Updated Hypothesis Testing Tool

t-Test Calculator

Run one-sample, two-sample (independent) and paired t-tests in a single place. Enter raw data or summary statistics to compute the t-statistic, degrees of freedom, approximate p-values, effect size and a quick decision for your chosen significance level.

One-Sample t-Test Two-Sample (Independent) Paired Samples t-Test Effect Size & p-Values

Calculate t-Statistics, p-Values & Effect Sizes

This t-Test Calculator combines multiple options in one interface. For each test you can either paste raw data or enter summary statistics, choose a null hypothesis, select a one-sided or two-sided alternative, and pick a significance level α. The calculator then returns the t-statistic, degrees of freedom, approximate p-values, Cohen’s d effect size and a simple “reject / do not reject H0” message.

p-values are computed using a normal-based approximation to the t-distribution. For exact values, compare t and df with a t-table or dedicated stats software.

Use the one-sample t-test when you compare a sample mean to a known or hypothesized population mean μ₀. You can either paste raw data or enter summary statistics.

Use the two-sample t-test to compare means from two independent groups. This calculator uses Welch’s t-test (does not assume equal variances) and supports both raw data and summary statistics.

Use the paired t-test when observations come in matched pairs, such as before/after measurements on the same subjects. The test is run on the difference scores D = Xafter − Xbefore.

t-Test Calculator – Complete Guide to One-Sample, Two-Sample and Paired t-Tests

The t-Test Calculator on MyTimeCalculator is designed to make classical hypothesis testing for means quick and transparent. It supports one-sample t-tests, two-sample (independent) t-tests and paired t-tests in a single, consistent interface. You can start from raw data or from summary statistics such as sample size, mean and standard deviation.

All t-tests are based on the Student’s t-distribution, which is used when the population standard deviation is unknown and the sample size is not extremely large. The calculator provides t-statistics, degrees of freedom, approximate p-values, effect size and a basic decision about H0 for your chosen significance level α.

1. Types of t-Tests Covered

The calculator implements three of the most common t-tests used in statistics, data analysis and research:

  • One-sample t-test: compares the mean of a single sample to a known or hypothesized population mean μ₀. Typical use cases include checking whether an average score is different from a benchmark or target.
  • Two-sample t-test (independent samples): compares the means of two independent groups, such as treatment and control, or two different populations. The calculator uses Welch’s t-test by default, which does not assume equal population variances.
  • Paired t-test: compares two related measurements on the same individuals or matched pairs, such as before/after measurements or repeated measures on the same subject.

2. Key t-Test Formulas

Below is a concise summary of the main formulas used by the calculator. In all cases, s denotes the sample standard deviation and n the sample size.

One-sample t-test

t = \(\dfrac{\bar{x} - \mu_0}{s / \sqrt{n}}\),   df = n − 1

Two-sample t-test (Welch’s version)

Let \(\bar{x}_1, s_1, n_1\) be the mean, standard deviation and size of group 1, and \(\bar{x}_2, s_2, n_2\) for group 2. With a hypothesized mean difference Δ₀ = μ₁ − μ₂ under H₀:

t = \(\dfrac{\bar{x}_1 - \bar{x}_2 - \Delta_0}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}\)
df ≈ \(\dfrac{\left(\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}\right)^2}{\dfrac{\left(\dfrac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \dfrac{\left(\dfrac{s_2^2}{n_2}\right)^2}{n_2 - 1}}\)

Paired t-test

For a paired t-test, you first compute the difference scores D = Xafter − Xbefore for each pair. Let \(\bar{d}, s_d, n\) be the mean, standard deviation and number of difference values, and μD,0 be the hypothesized mean difference under H₀ (often 0):

t = \(\dfrac{\bar{d} - \mu_{D,0}}{s_d / \sqrt{n}}\),   df = n − 1

3. Effect Size: Cohen’s d

Alongside significance testing, the calculator also reports Cohen’s d, a standardized effect size that measures how large the difference is in terms of standard deviations:

  • One-sample: d = (\(\bar{x} - \mu_0\)) / s.
  • Two-sample: d is based on the difference of means divided by a pooled or combined standard deviation.
  • Paired: d uses the mean of the difference scores divided by their standard deviation.

While rules of thumb vary, many analysts interpret |d| ≈ 0.2 as a small effect, 0.5 as medium and 0.8 or larger as a large effect. The calculator presents the numeric value so you can apply any interpretation guideline you prefer.

4. How to Use the t-Test Calculator

  1. Choose a test type: select the One-Sample t-Test, Two-Sample t-Test or Paired t-Test tab depending on your data structure and research question.
  2. Set the significance level α and tail type: use the global controls to pick α (for example 0.05) and whether your alternative hypothesis is two-sided (≠) or one-sided (< or >).
  3. Enter raw data or summary statistics: choose the input mode. For raw data, paste values separated by commas, spaces or line breaks. For summary statistics, provide sample sizes, means and standard deviations.
  4. Specify the null hypothesis: enter the hypothesized mean μ₀ for the one-sample test, the mean difference Δ₀ for the two-sample test, or the hypothesized mean difference for the paired test.
  5. Run the calculation: click the appropriate “Compute t-Test” button. The results section will show t, df, approximate p-values and Cohen’s d for your test.
  6. Interpret the decision: if the p-value is less than α, the calculator reports that the result is statistically significant at that level, indicating evidence against H0. You can still cross-check the result using a traditional t-table if desired.

5. Interpreting t-Statistics and p-Values

The t-statistic measures how many standard errors the observed difference is away from the null-hypothesis value. Larger absolute values of t usually indicate stronger evidence against H0, especially when the sample size is modest or large.

The p-value is the probability, under the assumption that H0 is true, of observing a test statistic as extreme as (or more extreme than) the one obtained. In a two-sided test, “extreme” means large positive or large negative values of t.

In this calculator, p-values are obtained from a normal-based approximation to the t-distribution for quick numerical intuition. For high-stakes analysis or very small samples, you may wish to confirm results using exact t-distribution software or statistical packages that implement precise special-function algorithms.

6. Assumptions Behind the t-Test

All t-tests rely on several assumptions. Understanding these conditions helps you decide whether your data and design are suitable for a classical t-test approach:

  • Independence: individual observations or paired differences should be approximately independent of each other.
  • Normality: the underlying population (or difference scores) is assumed to be roughly normal. The t-test is fairly robust to moderate deviations, especially for larger sample sizes (for example n ≥ 30).
  • Equal variances (for classical two-sample t): if you use a pooled-variance two-sample test, the variances in the two groups should be similar. Welch’s test, which is implemented here, relaxes this assumption by adjusting the degrees of freedom.
  • Measurement scale: the data should be on an interval or ratio scale so that means and standard deviations are meaningful summaries.

Related Tools from MyTimeCalculator

t-Test Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about one-sample, two-sample and paired t-tests, how to use this calculator and how to interpret the results.

Use a one-sample t-test when you compare a single sample mean to a known or hypothesized population mean (for example testing whether an average score differs from 50). Use a two-sample t-test when you compare means from two independent groups such as a treatment and a control group. Use a paired t-test when your observations come in matched pairs, such as measurements on the same person before and after an intervention.

Yes. For the one-sample and two-sample t-tests you can choose between Raw data and Summary statistics input modes. In raw mode, you paste data values directly and the calculator automatically computes the means and standard deviations. For the paired t-test, you always work from raw data in matched pairs, and the calculator computes difference scores internally.

A two-tailed t-test checks for any difference from the null value (for example H₁: μ ≠ μ₀), so extreme values in either direction count as evidence against H0. A one-tailed t-test only looks for differences in one direction (H₁: μ > μ₀ or μ < μ₀), which concentrates the rejection region in a single tail of the distribution. The calculator allows you to choose the tail type and adjusts the p-value accordingly.

The p-value measures how extreme your observed t-statistic is under the assumption that the null hypothesis is true. A small p-value (for example p < 0.05) suggests that such an extreme result would be unlikely if H0 were correct, providing evidence against the null. A large p-value indicates that the data are consistent with H0. The calculator uses a normal-based approximation to the t-distribution, which is typically accurate for moderate or large sample sizes.

Cohen’s d is a standardized effect size that expresses the difference between means in units of standard deviations. While statistical significance depends on sample size and variability, d focuses on the magnitude of the effect. Rough guidelines sometimes label |d| ≈ 0.2 as small, 0.5 as medium and 0.8 or higher as large, though context and domain knowledge should always guide interpretation.

Yes. You can enter the given sample statistics or raw data from a problem and compare the calculator’s t-statistic and p-value with your own work. This is especially useful for verifying multiple-choice answers or intermediate calculations. However, for written assignments you should still show your formulas and reasoning, since the calculator provides numerical results but not step-by-step algebraic derivations.

Related Calculators