t-Test Calculator

Fast, accurate and free online testu t student calculator tool running directly in your browser.

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Type of t-test

One trial x̄ vs μ₀ Two trials x̄₁ vs x̄₂ (independent)

Trial data

Mean x̄

Deviation p

Count n

Hypothesis (tail type)

Alternative hypothesis

Bilateral ≠ Lewostronny < Right sided >

Significance level α

Decimal places

Enter the data and click "Calculate t-test".

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Student's t-test calculator - verification of statistical hypotheses for small samples

In scientific research, medical analyses, psychology and marketing, there is often a need to compare average results from two different groups in order to check whether the observed difference is statistically significant or just a coincidence (random error). In situations where we have a small research sample (usually $n < $30) and we do not know the standard deviation in the entire population, the basic statistical tool to verify such assumptions is the **Student's t-test** (t-test). Created in the early 20th century by William Sealy Gosset (writing under the pseudonym "Student"), this test is based on the t-distribution. Our free **online Student's t-test calculator** allows you to quickly and error-free calculations for independent and dependent (related) samples, determining the $t$ statistic, degrees of freedom ($df$) and the value of the test probability $p$ (p-value).

Thanks to this, you will efficiently verify the null hypothesis $H_0$ at the selected significance level $\alpha$ (e.g. 0.05) and draw correct, mathematically proven conclusions from your research.

Types of Student's t-test and their application in research

Depending on the experimental design and method of data collection, the appropriate variant of Student's t-test should be used:

Variant of t-test	Specificity of compared groups	Example of research application	Degrees of freedom ($df$)
For One Sample(One-Sample t-test)	Comparison of the mean from one small group with a pre-established theoretical value (norm).	Checking whether the average reaction time of drivers after consuming the drug differs from the general norm of 1.5 seconds.	$$df = n - 1$$
For independent samples(Two-Sample Independent t-test)	Comparison of means from two separate, independent research groups.	Comparison of the level of job satisfaction among employees of the IT department (group A) and marketing department (group B).	$$df = n_1 + n_2 - 2$$
For dependent/related samples(Paired t-test)	Comparison of mean scores from the same group of subjects at two different times (e.g. before and after the experiment).	Measurement of stress levels in managers before and 2 weeks after training in stress management.	$$df = n - 1$$

Mathematical theory and Student's t-test formulas

Calculating the test statistic involves determining the ratio of the difference between the means to the standard error of this difference. For example, for one sample the formula for the $t$ statistic is:

$$t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$$

Where:

$\bar{X}$ – mean of the research sample.
$\mu_0$ – the hypothetical (reference) mean specified in the null hypothesis $H_0$.
$s$ – standard deviation calculated from the sample.
$n$ – sample size (number of observations).
$s / \sqrt{n}$ – standard error of the sample mean.

The obtained value $t$ is compared with the critical value read from the Student's t-distribution tables for a specific number of degrees of freedom ($df$) and the significance level $\alpha$. If calculated $|t| > t_{\text{crit}}$, we reject the null hypothesis in favor of the alternative hypothesis.

How to perform Student's t-test on a calculator? Instruction

Performing the analysis in our calculator is simple and does not require knowledge of statistical programs (such as SPSS or R):

Select the type of test:Select whether you are analyzing one sample, independent samples or dependent samples.
Enter input:Paste the raw data (numbers separated by commas or spaces) for each group or provide ready-made descriptive statistics (mean, standard deviation and count).
Specify the direction of the test:Indicate whether you are performing a two-sided test (checking for any difference) or one-sided test (checking if one mean is greater/smaller than the other).
Receive results:Click "Calculate". The system will display the value of the $t$ statistic, the degrees of freedom, the $p$ value and the statistical verdict (e.g."The difference is statistically significant at a significance level of 0.05").

Frequently Asked Questions (FAQ)

What is the difference between the Student's t-test for dependent and independent samples?

The independent samples test is used when we compare two completely different groups of people (e.g. women vs. men, control group vs. experimental group). The test for dependent (related) samples is used when we measure the same group of people twice (e.g. weight measurement before and after the diet) or when the tested people are matched in pairs (e.g. twins, married couples).

What are the key theoretical assumptions for conducting the Student's t-test?

The main assumptions are: 1. The dependent variable must be measured on a quantitative scale (interval or ratio). 2. The distribution of the studied variable in the population should be close to normal. 3. In the case of independent samples, homogeneity of variance in both groups is required (which is verified, for example, by Levene's test).

What are degrees of freedom (df) and how are they calculated in a t test?

Degrees of freedom ($df$) is the number of independent pieces of information in the sample that can change freely when estimating statistical parameters. For one-sample t-test and dependent samples $df = n - 1$. For a test for two independent samples $df = n_1 + n_2 - 2$, where $n$ are the sizes of the tested groups.

What does it mean that the variances in both compared groups are homogeneous?

Homogeneity of variance means that the dispersion (diversity) of results around the mean in both groups is similar. If the variances are extremely different (e.g. in one group everyone has similar results, and in the other they are very dispersed), the classic Student's t-test may give erroneous results. Then you need to apply Welch's correction (t-test with Welch's correction), which our calculator does automatically.

How to interpret the p-value result in the Student's t-test?

The $p$ value (p-value) is the probability of obtaining the observed differences assuming that the null hypothesis (no differences) is true. If $p \le 0.05$ (or another selected significance level $\alpha$), it means that the chance of a coincidence is negligible (below 5%) and **we reject the null hypothesis** - the difference between the means is statistically significant. If $p > 0.05$, there is no reason to reject the null hypothesis.