Two-sample t-test (Welch and pooled)

Compares the means of two independent groups; Welch's version (the engine default) does not assume equal variances, the pooled version does.

t = \frac{\xbar_1 - \xbar_2 - \Delta_0}{\sqrt{s_1^2/n_1 + s_2^2/n_2}}, \qquad \nu = \frac{(v_1 + v_2)^2}{v_1^2/(n_1-1) + v_2^2/(n_2-1)},\ v_i = \frac{s_i^2}{n_i}

Welch's t-test with Welch–Satterthwaite degrees of freedom (usually not a whole number).

$\xbar_i, s_i, n_i$: Mean, SD (n − 1) and size of group i.
$\Delta_0$: Difference stated by H₀ (usually 0).
$\nu$: Welch–Satterthwaite degrees of freedom.

t = \frac{\xbar_1 - \xbar_2 - \Delta_0}{s_p\sqrt{1/n_1 + 1/n_2}},\quad s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2},\quad df = n_1+n_2-2

Pooled-variance (Student) t-test: assumes equal population variances.

$s_p$: Pooled standard deviation.

When to use

A quantitative outcome compared between two independent groups (different people/units in each group). Use Welch by default; the pooled test is only marginally more powerful when variances really are equal and can be badly wrong when they are not (especially with unequal group sizes).

Assumptions

Independent observations within and between groups (random samples or random assignment).
Each group roughly normal, or large enough samples for the CLT; watch for outliers.
Pooled version only: equal population variances.

How to interpret

The estimate is $\xbar_1 - \xbar_2$ (first minus second); report it with its CI.
A small p-value is evidence that the population means differ. Whether a difference is caused by the grouping depends on the design (random assignment) — not on the test.
Effect size: Cohen's d with the pooled SD (the engine reports it for both versions).

Common mistakes

Using the two-sample test on paired data (it ignores the pairing and usually loses a lot of power).
Running a variance test first and switching between pooled and Welch based on it — just use Welch.
Reading the CI for the difference as "the overlap of two separate CIs": overlapping individual CIs do not imply a non-significant difference.

Example

Group A: 23, 25, 28, 30, 26, 27 (mean 26.5, SD 2.43); group B: 20, 22, 24, 21, 25, 19, 23 (mean 22.0, SD 2.16). Welch: $t = 3.50$ , $\nu = 10.18$ , $p = 0.0055$ , 95% CI for $\mu_A - \mu_B$ : 1.64 to 7.36. Pooled: $t = 3.54$ , df = 11, $p = 0.0047$ , CI 1.70 to 7.30. Cohen's $d = 1.97$ .