T-Test Calculator

How it works

A t-test asks whether a difference in means is larger than the random variation you would expect from samples of this size. The t statistic is the observed difference divided by its standard error: how many standard errors the difference is from zero. The p-value is the probability of a t at least this extreme if the true difference were zero, read from Student's t distribution with the appropriate degrees of freedom.

The one-sample test compares one sample mean with a fixed hypothesized value. The two-sample tests compare two independent samples. Welch's version uses each sample's own variance and adjusts the degrees of freedom with the Welch–Satterthwaite formula, so it stays valid when the two spreads differ; the pooled version assumes equal variances and combines them, gaining a little power when that assumption is true.

A two-tailed p-value counts extreme results in either direction and answers "are the means different?". A one-tailed p-value counts only the direction you observed and is half as large; use it only if you decided the direction before seeing the data. The p-value is computed from the regularized incomplete beta function, the same route as statistical tables and software.

Formula

one-sample:   t = (x̄ − μ0) / (s / √n)                    df = n − 1
Welch:        t = (x̄1 − x̄2) / √(s1²/n1 + s2²/n2)
              df = (s1²/n1 + s2²/n2)² / [ (s1²/n1)²/(n1 − 1) + (s2²/n2)²/(n2 − 1) ]
pooled:       sp² = [ (n1 − 1) s1² + (n2 − 1) s2² ] / (n1 + n2 − 2)
              t = (x̄1 − x̄2) / √(sp² (1/n1 + 1/n2))         df = n1 + n2 − 2
two-tailed p  = I_x(df/2, 1/2)   with x = df / (df + t²)   (regularized incomplete beta)
one-tailed p  = two-tailed p / 2

Example

A sample of 25 has mean 52.1 and standard deviation 6, and you want to know whether the population mean differs from 50. The standard error is 6/√25 = 1.2, so t = 2.1/1.2 = 1.75 with 24 degrees of freedom. The two-tailed p-value is 0.0929: not significant at the 5% level, though a one-tailed test in the observed direction would give 0.0464.

Two groups: 25 scores with mean 85 and SD 10 against 30 scores with mean 80 and SD 12. Welch's test gives a standard error of √(100/25 + 144/30) = 2.9665, t = 1.6855, 53.00 degrees of freedom and a two-tailed p of 0.0978. The pooled test on the same figures gives t = 1.6576, 53 degrees of freedom and p = 0.1033.

Assumptions and limitations

• Inputs are summary statistics: the sample mean, the sample standard deviation (with an n − 1 denominator) and the sample size. Raw data lists are not accepted; compute the summaries first.
• Samples are independent random samples from populations that are roughly normal, or large enough for the sample means to be. For two-sample tests the two samples are independent of each other; paired data needs a paired t-test on the differences, which can be run here as a one-sample test of the differences against 0.
• The pooled test assumes the two populations have the same variance. When in doubt use Welch.
• The p-value is a statement about the data under the null hypothesis, not the probability that the null hypothesis is true. The 5% threshold in the verdict is a convention, not a law.
• The p-value is computed numerically to about 1e-12 and agrees with published t-tables to the precision they print.

Frequently asked questions