q χ 2 , d ( α / 2 ) Q_{\chi^2, d}(\alpha/2) Q χ 2 , d ​ ( α /2 ) and Q χ 2 , d ( 1 − α / 2 ) Q_{\chi^2, d}(1 - \alpha/2) Q χ 2 , d ​ ( 1 − α /2 )

Several different tests lead to a χ²-score:

Goodness-of-fit test : does the empirical distribution agree with the expected distribution?

This test is right-tailed . Its test statistic follows the χ²-distribution with k − 1 k - 1 k − 1 degrees of freedom, where k k k is the number of classes into which the sample is divided.

Independence test : is there a statistically significant relationship between two variables?

This test is also right-tailed , and its test statistic is computed from the contingency table. There are ( r − 1 ) ( c − 1 ) (r - 1)(c - 1) ( r − 1 ) ( c − 1 ) degrees of freedom, where r r r is the number of rows, and c c c is the number of columns in the contingency table.

Test for the variance of normally distributed data : does this variance have some pre-determined value?

This test can be one- or two-tailed! Its test statistic has the χ²-distribution with n − 1 n - 1 n − 1 degrees of freedom, where n n n is the sample size.

Finally, choose F (Fisher-Snedecor) if your test statistic follows the F-distribution . This distribution has a pair of degrees of freedom .

Let us see how those degrees of freedom arise. Assume that you have two independent random variables, X X X and Y Y Y , that follow χ²-distributions with d 1 d_1 d 1 ​ and d 2 d_2 d 2 ​ degrees of freedom, respectively. If you now consider the ratio ( X d 1 ) : ( Y d 2 ) (\frac{X}{d_1}):(\frac{Y}{d_2}) ( d 1 ​ X ​ ) : ( d 2 ​ Y ​ ) , it turns out it follows the F-distribution with ( d 1 , d 2 ) (d_1, d_2) ( d 1 ​ , d 2 ​ ) degrees of freedom. That's the reason why we call d 1 d_1 d 1 ​ and d 2 d_2 d 2 ​ the numerator and denominator degrees of freedom , respectively.

In the formulae below, Q F , d 1 , d 2 Q_{\text{F}, d_1, d_2} Q F , d 1 ​ , d 2 ​ ​ stands for the quantile function of the F-distribution with ( d 1 , d 2 ) (d_1, d_2) ( d 1 ​ , d 2 ​ ) degrees of freedom:

Left-tailed F critical value: Q F , d 1 , d 2 ( α ) Q_{\text{F}, d_1, d_2}(\alpha) Q F , d 1 ​ , d 2 ​ ​ ( α )

Right-tailed F critical value: Q F , d 1 , d 2 ( 1 − α ) Q_{\text{F}, d_1, d_2}(1 - \alpha) Q F , d 1 ​ , d 2 ​ ​ ( 1 − α )

Two-tailed F critical values: Q F , d 1 , d 2 ( α / 2 ) Q_{\text{F}, d_1, d_2}(\alpha/2) Q F , d 1 ​ , d 2 ​ ​ ( α /2 ) and Q F , d 1 , d 2 ( 1 − α / 2 ) Q_{\text{F}, d_1, d_2}(1 -\alpha/2) Q F , d 1 ​ , d 2 ​ ​ ( 1 − α /2 )

Here we list the most important tests that produce F-scores: each of them is right-tailed .

ANOVA : tests the equality of means in three or more groups that come from normally distributed populations with equal variances. There are ( k − 1 , n − k ) (k - 1, n - k) ( k − 1 , n − k ) degrees of freedom, where k k k is the number of groups, and n n n is the total sample size (across every group).

Overall significance in regression analysis . The test statistic has ( k − 1 , n − k ) (k - 1, n - k) ( k − 1 , n − k ) degrees of freedom, where n n n is the sample size, and k k k is the number of variables (including the intercept).

Compare two nested regression models . The test statistic follows the F-distribution with ( k 2 − k 1 , n − k 2 ) (k_2 - k_1, n - k_2) ( k 2 ​ − k 1 ​ , n − k 2 ​ ) degrees of freedom, where k 1 k_1 k 1 ​ and k 2 k_2 k 2 ​ are the number of variables in the smaller and bigger models, respectively, and n n n is the sample size.

The equality of variances in two normally distributed populations . There are ( n − 1 , m − 1 ) (n - 1, m - 1) ( n − 1 , m − 1 ) degrees of freedom, where n n n and m m m are the respective sample sizes.

I'm Anna, the mastermind behind the critical value calculator and a PhD in mathematics from Jagiellonian University .

The idea for creating the tool originated from my experiences in teaching and research. Recognizing the need for a tool that simplifies the critical value determination process across various statistical distributions, I built a user-friendly calculator accessible to both students and professionals. After publishing the tool, I soon found myself using the calculator in my research and as a teaching aid.

Trust in this calculator is paramount to me. Each tool undergoes a rigorous review process , with peer-reviewed insights from experts and meticulous proofreading by native speakers. This commitment to accuracy and reliability ensures that users can be confident in the content. Please check the Editorial Policies page for more details on our standards.

What is a Z critical value?

A Z critical value is the value that defines the critical region in hypothesis testing when the test statistic follows the standard normal distribution . If the value of the test statistic falls into the critical region, you should reject the null hypothesis and accept the alternative hypothesis.

How do I calculate Z critical value?

To find a Z critical value for a given confidence level α :

Check if you perform a one- or two-tailed test .

For a one-tailed test:

Left -tailed: critical value is the α -th quantile of the standard normal distribution N(0,1).

Right -tailed: critical value is the (1-α) -th quantile.

Two-tailed test: critical value equals ±(1-α/2) -th quantile of N(0,1).

No quantile tables ? Use CDF tables! (The quantile function is the inverse of the CDF.)

Verify your answer with an online critical value calculator.

Is a t critical value the same as Z critical value?

In theory, no . In practice, very often, yes . The t-Student distribution is similar to the standard normal distribution, but it is not the same . However, if the number of degrees of freedom (which is, roughly speaking, the size of your sample) is large enough (>30), then the two distributions are practically indistinguishable , and so the t critical value has practically the same value as the Z critical value.

What is the Z critical value for 95% confidence?

The Z critical value for a 95% confidence interval is:

• 1.96 for a two-tailed test;
• 1.64 for a right-tailed test; and
• -1.64 for a left-tailed test.

99% confidence interval

Chilled drink, grams to cups, stem and leaf plot.

• Biology (100)
• Chemistry (100)
• Construction (144)
• Conversion (294)
• Ecology (30)
• Everyday life (262)
• Finance (569)
• Health (440)
• Physics (509)
• Sports (104)
• Statistics (182)
• Other (181)
• Discover Omni (40)