T- Test for Two Independent Groups

The two-sample (independent-groups) t-test extends the same logic to comparing two groups against each other — an experimental group against a control group, say — to check whether their means differ by more than random chance would explain. A typical hypothesis pair looks like:

$H_0: \mu_1 = \mu_2$ (the two population means are equal)

$H_1: \mu_1 \neq \mu_2$ (the two population means are not equal)

The exact test statistic depends on whether the two populations are assumed to share the same variance.

Case 1 — variances assumed equal. The two samples are pooled into a single combined estimate of variability, $S_p$, before computing $t$:

where $s_1$ and $s_2$ are the two sample standard deviations, and $S_p$ (the pooled standard deviation) blends them into one combined estimate.

Case 2 — variances assumed unequal. No pooling is done; each sample's own variance is used directly:

Degrees of freedom also work differently in the two cases. Under equal variances, $df = n_1 + n_2 - 2$. Under unequal variances, $df$ is computed from a more elaborate expression combining both sample variances and sizes, and is generally rounded down to a whole number.

As with the one-sample test, the calculated $t$ is compared against the critical $t$ value from the t-distribution table at the chosen confidence level and degrees of freedom — if the calculated value exceeds the critical value, the null hypothesis of equal means is rejected.

Critical t-Values (t-Distribution Table)

The table below lists the critical values of the $t$-distribution, transcribed from the t-table printed in the chapter. The top row is the area in one tail (the significance level $\alpha$ for a one-tailed test); the italic row beneath it is the matching area in two tails (use it for a two-tailed test). Each row is a value of the degrees of freedom $df$. To find a critical $t$, read down to your $df$ and across to the appropriate $\alpha$ column; for a left-tailed test the value is negative, for a right-tailed test positive, and for a two-tailed test both signs ($\pm$) apply. For example, at $df = 29$ the two-tailed $0.05$ critical value is $\pm 2.045$, and the one-tailed $0.05$ value is $1.699$.