hypothesis testing t distribution 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.

Student's t-distribution calculator with graph generator

Critical value calculator - student's t-distribution.

This statistical calculator allows you to calculate the critical value corresponding to the Student's t-distribution, you can also see the result in a graph through our online graph generator and if you wish you can download the graph. Just enter the significance value (alpha), degrees of freedom, and left, right, or both tails.

Critical value result

P-value calculator - student's t distribution.

Use our online statistical calculator to calculate the p-value of the Student's t-distribution. You just need to enter the t-value and degrees of freedom and specify the tail. In addition to the p-value, you can get and download the graph created with our graph generator

p-value result

One sample t-test calculator.

The one sample t-test is a statistical hypothesis test calculator, use our calculator to check if you get a statistically significant result or not. To obtain it, fill in the corresponding fields and you will obtain the value of the t-score, p-value, critical value, and the degrees of freedom. You can also download a graph that will display your results in the form of the Student's t-distribution.

T-score result

Two sample t-test calculator.

To determine whether or not the means of two groups are equal, you can use our two-sample t-test calculator that applies the t-test. The results are displayed in a Student's t-distribution plot that you can download. To complete the form, you must include information for both groups, including the mean, standard deviation, sample size, significance level,and whether the test is left, right, or two-tailed.

Common questions related to the Student's t-distribution

In this section, we will try to address the most frequently asked questions about the Student's t-distribution. To give you a fundamental and complementary understanding, we will try to dive into the underlying ideas of the t-distribution. The approach we want to take is to answer the most common questions from students with relevant information. Let's tackle problems simply and offer short and understandable solutions.

Questions related to the student's t-distribution

The formula in relation to the probability density function (pdf) for Student's t-distribution, is given as follows:

Where: π is the pi (approximately 3.14), ν correspond to the degrees of freedom, and Γ is the Euler Gamma function.

A distribution of mean estimates derived from samples taken from a population is what is, by definition, the Student's t-distribution. The t-distribution, commonly known as the Student's t-distribution, is a type of symmetric bell-shaped distribution, it has a lower height but a wider spread than the normal distribution. It is symmetric around 0, but the t-distribution has a wider spread than the typical normal distribution curve, or put another way, the t-distribution has a high standard deviation. The variability of individual observations around their mean is measured by a standard deviation. The degrees of freedom (df) are n - 1. So, df is equal to n – 1, where n is the sample size. The degrees of freedom affect the shape of each t distribution curve.

When the sample size is less than 30 and the population standard deviation is unknown, the t-distribution is utilized in hypothesis testing. It is helpful when the sample size is relatively small or the population standard deviation is unknown. It resembles the normal distribution more closely as sample size grows.

A statistical metric known as the standard deviation is used to quantify the distances between each observation and the mean in a set of data. The standard deviation calculates the degree of dispersion or variability. In other words, it's used to calculate how much a random variable deviates from the mean.

The t-value and t-score have the same meanings. It is one of the relative position measurements. By definition, a value of t defines the location of a continuous random variable, X, in relation to the number of standard deviations from the mean.

The significance level is a point in the normal distribution that must be understood in order to either reject or fail to reject the null hypothesis and to assess whether or not the results are statistically significant. If you decide to make use of our t distribution calculator , you must enter the alpha value corresponding to the significance level. The most common alpha values are 0.1, 0.05 or 0.01. Generally, the most common confidence intervals are: 90%, 95% and 99% (1 − α is the confidence level).

The p-value is a probability with a value ranging from 0 to 1. It is used to test a hypothesis. As an example, in some experiment, we choose the significance level value as 0.05, in this case, the alternative hypothesis is more likely to be supported by stronger evidence when the p-value is less than 0.05 (p-value < 0.05), in case the p-value is high (p-value > 0.05), the probability of accepting the null hypothesis is also high.

The z and t distributions are symmetric and bell-shaped. However, what most characterizes the t distribution are its tails, since they are heavier than in the normal distribution. Furthermore, it can be seen that there are more values in the t-distribution located at the ends of the tail instead of the center of the distribution. You must have the population standard deviation to use the standard normal or z distribution. On the other hand, one of the important conditions for adopting the t distribution is that the population variance is unknown

The t-test , it is a parametric comparison test, is used if the means of two samples are compared using a hypothesis test, if they are independent, from two separate samples, or dependent, a sample evaluated at two different times. The procedure is carried out to evaluate if the differences between the means are significant, determining that they are not due to chance.

To interpret the results of a t-test, you can compare the t-score to the critical value and consider the p-value. A high t-score and low p-value indicate that there is a statistically significant difference between the two means, while a low t-score and high p-value indicate that the difference is not statistically significant. The degrees of freedom and the significance level (alpha) also play a role in determining the critical value and the p-value.

A one sample t-test is a statistical procedure used to test whether the mean of a single sample is significantly different from a hypothesized mean. It is used to determine whether the sample comes from a population with a mean that is different from the hypothesized mean. To perform a one sample t-test using a calculator, you need to input the following information: The sample data, including the mean and standard deviation. The hypothesized mean. The significance level (alpha). The type of tail (left, right, or two-tailed). The calculator will then calculate the t-score and p-value based on this information, and will also provide the critical value and degrees of freedom. To interpret the results, you can compare the t-score to the critical value and consider the p-value. If the t-score is greater than the critical value and the p-value is less than the significance level, you can reject the null hypothesis and conclude that the sample mean is significantly different from the hypothesized mean. If the t-score is less than the critical value or the p-value is greater than the significance level, you cannot reject the null hypothesis and must conclude that the sample mean is not significantly different from the hypothesized mean.

A two-sample t-test is a statistical procedure used to determine whether there is a significant difference between the means of two groups. It is often used to compare the means of two groups in order to determine whether a difference exists between them. For example, a researcher might use a two-sample t-test to determine whether there is a significant difference in the average scores on a test between males and females, or between two different treatment groups in a medical study. The t-test is based on the t-statistic , which is calculated from the sample data and represents the difference between the two groups in relation to the variation within the groups. The t-test is used to determine whether this difference is statistically significant, meaning that it is unlikely to have occurred by chance.

Teach yourself statistics

T Distribution Calculator

The t distribution calculator makes it easy to compute the cumulative probability associated with a t score or with a sample mean. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems .

To learn more about Student's t distribution, go to Stat Trek's tutorial on the t distribution .

In the dropdown box, select the statistic of interest.
Enter a value for degrees of freedom.
Enter a value for all but one of the remaining textboxes.
Click the Calculate button to compute a value for the blank textbox.

Note : Both the t distribution and the standard normal distribution assume that observations are normally distributed in the population . And as sample size increases, the t distribution becomes increasingly similar to the standard normal distribution. So when would a researcher choose the t distribution over the standard normal distribution? A common rule of thumb is to choose the t distribution when (1) the sample size is small and/or (2) the population standard deviation is unknown.

Frequently-Asked Questions

Instructions: To find the answer to a frequently-asked question, simply click on the question.

Which statistic should I use - the t score or the mean score"?

The t distribution calculator accepts two statistics as input: a t score or a sample mean. Choose the option that is easiest. Here are some things to consider.

t = [ x - μ ] / [ s / sqrt( n ) ]

If you choose to work with the sample mean, you can avoid the "transformation" step. But you will need to provide additional input in the form of the population mean and/or the sample standard deviation.

For an example that uses the t score, see Sample Problem 1 . For an example that uses the sample mean, see Sample Problem 2

What are degrees of freedom?

Degrees of freedom can be described as the number of scores that are free to vary. For example, suppose you tossed three dice. The total score adds up to 12. If you rolled a 3 on the first die and a 5 on the second, then you know that the third die must be a 4 (otherwise, the total would not add up to 12). In this example, 2 die are free to vary while the third is not. Therefore, there are 2 degrees of freedom.

In many situations, the degrees of freedom are equal to the number of observations minus one. Thus, if the sample size were 20, there would be 20 observations; and the degrees of freedom would be 20 minus 1 or 19.

What is a standard deviation?

The standard deviation is a numerical value used to indicate how widely individuals in a group vary. It is a measure of the average distance of individual observations from the group mean.

What is a t statistic?

A t statistic is a statistic whose values are given by

t = [ x - μ> ] / [ s / sqrt( n ) ]

where x is the sample mean, μ is the population mean, s is the standard deviation of the sample, n is the sample size, and t is the t statistic.

What is a population mean?

A mean score is an average score. It is the sum of individual scores divided by the number of individuals. A population mean is the mean score of a population .

What is a sample mean?

A mean score is an average score. It is the sum of individual scores divided by the number of individuals. A sample mean is the mean score of a sample .

What is a probability?

A probability is a number expressing the chances that a specific event will occur. This number can take on any value from 0 to 1. A probability of 0 means that there is zero chance that the event will occur; a probability of 1 means that the event is certain to occur. Numbers between 0 and 1 quantify the uncertainty associated with the event. For example, the probability of a coin flip resulting in Heads (rather than Tails) would be 0.50. Fifty percent of the time, the coin flip would result in Heads; and fifty percent of the time, it would result in Tails.

What is a cumulative probability?

A cumulative probability is a sum of probabilities. In connection with the t distribution calculator, a cumulative probability refers to the probability that a t score or a raw score will be less than or equal to a specified value.

Suppose, for example, that we sample 100 first-graders. If we ask about the probability that the average first grader weighs exactly 70 pounds, we are asking about a simple probability - not a cumulative probability.

But if we ask about the probability that average weight is less than or equal to 70 pounds, we are really asking about a sum of probabilities (i.e., the probability that the average weight is exactly 70 pounds plus the probability that it is 69 pounds plus the probability that it is 68 pounds, etc.). Thus, we are asking about a cumulative probability.

Note: The t distribution calculator only reports cumulative probabilities (e.g., the probability that a t score is less than or equal to a specified value.)

Sample Problems

Compute a t statistic, assuming that the breaking strength for the customer's chains is 19,800 pounds.
Determine the cumulative probability for that t statistic.
The t statistic is equal to -0.4276.
The number of degrees of freedom is equal to 13. (In situations like this, the number of degrees of freedom is equal to number of observations minus 1. Hence, the number of degrees of freedom is equal to 14 - 1 or 13.)

Now, we are ready to use the T Distribution Calculator . Since we have already computed the t statistic, we select "t score" from the drop-down box. Then, we enter the t statistic (-0.4276) and the degrees of freedom (13) into the calculator, and hit the Calculate button. The calculator reports that the cumulative probability is 0.338.

Therefore, there is a 33.8% chance that the average breaking strength for the customer's chains will be no more than 19,800 pounds.

The population mean is 20,000.
The standard deviation is 1750.
The sample mean, for which we want to find a cumulative probability, is 19,800.
The number of degrees of freedom is 13. (In situations like this, the number of degrees of freedom is equal to number of observations minus 1. Hence, the number of degrees of freedom is equal to 14 - 1 or 13.)

First, we select "mean score" from the dropdown box in the T Distribution Calculator . Then, we plug our known inputs (degrees of freedom, sample mean, standard deviation, and population mean) into the T Distribution Calculator and hit the Calculate button. The calculator reports that the cumulative probability is 0.338. Thus, there is a 33.8% probability that the average breaking strength for the customer's chains will be 19,800 pounds or less.

Note: This is the same answer that we found in Example 1. However, the approach that we followed in this example may be a little bit easier than the approach that we used in the previous example, since this approach does not require manual computation of a t statistic.

The cumulative probability is 0.90.
The standard deviation is 11.
The sample mean is 115.
The number of degrees of freedom is 14. (In situations like this, the number of degrees of freedom is equal to number of observations minus 1. Hence, the number of degrees of freedom is equal to 15 - 1 or 14.)

First, we select "mean score" from the dropdown box in the T Distribution Calculator . Then, we plug the known inputs (cumulative probability, standard deviation, sample mean, and degrees of freedom) into the calculator and hit the Calculate button. The calculator reports that the population mean is 111.2.

Thus, if the principal's assessment of the IQ of his faculty is correct, we conclude that the average IQ of a teacher in the district is 111.2.

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T Statistic Calculator (T-Value)

Introduction.

The T-statistic, also known as the T-value, is a statistical measure used to assess whether the means of two groups are significantly different from each other. It is a fundamental tool in hypothesis testing, helping researchers determine whether the differences observed in a sample are likely due to random chance or if they represent a real effect. The T-statistic is particularly useful when dealing with small sample sizes where the distribution of data may not be perfectly normal.

In this article, we will explore the T-Statistic Calculator, its formula, how to use it, provide an example, and answer some frequently asked questions to help you understand this crucial statistical concept.

The formula for calculating the T-statistic depends on the context of the analysis. There are two main scenarios: when you have two independent sample groups or when you have one sample group and you want to compare it to a known population mean. Here are the formulas for both scenarios:

1. Independent Sample T-Test:

For comparing the means of two independent sample groups, the T-statistic formula is as follows:

T = (x̄₁ – x̄₂) / (s√((1/n₁) + (1/n₂)))

x̄₁ and x̄₂ are the sample means of the two groups.
s is the pooled standard deviation of the two groups.
n₁ and n₂ are the sample sizes of the two groups.

2. One-Sample T-Test:

For comparing the mean of a single sample group to a known population mean, the T-statistic formula is as follows:

T = (x̄ – μ) / (s / √n)

x̄ is the sample mean.
μ is the known population mean.
s is the sample standard deviation.
n is the sample size.

How to Use?

Using the T-Statistic Calculator is relatively straightforward:

Identify the type of analysis you are conducting: independent sample T-test or one-sample T-test.
For the independent sample T-test, you need data from two separate groups.
For the one-sample T-test, you need data from a single group and a known population mean for comparison.
For the independent sample T-test, input the means, standard deviations, and sample sizes of both groups.
For the one-sample T-test, input the sample mean, known population mean, sample standard deviation, and sample size.
Click the “Calculate” button.
The calculator will provide you with the T-statistic value.
Compare the calculated T-statistic to a critical value from the T-distribution table or use it to calculate a p-value.

Let’s walk through an example of a one-sample T-test using the T-Statistic Calculator:

Suppose you are a manufacturer of light bulbs, and you claim that your bulbs last, on average, 1200 hours. You want to test this claim using a sample of 30 light bulbs, and you find that the sample has a mean lifespan of 1150 hours with a standard deviation of 100 hours.

Identify the type of analysis: one-sample T-test.
Sample mean ( x̄ ): 1150 hours
Known population mean ( μ ): 1200 hours
Sample standard deviation ( s ): 100 hours
Sample size ( n ): 30 bulbs
Enter the values into the T-Statistic Calculator.
Click “Calculate.”
The calculator provides you with the T-statistic value, let’s say it’s -2.0.
You can now compare this T-statistic to a critical value or calculate a p-value. In this case, you might find that the T-statistic corresponds to a p-value of 0.029.

Q1: What is the T-distribution?

The T-distribution is a probability distribution used in hypothesis testing when the sample size is small, and the population standard deviation is unknown. It resembles a normal distribution but has heavier tails.

Q2: What is a p-value, and how is it related to the T-statistic?

The p-value is a probability that measures the evidence against a null hypothesis. In T-testing, a smaller p-value suggests stronger evidence against the null hypothesis. You can calculate the p-value using the T-statistic and degrees of freedom.

Q3: What is the significance level (alpha) in hypothesis testing?

The significance level (alpha) is the threshold value used to determine the statistical significance of results. Common choices for alpha are 0.05 and 0.01. If the p-value is less than alpha, you reject the null hypothesis.

Conclusion:

The T-Statistic Calculator, also known as the T-value calculator, is an essential tool in statistical analysis, helping researchers assess the significance of differences between sample means or compare a sample mean to a known population mean. Understanding how to calculate and interpret the T-statistic is crucial for making informed decisions in various fields, including science, engineering, and business. By following the provided formula and guidelines, you can use this calculator to perform hypothesis tests and draw meaningful conclusions from your data.

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Here you can find out everything about location parameters and dispersion parameters and how you can describe and clearly present your data using characteristic values.

Hypothesis Test

Here you will find everything about hypothesis testing: One sample t-test , Unpaired t-test , Paired t-test and Chi-square test . You will also find tutorials for non-parametric statistical procedures such as the Mann-Whitney u-Test and Wilcoxon-Test . mann-whitney-u-test and the Wilcoxon test

The regression provides information about the influence of one or more independent variables on the dependent variable. Here are simple explanations of linear regression and logistic regression .

Correlation

Correlation analyses allow you to analyze the linear association between variables. Learn when to use Pearson correlation or Spearman rank correlation . With partial correlation , you can calculate the correlation between two variables to the exclusion of a third variable.

Partial Correlation

The partial correlation shows you the correlation between two variables to the exclusion of a third variable.

Levene Test

The Levene Test checks your data for variance equality. Thus, the levene test is used as a prerequisite test for many hypothesis tests .

The p-value is needed for every hypothesis test to be able to make a statement whether the null hypothesis is accepted or rejected.

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DATAtab provides you with tables with distributions and helpful explanations of the distribution functions. These include the Table of t-distribution and the Table of chi-squared distribution

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With a contingency table you can get an overview of two categorical variables in the statistics.

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In an equivalence trial, the statistical test aims at showing that two treatments are not too different in characteristics and a non-inferiority trial wants to show that an experimental treatment is not worse than an established treatment.

If there is a clear cause-effect relationship between two variables, then we can speak of causality. Learn more about causality in our tutorial.

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Multicollinearity is when two or more independent variables have a high correlation.

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Learn how to calculate the effect size for the t-test for independent samples.

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On DATAtab, Cohen's Kappa can be easily calculated online in the Cohen’s Kappa Calculator . there is also the Fleiss Kappa Calculator . Of course, the Cronbach's alpha can also be calculated in the Cronbach's Alpha Calculator .

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Repeated measures ANOVA tests whether there are statistically significant differences in three or more dependent samples.

Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net

Hypothesis Test Graph Generator

Note: After clicking "Draw here", you can click the "Copy to Clipboard" button (in Internet Explorer), or right-click on the graph and choose Copy. In your Word processor, choose Paste-Special from the Edit menu, and select "Bitmap" from the choices

Note: This creates the graph based on the shape of the normal curve, which is a reasonable approximation to the t-distribution for a large sample size. These graphs are not appropriate if you are doing a t-distribution with small sample size (less than 30).

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T test calculator

A t test compares the means of two groups. There are several types of two sample t tests and this calculator focuses on the three most common: unpaired, welch's, and paired t tests. Directions for using the calculator are listed below, along with more information about two sample t tests and help on which is appropriate for your analysis. NOTE: This is not the same as a one sample t test; for that, you need this One sample t test calculator .

1. Choose data entry format

Caution: Changing format will erase your data.

2. Choose a test

Help me choose

3. Enter data

Help me arrange the data

4. View the results

What is a t test.

A t test is used to measure the difference between exactly two means. Its focus is on the same numeric data variable rather than counts or correlations between multiple variables. If you are taking the average of a sample of measurements, t tests are the most commonly used method to evaluate that data. It is particularly useful for small samples of less than 30 observations. For example, you might compare whether systolic blood pressure differs between a control and treated group, between men and women, or any other two groups.

This calculator uses a two-sample t test, which compares two datasets to see if their means are statistically different. That is different from a one sample t test , which compares the mean of your sample to some proposed theoretical value.

The most general formula for a t test is composed of two means (M1 and M2) and the overall standard error (SE) of the two samples:

See our video on How to Perform a Two-sample t test for an intuitive explanation of t tests and an example.

How to use the t test calculator

Choose your data entry format . This will change how section 3 on the page looks. The first two options are for entering your data points themselves, either manually or by copy & paste. The last two are for entering the means for each group, along with the number of observations (N) and either the standard error of that mean (SEM) or standard deviation of the dataset (SD) standard error. If you have already calculated these summary statistics, the latter options will save you time.
Choose a test from the three options: Unpaired t test, Welch's unpaired t test, or Paired t test. Use our Ultimate Guide to t tests if you are unsure which is appropriate, as it includes a section on "How do I know which t test to use?". Notice not all options are available if you enter means only.
Enter data for the test, based on the format you chose in Step 1.
Click Calculate Now and View the results. All options will perform a two-tailed test .

Performing t tests? We can help.

Common t test confusion

In addition to the number of t test options, t tests are often confused with completely different techniques as well. Here's how to keep them all straight.

Correlation and regression are used to measure how much two factors move together. While t tests are part of regression analysis, they are focused on only one factor by comparing means in different samples.

ANOVA is used for comparing means across three or more total groups. In contrast, t tests compare means between exactly two groups.

Finally, contingency tables compare counts of observations within groups rather than a calculated average. Since t tests compare means of continuous variable between groups, contingency tables use methods such as chi square instead of t tests.

Assumptions of t tests

Because there are several versions of t tests, it's important to check the assumptions to figure out which is best suited for your project. Here are our analysis checklists for unpaired t tests and paired t tests , which are the two most common. These (and the ultimate guide to t tests ) go into detail on the basic assumptions underlying any t test:

Exactly two groups
Sample is normally distributed
Independent observations
Unequal or equal variance?
Paired or unpaired data?

Interpreting results

The three different options for t tests have slightly different interpretations, but they all hinge on hypothesis testing and P values. You need to select a significance threshold for your P value (often 0.05) before doing the test.

While P values can be easy to misinterpret , they are the most commonly used method to evaluate whether there is evidence of a difference between the sample of data collected and the null hypothesis. Once you have run the correct t test, look at the resulting P value. If the test result is less than your threshold, you have enough evidence to conclude that the data are significantly different.

If the test result is larger or equal to your threshold, you cannot conclude that there is a difference. However, you cannot conclude that there was definitively no difference either. It's possible that a dataset with more observations would have resulted in a different conclusion.

Depending on the test you run, you may see other statistics that were used to calculate the P value, including the mean difference, t statistic, degrees of freedom, and standard error. The confidence interval and a review of your dataset is given as well on the results page.

Graphing t tests

This calculator does not provide a chart or graph of t tests, however, graphing is an important part of analysis because it can help explain the results of the t test and highlight any potential outliers. See our Prism guide for some graphing tips for both unpaired and paired t tests.

Prism is built for customized, publication quality graphics and charts. For t tests we recommend simply plotting the datapoints themselves and the mean, or an estimation plot . Another popular approach is to use a violin plot, like those available in Prism.

For more information

Our ultimate guide to t tests includes examples, links, and intuitive explanations on the subject. It is quite simply the best place to start if you're looking for more about t tests!

If you enjoyed this calculator, you will love using Prism for analysis. Take a free 30-day trial to do more with your data, such as:

Clear guidance to pick the right t test and detailed results summaries
Custom, publication quality t test graphics, violin plots, and more
More t test options, including normality testing as well as nested and multiple t tests
Non-parametric test alternatives such as Wilcoxon, Mann-Whitney, and Kolmogorov-Smirnov

Check out our video on how to perform a t test in Prism , for an example from start to finish!

Remember, this page is just for two sample t tests. If you only have one sample, you need to use this calculator instead.

We Recommend:

Analyze, graph and present your scientific work easily with GraphPad Prism. No coding required.

T-Value Calculator / Critical Value Calculator

A T value is the “cut-off point” on a T distribution. The T value is almost the same with the Z value which is the “cut-off point” on a normal distribution. The only variation between these two is that they have different shapes. Therefore, the values for their cut-off points vary slightly too. When conducting a hypothesis test, you can use the T value to compare against a T score that you’ve calculated. The easiest way to get the T value is by using this T value calculator.

Table of Contents

How to use the T value calculator?

Calculations can be quite intimidating for a lot of people, especially if the formulas have a very complex nature. Fortunately, there are online tools such as this critical value calculator which can do the computations for you. Here are the steps to use this calculator:

First, enter the value for the Degrees of Freedom.
Then, enter the value for the Significance level. This value should be between 0 and 1 only.
After entering these values, the T score calculator will generate the T value (right-tailed) and the T value (two-tailed).

How do you calculate the T value?

There are two main ways you can calculate the T value without using the T value calculator:

Perform the calculation using Excel

You can calculate the T value using Microsoft Excel by combining custom formulas with its built-in functions. Aside from the T value, you can also get other values such as the degrees of freedom, standard deviation, and the means.

Perform the calculation by hand

Start with the value of the sample size then subtract one to get the degrees of freedom.

Select an alpha level. Usually, you would get this value in the problem, but the most common value is 0.05 or 5%.

Determine which distribution table you will use. This will depend on whether you will run a one or two-tailed test. Use the distribution table to find the intersection of the column and the row. Then you can check the correctness of your answer using the T distribution calculator.

What is T in confidence interval?

There are other concepts to learn about when using a T distribution calculator one of which is the T value in confidence interval. This is a type of function that falls under the statistical functions category.

Use this function to calculate the confidence value which you can use to build the confidence interval. This is very useful for population means for sample size and supplied probability. It’s also very useful when you’re trying to determine the T value for a confidence interval of 95. The T in confidence interval has the following formula:

T Confidence Interval Formula = CONFIDENCE.T(alpha,standard_dev,size)

alpha refers to the significance level you use when computing the confidence level

standard_dev refers to the data range’s population standard deviation

size refers to the standard size

What does T score mean in statistics?

In statistics, the T score is inextricably linked to the P score. When performing a T-test, it means that you’re attempting to obtain evidence of a significant variation between a hypothesized value and a population means or between two population means.

In statistics, you use the T value or T score to measure how big the difference is in relation to the variation in your data sample. In other words, the T score is the difference which you’ve calculated, and you represent this in units of standard error.

You can calculate the T score in the output from a single sample taken from the whole population. If you take recurring samples of random data from a single population, you will obtain T scores which are slightly different every time. This is because of a random sampling error. This isn’t really a mistake, it’s simply a random variation that you would expect in the data.

But when you think about it, how different should the T scores be from the random samples you take from just one population? Also, how does the T score from the sample data compare to the T scores you’re expecting? To answer these questions, you can use a T-distribution .

T scores with higher magnitudes whether positive or negative, aren’t very likely. This is because the far right and left tails of the distribution curves correspond to situations where you obtain extreme T scores which are very far from zero. On the other hand, if you have a P score that’s very low, you can reject the null hypothesis. Also, you can conclude that, in fact, there’s a difference that’s statistically significant.

Going back to the link between the T score and P score , we’ve mentioned that both of these are inextricably linked. You can consider them as various ways you can use to quantify your results’ extremeness under a null hypothesis. Because of their link, it’s not possible to change one of the values without also altering the other. If you have a large absolute T score value, you would have a smaller P score value. Also, you would have greater evidence against the nullity of your hypothesis.

What is the critical value for the test statistic?

We define a critical value for the test statistic using the context of probability and population distribution. It can be mathematically expressed using the following formula:

Pr[X <= critical value] = probability

Pr refers to the probability calculation

X refers to the population observations

critical value refers to the calculated critical value

probability refers to the selected probability

You can calculate critical values using a critical value calculator. You can also perform the calculation using the mathematical formula above. For the distributions which are very common, you can’t calculate the value analytically. Instead, you can estimate the value through numerical methods.

You can also use critical values in statistical significance testing. We often express the probability as the “significance” with the Greek letter alpha as the symbol. Use the standard alpha values which you’re computing critical values. These values provide an equivalent and alternative way for you to interpret the hypothesis of your statistical tests.

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Student t-Value Calculator Online

Student t-value calculator.

In order to calculate the Student T Value for any degrees of freedom and given probability. The calculator will return Student T Values for one tail (right) and two tailed probabilities. Please input degrees of freedom and probability level and then click “CALCULATE”

Descriptive Statistics

Hypothesis test, independent t-test calculator.

To calculate an independent t-Test online just select one metric Variable and one nominal Variable with two values.

If you want to use your own data, just copy your data into the upper table and make sure that the variable name is in the first row. The results of independent t tests are then displayed clearly.

At the beginning of the independent t-test calculator you can choose what your alternative hypothesis is. Then you can test the assumptions for the t-test and you will get the null and alternative hypotheses. Then you get the relative results.

Independent t-Test

An independent t-test is a statistical hypothesis test that is used to determine if there is a significant difference between the means of two independent groups or conditions. It is based on the t-distribution, which is a probability distribution that takes into account the sample size and the variability of the data.

The t-test works by comparing the t-value, which is a measure of the difference between the means, to a critical value based on the desired level of significance (usually denoted as α). The critical value is determined by the degrees of freedom, which depend on the sample sizes of the groups being compared.

If your data are not normally distributed and therefore do not meet the requirements for the independent t-test, you can simply calculate the Mann-Whitney U test online .

Independent T-test Hypotheses

In the independent t-test (also known as the independent samples t-test or two-sample t-test), we compare the means of two independent groups to see if there is a statistically significant difference between them.

The hypotheses for the independent t-test are:

Null Hypothesis (H 0 ): The population means of the two groups are equal. This suggests that there is no significant difference between the two group means. [ H 0 : μ 1 = μ 2 ]
Two-tailed test: We are interested in any difference between the group means, but we don't specify a direction. [ H a : μ 1 ≠ μ 2 ]
One-tailed test (upper-tailed): We are specifically interested in whether the mean of group 1 is greater than the mean of group 2. [ H a : μ 1 > μ 2 ]
One-tailed test (lower-tailed): We are specifically interested in whether the mean of group 1 is less than the mean of group 2. [ H a : μ 1 2 ]

When conducting an independent t-test, you'd choose the form of the alternative hypothesis based on your research question or specific hypothesis. After performing the test, if the test statistic (t-value) is found to be in the critical region (typically using a significance level like 0.05), you'd reject the null hypothesis in favor of the alternative hypothesis. If the t-value is not in the critical region, you'd fail to reject the null hypothesis, indicating that you didn't find enough evidence to suggest a significant difference between the two group means.

How to calulate an independent t-test

Formulate the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis typically assumes that there is no significant difference between the means of the two groups, while the alternative hypothesis assumes that there is a significant difference.
Collect data from the two groups being compared, ensuring that the samples are independent and representative of the populations being studied.
Calculate the sample means (x̄1 and x̄2), the sample standard deviations (s1 and s2), and the sample sizes (n1 and n2) for each group.
Compute the t-value using the formula: t = (x̄1 - x̄2) / sqrt((s1^2/n1) + (s2^2/n2))
Determine the degrees of freedom for the t-distribution, which is calculated using the formula: df = n1 + n2 - 2
Compare the calculated t-value to the critical value from the t-distribution table (or use statistical software) at the desired level of significance (α).
If the calculated t-value exceeds the critical value (i.e., it falls within the rejection region), then the null hypothesis is rejected, indicating that there is a significant difference between the means. If the calculated t-value does not exceed the critical value (i.e., it falls within the non-rejection region), then the null hypothesis is not rejected, suggesting that there is no significant difference between the means.

The p-value is important to reported in a independent samples t-tests, representing the probability of obtaining a difference as extreme as the observed one, assuming the null hypothesis is true. If the p-value is below the chosen significance level (α), typically 0.05, then the null hypothesis is rejected.

Calculate Independent t-test online

Therefore, to calculate an independent t-test online, you only need two independent samples. The samples should be normally distributed. Then simply copy the data into the table above, making sure that the first row of variables contains the name! To calculate a paired samples t-test , simply select two metric variables.

Assumptions for Independent Samples T-test

Independence: The observations in each group are independent of each other. This means that the values in one group are not influenced by or related to the values in the other group.

Normally Distributed Data: The data in each group should be approximately normally distributed. If the sample size is large enough (typically >30 observations in each group), the t-test can be robust to moderate deviations from normality. However, for smaller sample sizes, normality assumption becomes more critical.

Homogeneity of Variance: The variances of the two groups should be approximately equal. This assumption is known as homoscedasticity. Unequal variances between groups can affect the accuracy of the t-test.

Calculate the Effect Size for the Independent t-Test

The effect size for an independent t-test is a measure of the magnitude or strength of the difference between the means of two independent groups. It provides information about the practical significance of the observed difference, which can be valuable in addition to the statistical significance determined by the t-test. There are several common effect size measures for independent t-tests, and two of the most widely used ones are Cohen's d and Hedges' g.

Cohen's d is a standardized measure of effect size that expresses the difference between the means in terms of standard deviation units.

The formula for Cohen's d is as follows:

Cohen's d = (M1 - M2) / S_pooled

M1 is the mean of Group 1.
M2 is the mean of Group 2.
S_pooled is the pooled standard deviation, calculated as:

S_pooled = √((s1^2 + s2^2) / 2)

Where s1 and s2 are the standard deviations of Group 1 and Group 2, respectively.

Hedges' g is similar to Cohen's d but includes a correction factor for small sample sizes. It's useful when you have unequal group sizes or small sample sizes.

The formula for Hedges' g is as follows:

Hedges' g = (M1 - M2) / S_pooled * (1 - (3 / (4 * (N1 + N2) - 9)))

S_pooled is the pooled standard deviation, calculated as described above.
N1 is the sample size of Group 1.
N2 is the sample size of Group 2.

Once you have calculated Cohen's d or Hedges' g, you can interpret the effect size using general guidelines. Cohen's original guidelines suggest that:

Small effect size: d ≈ 0.2
Medium effect size: d ≈ 0.5
Large effect size: d ≈ 0.8

However, it's important to note that the interpretation of effect size can be context-dependent and may vary by field or research area. In some cases, even a small effect size can be practically significant, while in others, a large effect size may not have much practical relevance. Therefore, it's essential to consider the specific context of your study when interpreting the effect size.

Dive Deep into Data with the Independent t-test Calculator

The statistical realm can seem intimidating, with its barrage of numbers and terms. But when armed with the right tools, it becomes a fascinating world to explore. Introducing the independent t-test calculator - a beacon for those navigating the seas of independent data sets!

Decoding the Independent t-test

Before we delve into the intricacies of the calculator, let's decode the basics. The independent t-test, often referred to as the two-sample t-test, determines whether there's a significant difference between the means of two unrelated groups. For instance, comparing the exam scores of students from two separate classes is a fitting application of this test.

Why Choose the Independent t-test Calculator?

Efficiency : Bypass time-consuming manual calculations and swiftly gain insights.
Accuracy : Harness the power of precise algorithms to ensure error-free results.
User-friendly : Designed with simplicity in mind, the tool is accessible for novices and pros alike.

How to Use Our Calculator Efficiently

Enter Data : Populate the given fields with the data sets for both groups.
Select Significance Level : Commonly set at 0.05, but can be tweaked based on your research needs.
Compute : Hit 'Calculate' and witness the tool's prowess.

Deciphering the Output

Post-calculation, you'll be greeted with a t-value and a p-value.

t-value : Signifies the size of the difference relative to the variance within the samples.
p-value : Denotes the likelihood of observing your specific results (or more extreme) under the assumption that the null hypothesis is true.

A p-value smaller than your chosen significance level means you can reject the null hypothesis, pointing towards a significant difference between the two groups.

Final Thoughts

The independent t-test calculator stands as an indispensable ally for all venturing into the comparison of unrelated groups. For students, researchers, and data enthusiasts, this tool demystifies complex calculations, leading to enlightened data-driven decisions.

Embrace the convenience and clarity of the independent t-test calculator and elevate your analytical journey!

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Knowledge Base
T-distribution: What it is and how to use it

T-Distribution | What It Is and How To Use It (With Examples)

Published on August 28, 2020 by Rebecca Bevans . Revised on June 21, 2023.

The t -distribution, also known as Student’s t -distribution, is a way of describing data that follow a bell curve when plotted on a graph, with the greatest number of observations close to the mean and fewer observations in the tails.

It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.

The t-distribution follows a bell curve, with the most likely observations close to the mean and less likely observations in the tails.

In statistics, the t -distribution is most often used to:

Find the critical values for a confidence interval when the data is approximately normally distributed.
Find the corresponding p -value from a statistical test that uses the t -distribution ( t -tests , regression analysis ).

What is a t-distribution?
T-distribution and the standard normal distribution
T-distribution and t-scores

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T -distribution and the standard normal distribution

As the degrees of freedom (total number of observations minus 1) increases, the t -distribution will get closer and closer to matching the standard normal distribution, a.k.a. the z -distribution, until they are almost identical.

Above 30 degrees of freedom, the t -distribution roughly matches the z -distribution. Therefore, the z -distribution can be used in place of the t -distribution with large sample sizes.

The z -distribution is preferable over the t -distribution when it comes to making statistical estimates because it has a known variance. It can make more precise estimates than the t -distribution, whose variance is approximated using the degrees of freedom of the data.

Student’s t-distribution at 1, 3, 8, and 20 degrees of freedom, and compared to the z-distribution.

T -distribution and t -scores

A t -score is the number of standard deviations from the mean in a t -distribution. You can typically look up a t -score in a t -table , or by using an online t -score calculator.

In statistics, t -scores are primarily used to find two things:

The upper and lower bounds of a confidence interval when the data are approximately normally distributed.
The p -value of the test statistic for t -tests and regression tests.

T -scores and confidence intervals

Confidence intervals use t -scores to calculate the upper and lower bounds of the prediction interval. The t -score used to generate the upper and lower bounds is also known as the critical value of t , or t *.

Using a two-tailed t -test, you generate an estimate of the difference between the two classes and a confidence interval around that estimate. From the t -test you find the difference in average score between class 1 and class 2 is 4.61, with a 95% confidence interval of 3.87 to 5.35.

Because the confidence interval does not cross zero, and is in fact quite far from zero, it is unlikely that this difference in test scores could have occurred under the null hypothesis of no difference between groups.

A t-distribution showing the upper and lower bounds of a 95% confidence interval.

T -scores and p -values

Statistical tests generate a test statistic showing how far from the null hypothesis of the statistical test your data is. They then calculate a p -value that describes the likelihood of your data occurring if the null hypothesis were true.

The test statistic for t -tests and regression tests is the t -score. While most statistical programs will automatically calculate the corresponding p -value for the t -score, you can also look up the values in a t -table, using your degrees of freedom and t -score to find the p -value.

The t -score which generates a p -value below your threshold for statistical significance is known as the critical value of t , or t *.

The degrees of freedom is 38 (n–1 for each group). Looking this up in a t -table (or calculating it in your favorite stats program) you find a p -value < 0.001.

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.

Student’s t table
Descriptive statistics
Measures of central tendency
Correlation coefficient

Methodology

Cluster sampling
Stratified sampling
Types of interviews
Cohort study
Thematic analysis

Research bias

Implicit bias
Cognitive bias
Survivorship bias
Availability heuristic
Nonresponse bias
Regression to the mean

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The t -distribution is a way of describing a set of observations where most observations fall close to the mean , and the rest of the observations make up the tails on either side. It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.

The t -distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation .

The t -distribution gives more probability to observations in the tails of the distribution than the standard normal distribution (a.k.a. the z -distribution).

In this way, the t -distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance , you will need to include a wider range of the data.

A t -score (a.k.a. a t -value) is equivalent to the number of standard deviations away from the mean of the t -distribution .

The t -score is the test statistic used in t -tests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a t -distribution.

A test statistic is a number calculated by a statistical test . It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.

The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.

A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval , or which defines the threshold of statistical significance in a statistical test. It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data (i.e. 90%, 95%, 99%).

If you are constructing a 95% confidence interval and are using a threshold of statistical significance of p = 0.05, then your critical value will be identical in both cases.

Cite this Scribbr article

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9.4: Distribution Needed for Hypothesis Testing

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Earlier in the course, we discussed sampling distributions. Particular distributions are associated with hypothesis testing. Perform tests of a population mean using a normal distribution or a Student's $t$-distribution. (Remember, use a Student's $t$-distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal.) We perform tests of a population proportion using a normal distribution (usually $n$ is large or the sample size is large).

If you are testing a single population mean, the distribution for the test is for means :

\[\bar{X} - N\left(\mu_{x}, \frac{\sigma_{x}}{\sqrt{n}}\right)\]

The population parameter is $\mu$. The estimated value (point estimate) for $\mu$ is $\bar{x}$, the sample mean.

If you are testing a single population proportion, the distribution for the test is for proportions or percentages:

\[P' - N\left(p, \sqrt{\frac{p-q}{n}}\right)\]

The population parameter is $p$. The estimated value (point estimate) for $p$ is $p′$. $p' = \frac{x}{n}$ where $x$ is the number of successes and n is the sample size.

Assumptions

When you perform a hypothesis test of a single population mean $\mu$ using a Student's $t$-distribution (often called a $t$-test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation. (Note that if the sample size is sufficiently large, a $t$-test will work even if the population is not approximately normally distributed).

When you perform a hypothesis test of a single population mean $\mu$ using a normal distribution (often called a $z$-test), you take a simple random sample from the population. The population you are testing is normally distributed or your sample size is sufficiently large. You know the value of the population standard deviation which, in reality, is rarely known.

When you perform a hypothesis test of a single population proportion $p$, you take a simple random sample from the population. You must meet the conditions for a binomial distribution which are: there are a certain number $n$ of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success $p$. The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities $np$ and $nq$ must both be greater than five $(np > 5$ and $nq > 5)$. Then the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with $\mu = p$ and $\sigma = \sqrt{\frac{pq}{n}}$. Remember that $q = 1 – p$.

In order for a hypothesis test’s results to be generalized to a population, certain requirements must be satisfied.

When testing for a single population mean:

A Student's $t$-test should be used if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation.
The normal test will work if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with a known standard deviation.

When testing a single population proportion use a normal test for a single population proportion if the data comes from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of successes and the mean number of failures satisfy the conditions: $np > 5$ and $nq > 5$ where $n$ is the sample size, $p$ is the probability of a success, and $q$ is the probability of a failure.

Formula Review

If there is no given preconceived $\alpha$, then use $\alpha = 0.05$.

Types of Hypothesis Tests

Single population mean, known population variance (or standard deviation): Normal test .
Single population mean, unknown population variance (or standard deviation): Student's $t$-test .
Single population proportion: Normal test .
For a single population mean , we may use a normal distribution with the following mean and standard deviation. Means: $\mu = \mu_{\bar{x}}$ and $\\sigma_{\bar{x}} = \frac{\sigma_{x}}{\sqrt{n}}$
A single population proportion , we may use a normal distribution with the following mean and standard deviation. Proportions: $\mu = p$ and $\sigma = \sqrt{\frac{pq}{n}}$.
It is continuous and assumes any real values.
The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
It approaches the standard normal distribution as $n$ gets larger.
There is a "family" of $t$-distributions: every representative of the family is completely defined by the number of degrees of freedom which is one less than the number of data items.

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Single-Sample Confidence Interval Calculator

This simple confidence interval calculator uses a t statistic and sample mean ( M ) to generate an interval estimate of a population mean (μ).

The formula for estimation is:

M = sample mean t = t statistic determined by confidence level s M = standard error = √( s 2 / n )

As you can see, to perform this calculation you need to know your sample mean, the number of items in your sample, and your sample's standard deviation. (If you need to calculate mean and standard deviation from a set of raw scores, you can do so using our descriptive statistics tools .)

The Calculation

Please enter your data into the fields below, select a confidence level (the calculator defaults to 95%), and then hit Calculate. Your result will appear at the bottom of the page.

Please enter your values above, and then hit the calculate button.

T Distribution Graph Generator

Instructions: Make a t-distribution graph using the form below. Please type the number of degrees of freedom associated to the t-distribution, and provide details about the event you want to graph:

More About this T-distribution Graph Maker

The t-distribution is a type of continuous probability distribution that takes random values on the whole real line. The main properties of the t-distribution are:

It is continuous (and as a consequence, the probability of getting any single, specific outcome is zero)
It is "bell shaped", in the same way the normal curves are bell-shaped
It is determined by one parameter: the number of degrees of freedom (df). For one sample, the number of degrees of freedom is df = n - 1, where n is the sample size
It is symmetric with respect to 0
The t-distribution "converges" to the standard normal distribution as the number of degrees of freedom (df) converges to infinity ($+\infty$), in the sense that its shapes resembles more and more that of the standard normal distribution when the number of degrees of freedom becomes larger and larger.

In order to compute probabilities associated to the t-distribution we can either use specialized software such as Excel, etc, or we can use t-distribution tables (normally available at college statistics textbooks. The use of the t-distribution arises when performing hypothesis testing (for the case when the population standard deviation is not known).

In case you are rather interested in the normal distribution, you can try our normal distribution graph generator

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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 .

Critical values for symmetric distribution

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.

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Hypothesis Testing Calculator with Steps
Hypothesis Testing Calculator. 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 ...
t-test Calculator
Recall, that in the critical values approach to hypothesis testing, you need to set a significance level, α, before computing the critical values, which in turn give rise to critical regions (a.k.a. rejection regions). Formulas for critical values employ the quantile function of t-distribution, i.e., the inverse of the cdf:. Critical value for left-tailed t-test:
Hypothesis Test Calculator
Calculation Example: There are six steps you would follow in hypothesis testing: Formulate the null and alternative hypotheses in three different ways: H 0: θ = θ 0 v e r s u s H 1: θ ≠ θ 0. H 0: θ ≤ θ 0 v e r s u s H 1: θ > θ 0. H 0: θ ≥ θ 0 v e r s u s H 1: θ < θ 0.
Student's t-distribution calculator with graph generator
One sample T-test calculator. The one sample t-test is a statistical hypothesis test calculator, use our calculator to check if you get a statistically significant result or not. To obtain it, fill in the corresponding fields and you will obtain the value of the t-score, p-value, critical value, and the degrees of freedom.
T-Distribution Probability Calculator
Solution: We need to compute \Pr (-1 \leq T \leq 0.8) Pr(−1 ≤ T ≤ 0.8), where T T has a t-distribution with df = 34 df = 34 degrees of freedom. Therefore, the probability is computed as: Use this T-Distribution Probability Calculator toc ompute t-distribution probabilities. Type the degrees of freedom and the probability event.
T Distribution Calculator
First, we select "mean score" from the dropdown box in the T Distribution Calculator. Then, we plug the known inputs (cumulative probability, standard deviation, sample mean, and degrees of freedom) into the calculator and hit the Calculate button. The calculator reports that the population mean is 111.2.
T Statistic Calculator (T-Value)
For the one-sample T-test, input the sample mean, known population mean, sample standard deviation, and sample size. Click the "Calculate" button. The calculator will provide you with the T-statistic value. Compare the calculated T-statistic to a critical value from the T-distribution table or use it to calculate a p-value.
Hypothesis Testing for the Mean: T-test
Using a t distribution table for a 1-tailed test with = 0.10 and d.f.= 17, we find that the critical t value is t = -1.333. This is a left-tailed test, so since -1.381 < -1.333, we are in the rejection region, so reject : 80, and accept the alternative hypothesis : < 80. Conclusions based on P-value: The test p value shown above is p = 0. ...
Online Statistics Calculator: Hypothesis testing, t-test, chi-square
Hypothesis Test. Here you will find everything about hypothesis testing: One sample t-test, Unpaired t-test, Paired t-test and Chi-square test. You will also find tutorials for non-parametric statistical procedures such as the Mann-Whitney u-Test and Wilcoxon-Test. mann-whitney-u-test and the Wilcoxon test
How t-Tests Work: t-Values, t-Distributions, and Probabilities
Hypothesis tests work by taking the observed test statistic from a sample and using the sampling distribution to calculate the probability of obtaining that test statistic if the null hypothesis is correct. In the context of how t-tests work, you assess the likelihood of a t-value using the t-distribution.
Hypothesis Test Graph Generator
Hypothesis Test Graph Generator. Note: After clicking "Draw here", you can click the "Copy to Clipboard" button (in Internet Explorer), or right-click on the graph and choose Copy. In your Word processor, choose Paste-Special from the Edit menu, and select "Bitmap" from the choices. Note: This creates the graph based on the shape of the normal ...
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Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.
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A t test compares the means of two groups. There are several types of two sample t tests and this calculator focuses on the three most common: unpaired, welch's, and paired t tests. Directions for using the calculator are listed below, along with more information about two sample t tests and help on which is appropriate for your analysis. NOTE: This is not the same as a one sample t test; for ...
9.4: A Single Population Mean using the Student t-Distribution
Since the test statistic, $t=1.979$ is greater than the critical value, $\text{CV}=1.8331$, the decision will be to Reject the Null Hypothesis. Conclusion (Same as the $P$-value solution) : At a 5% level of significance, the sample data show sufficient evidence that the mean (average) test score is more than 65, just as the math ...
T-Value Calculator / Critical Value Calculator
Here are the steps to use this calculator: First, enter the value for the Degrees of Freedom. Then, enter the value for the Significance level. This value should be between 0 and 1 only. After entering these values, the T score calculator will generate the T value (right-tailed) and the T value (two-tailed).
Student T-Value Calculator
The calculator will return Student T Values for one tail (right) and two tailed probabilities. Please input degrees of freedom and probability level and then click "CALCULATE". Use this t score calculator to calculate t critical value by confidence level & degree of freedom for the Student's t distribution.
Independent t-Test Calculator
Independent t-test Calculator. ... An independent t-test is a statistical hypothesis test that is used to determine if there is a significant difference between the means of two independent groups or conditions. It is based on the t-distribution, which is a probability distribution that takes into account the sample size and the variability of ...
T-Distribution
T-distribution and t-scores. A t-score is the number of standard deviations from the mean in a t-distribution.You can typically look up a t-score in a t-table, or by using an online t-score calculator.. In statistics, t-scores are primarily used to find two things: The upper and lower bounds of a confidence interval when the data are approximately normally distributed.
Statistical Power Calculator using the t-distribution*
Statistical Power Calculator using the t-distribution*. Interactive calculator for illustrating power of a statistical hypothesis test. alpha α : 0.01 0.2 0 0 0.05 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.2. Difference in means δ : 0.1 2 0 0 0.5 0.1 0.29 0.48 0.67 0.86 1.05 1.24 1.43 1.62 1.81 2. Sample size in each group n : 12 ...
9.4: Distribution Needed for Hypothesis Testing
This page titled 9.4: Distribution Needed for Hypothesis Testing is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. When testing for a single population mean: A ...
Confidence Interval Calculator: Single-Sample T Statistic
This simple confidence interval calculator uses a t statistic and sample mean (M) to generate an interval estimate of a population mean (μ). The formula for estimation is: μ = M ± t ( s M )
T Distribution Graph Generator
The t-distribution is a type of continuous probability distribution that takes random values on the whole real line. The main properties of the t-distribution are: It is continuous (and as a consequence, the probability of getting any single, specific outcome is zero) It is "bell shaped", in the same way the normal curves are bell-shaped.
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?