Should I use Welch's or Student's t-test?

Use Welch's t-test by default for two independent groups. It does not assume equal variances and loses almost nothing when variances are equal. Student's t-test is shown as a comparison and matches SPSS's "Equal variances assumed" row.

What is the difference between a paired and an independent t-test?

A paired t-test is for the same people measured twice, such as before and after an intervention. An independent t-test is for two different groups of people. Using the wrong one gives incorrect p values.

How do I calculate a t-test p value?

The p value is the probability of a t statistic at least as extreme as yours under the t distribution with your degrees of freedom. This calculator computes the exact two-tailed p value, so you do not need a t table.

What if my data are not normal?

With about 30+ per group the t-test is fairly robust. For small, skewed, or ordinal data, use the Mann-Whitney U test (independent groups) or the Wilcoxon signed-rank test (paired data).

T-TEST CALCULATOR

t-test calculator: independent, paired, and one-sample.

Paste your data and get t, degrees of freedom, the p value, the mean difference with a 95% confidence interval, Cohen's d, a Levene check for equal variances, and an APA-style sentence.

DATA

Two independent groups

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GROUP 1 NAMEGROUP 2 NAME

GROUP 1 VALUESGROUP 2 VALUES

Use Welch's t-test (does not assume equal variances, recommended)

t-test formulas, which version to use, and how to report it

Which t-test should I use?

Independent-samples t-test: two separate groups of people (e.g. online vs in-person students)
Paired-samples t-test: the same people measured twice (pre-test and post-test, before and after)
One-sample t-test: one group's mean compared with a known or hypothesised value (e.g. a scale midpoint of 3)
Outcome ordinal or clearly skewed? Use Mann-Whitney U (independent) or Wilcoxon signed-rank (paired) instead

Welch's t-test (default for two independent groups)

t = (M₁ − M₂) ÷ √(s₁²/n₁ + s₂²/n₂),   df = (s₁²/n₁ + s₂²/n₂)² ÷ [ (s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1) ]

Welch's version does not assume equal variances. Delacre, Lakens and Leys (2017) recommend it as the default because it performs as well as Student's t-test when variances are equal and much better when they are not. That is why the df is often a decimal, such as t(18.30).

Student's t-test (pooled variance)

sp = √{ [(n₁−1)s₁² + (n₂−1)s₂²] ÷ (n₁+n₂−2) },   t = (M₁ − M₂) ÷ (sp·√(1/n₁ + 1/n₂)),   df = n₁+n₂−2

This is the classic version SPSS shows in the "Equal variances assumed" row. Switch off Welch above to use it. The calculator also runs Levene's test (Brown-Forsythe, median-centred) so you can see whether the equal-variance assumption looks reasonable.

Paired and one-sample t-tests

t = M_d ÷ (s_d ÷ √n), df = n − 1

For a paired test, M_d and s_d are the mean and SD of the differences (after − before). For a one-sample test, M_d is the sample mean minus the test value μ₀. Cohen's d is M_d ÷ s_d (often written d_z for paired data).

Assumptions to check

Continuous outcome (scores, times, summed scale totals)
Independent observations (each person counted once per group)
Roughly normal data in each group, or normal differences for paired data: check with the Shapiro-Wilk test or a Q-Q plot
No extreme outliers; with about 30+ per group the t-test is fairly robust to mild non-normality

How to report it (APA style)

“Students who used practice quizzes (M = 84.20, SD = 4.94) scored higher than students who did not (M = 75.82, SD = 4.49), t(18.30) = 4.05, p < .001, 95% CI [4.04, 12.72], d = 1.78.” Report means and SDs, t with df, the exact p value (p < .001 when very small), the confidence interval, and an effect size.

References

Student. (1908). The probable error of a mean. Biometrika, 6(1), 1–25.
Welch, B. L. (1947). The generalization of 'Student's' problem when several different population variances are involved. Biometrika, 34(1–2), 28–35.
Brown, M. B., & Forsythe, A. B. (1974). Robust tests for the equality of variances. Journal of the American Statistical Association, 69(346), 364–367.
Delacre, M., Lakens, D., & Leys, C. (2017). Why psychologists should by default use Welch's t-test instead of Student's t-test. International Review of Social Psychology, 30(1), 92–101.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum.

t-Test Calculator

Paste raw scores and get a complete t-test: group means and SDs, t, degrees of freedom, exact p value, confidence interval, Cohen's d, Levene's test, and a ready APA sentence.

Compare two separate groups' mean scores in a thesis or survey

Test a pre-test vs post-test change on the same participants

Compare a sample mean with a scale midpoint or benchmark value

KEY FUNCTIONS

Independent (Welch or Student), paired, and one-sample modesExact two-tailed p value and 95% confidence intervalCohen's d with interpretationLevene's (Brown-Forsythe) equal-variance checkAPA 7 report sentence

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