t-Test Calculator

A t-test is a statistical hypothesis test that determines whether an observed difference between means is likely to be real or simply the result of random sampling variation. Unlike a z-test, which requires a known population standard deviation and large samples, the t-test works with small samples by using an estimated standard deviation and the heavier-tailed t-distribution.

The test produces a t-statistic — a standardised measure of how many standard errors the observed mean is from the null hypothesis value. The t-statistic is then compared against the t-distribution with the appropriate degrees of freedom to compute a p-value: the probability of observing a result at least this extreme if the null hypothesis were true.

t-tests are among the most widely used statistical procedures in social science research, clinical trials, quality control, and A/B testing. They are appropriate when the data are approximately normally distributed (or the sample size is large enough for the Central Limit Theorem to apply) and the observations are independent.

The one-sample t-test compares the mean of a single sample against a known or hypothesized population mean (μ₀). Use it when you have one group of measurements and want to test whether that group is consistent with a benchmark value.

Common applications: comparing a class's average test score against the national average, checking whether a batch of products meets a target weight specification, testing whether a coin-flip experiment deviates from 50% heads, or verifying that a treatment group's resting heart rate matches the healthy population norm.

Where x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and μ₀ is the hypothesized population mean. With 25 observations, a sample mean of 5.2, a hypothesized mean of 5.0, and a standard deviation of 0.4, you get t = 2.500 on 24 degrees of freedom — a p-value of roughly 0.020, which is statistically significant at the 5% level.

The two-sample t-test determines whether two independent groups have different population means. This calculator uses Welch's t-test, which does not assume that the two groups have equal variances. This makes it more broadly applicable than Student's classic equal-variance t-test.

Common applications: A/B testing two versions of a landing page, comparing exam scores between two teaching methods, testing whether a drug lowers blood pressure more than a placebo, or comparing average transaction values between two customer segments.

The Welch–Satterthwaite equation adjusts the degrees of freedom downward when the group variances are unequal, giving a more conservative (and correct) test. When both groups have identical variance and sample size, Welch's degrees of freedom equals the Student's t-test value of n₁ + n₂ − 2.

The p-value is the probability of observing a t-statistic at least as extreme as the one you computed, assuming the null hypothesis is true. It is not the probability that the null hypothesis is correct.

A low p-value does not mean the effect is large or practically meaningful. With a very large sample, even a trivially small difference can yield p < 0.001. Always pair p-values with effect sizes (such as Cohen's d) and confidence intervals for a complete picture.

The t-test produces valid results when several conditions are met. Violating these assumptions can inflate the Type I error rate or reduce statistical power.