Student’s t-test

The t-test formula helps us to compare the average values of two data sets and determine if they belong to the same population or are they different. The t-score is compared with the critical value obtained from the t-table. The large t-score indicates that the groups are different and a small t-score indicates that the groups are similar.

Contents

Student’s t-test

A t-test (also known as Student’s t-test) is a tool for evaluating the means of one or two populations using hypothesis testing. A t-test may be used to evaluate whether a single group differs from a known value (a one-sample t-test), whether two groups differ from each other (an independent two-sample t-test), or whether there is a significant difference in paired measurements (a paired, or dependent samples t-test). [3]

Here are some common scenarios for its use:

Comparing Two Independent Groups: When you want to compare the means of two different groups, such as test scores of students from two different classes.
Comparing Two Related Groups: When you have paired data, such as measurements taken from the same subjects before and after a treatment (e.g., weight loss before and after a diet).
Small Sample Sizes: When the sample size is small (typically less than 30) and the population standard deviation is unknown.
Normal Distribution: When the data in each group is approximately normally distributed. If the sample size is large, the t-test is robust to violations of normality.
Equality of Variances: When you want to test the means of two groups and can assume that the variances of the two groups are equal (using the equal variance t-test) or when they are not (using the Welch’s t-test).

A Student’s t-test is appropriate when comparing means between groups under certain conditions, especially with smaller sample sizes and unknown population variances.

How are 𝑡-tests used?

First, you define the hypothesis you are going to test and specify an acceptable risk of drawing a faulty conclusion. For example, when comparing two populations, you might hypothesize that their means are the same, and you decide on an acceptable probability of concluding that a difference exists when that is not true. Next, you calculate a test statistic from your data and compare it to a theoretical value from a t- distribution. Depending on the outcome, you either reject or fail to reject your null hypothesis. [3]

t-Test assumptions

While t-tests are relatively robust to deviations from assumptions, t-tests do assume that: [3]

The data are continuous.
The sample data have been randomly sampled from a population.
There is homogeneity of variance (i.e., the variability of the data in each group is similar).
The distribution is approximately normal.

For two-sample t-tests, we must have independent samples. If the samples are not independent, then a paired t-test may be appropriate.

How does a t-test work?

A t-test checks whether two means really differ. To do this, it compares two things: [1]

The size of the difference between the means (the signal), and
The variation or spread of the values within the groups (the noise).

If the difference is large compared to the natural variation, it suggests the groups likely differ in the real world, not just in your sample. The test produces a p-value to help you judge the strength of this evidence.

Types of t-test

There are three main versions of the test depending on what you are comparing: [1]

One-Sample t-test: Compares one group’s mean against a known value or standard. In a one sample t-test, the average mean value of the sample is compared against a known threshold value. This threshold value can be a legal requirement or the existing industry standard. For example, Suppose the average number of grains per barley pod is known to be 24. So, one sample t-test can be used to test if the barley yield differs significantly from the above known value. [2]
Independent Two-Sample t-test: Compares the means of two entirely separate groups, such as a treatment group vs. a control group. Let us suppose, we have 2 different types of barley and we want to compare the yields of both the types. We can do an Independent two sample t-test to know if the yield of one type is significantly better than the other. [2]
Paired Samples t-test: Compares means from the same group at two different times, such as “Before” vs. “After” an intervention. Suppose, a nitrogen fertilizer is used on the same crop and we would like to test if adding the fertilizer had a significant impact on the yield of the crop. Paired sample t-test can be used to test if adding the fertilizer resulted in any significant impact to the crop yield. [2]

References

[1] ⭐ Jesussek, Mathias . “T-Test: A Comprehensive Tutorial for Beginners.” numiqo, January 21, 2026. https://numiqo.com/tutorial/t-test.

[2] Kolamanvitha. “Introduction to Student’s t-Test.” Nerd For Tech, June 9, 2021. https://kolamanvitha.medium.com/introduction-to-students-t-test-7e27098a8fcb.

[3] “The T-Test.” JMP. Accessed June 23, 2026. https://www.jmp.com/en/statistics-knowledge-portal/inferential-statistics/hypothesis-testing/t-test.

Additional Reading

Buchan, Iain. “StatisticalHelp from StatsDirect.” StatisticalHelp, 2025. https://www.statsdirect.com/help/contents.htm.

If you are unsure which statistical method is appropriate to your analysis, you may find the statistical method selection guide helpful. You can install the free statistical tools from www.statsdirect.com.

Dementyev, Artem. “T-Test and Hypothesis Testing (Explained Simply).” Towards Data Science, August 5, 2022. https://towardsdatascience.com/t-test-and-hypothesis-testing-explained-simply-1cff6358633e/.

Student’s t-tests are commonly used in inferential statistics for testing a hypothesis on the basis of a difference between sample means. However, people often misinterpret the results of t-tests, which leads to false research findings and a lack of reproducibility of studies. This problem exists not only among students. Even instructors and “serious” researchers fall into the same trap. To prove my words, I can link this article, but there are others.

Navarro, Danielle. “11.3: The Independent Samples t-Test (Student Test).” LibreTexts, April 27, 2023. https://stats.libretexts.org/Courses/Cerritos_College/Introduction_to_Statistics_with_R/11%3A_Comparing_Two_Means/11.03%3A_The_Independent_Samples_t-test_(Student_Test).

Although the one sample t-test has its uses, it’s not the most typical example of a t-test. A much more common situation arises when you’ve got two different groups of observations. In psychology, this tends to correspond to two different groups of participants, where each group corresponds to a different condition in your study. For each person in the study, you measure some outcome variable of interest, and the research question that you’re asking is whether or not the two groups have the same population mean. This is the situation that the independent samples t-test is designed for.

Plevris, Vagelis. “The Student Who Was Not A Student.” Medium, June 18, 2026. https://vplevris.medium.com/the-student-who-was-not-a-student-ddde2205ebf0.

William Sealy Gosset, a Guinness brewer, revolutionized statistics by addressing the limitations of the normal distribution when handling small sample sizes. Publishing under the pseudonym “Student,” Gosset introduced the t-test to account for increased uncertainty in limited datasets. By incorporating “heavier tails” into his distribution, he taught scientists to prioritize humility over overconfidence, ensuring that industrial and experimental decisions remain grounded in the reality of limited evidence.

“Student’s t-Test.” Wikipedia, June 1, 2026. https://en.wikipedia.org/wiki/Student%27s_t-test.

⭐ “T-test.” GeeksforGeeks, November 25, 2020. https://www.geeksforgeeks.org/data-science/t-test/.

“T Test (Student’s T-Test): Definition and Examples.” Statistics How To. February 28, 2025. https://www.statisticshowto.com/probability-and-statistics/t-test/.

A t-test (also called Student’s t-test) is a statistical method used to assess the difference between the means of exactly two groups. It concentrates on a single numerical variable, rather than focusing on counts or relationships among multiple variables. When analyzing the average of a sample of measurements, t-tests are the most frequently used technique for data evaluation.

“T Test Calculator.” GraphPad. Accessed June 23, 2026. https://www.graphpad.com/quickcalcs/ttest1/.

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.

⭐ “T-Test Formula – Derivation, Examples.” CUEMATH. Accessed June 23, 2026. https://www.cuemath.com/t-test-formula/.

Turney, Shaun. “How to Find Degrees of Freedom.” Scribbr, July 7, 2022. https://www.scribbr.com/statistics/degrees-of-freedom/.

Degrees of freedom, often represented by v or df, is the number of independent pieces of information used to calculate a statistic. It’s calculated as the sample size minus the number of restrictions. Degrees of freedom are normally reported in brackets beside the test statistic, alongside the results of the statistical test.

Turney, Shaun. “Student’s t Table (Free Download).” Scribbr, April 29, 2022. https://www.scribbr.com/statistics/students-t-table/.

Student’s t table is a reference table that lists critical values of t. Student’s t table is also known as the t table, t-distribution table, t-score table, t-value table, or t-test table. A critical value of t defines the threshold for significance for certain statistical tests and the upper and lower bounds of confidence intervals for certain estimates. The critical values of t are calculated from Student’s t distribution. Student’s t distribution is the distribution of the test statistic t. The critical values of t are difficult to calculate by hand, which is why most people use a t table or computer software instead.

“Ultimate Guide to T Tests.” GraphPad. Accessed June 23, 2026. https://www.graphpad.com/guides/the-ultimate-guide-to-t-tests.

Videos

What is a t-test and when is it used? What types of t-tests are there? What are hypotheses and prerequisites in a t-test? How is a t-test calculated and how are the results interpreted? The t-test is a statistical test procedure and checks whether there is a significant difference between the means of two groups. There are three types of t-test: the one-sample t-test, the independent samples t-test and the paired samples t-test. We go through all three types in this video.

Are you relying on the t-test in every situation? It might be time to rethink that approach! In this video, we dive into three specific cases where the t-test can lead you astray, and explain why alternative methods are better suited for these scenarios. Whether you’re in research, data analysis, or just curious about statistical techniques, understanding when not to use the t-test is crucial.

In this video, we will explore the T-Distribution, a fundamental concept in statistics. We will also explain the different types of t-test, such as One Sample t-Test, Paired t-Test.

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⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.

The featured image on this page is from the YouTube video What is Student’s t-test in Statistics ? | Student’s t -distribution ? | Explained with Examples.