Confidence intervals for a difference in means (Welch, pooled, paired)

Intervals for μ₁ − μ₂ from independent samples (Welch by default, pooled when variances are assumed equal) or for the mean difference of paired data.

(\xbar_1 - \xbar_2) \pm t_{1-\alpha/2,\,\nu}\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}, \quad \nu = \frac{(v_1 + v_2)^2}{\frac{v_1^2}{n_1-1} + \frac{v_2^2}{n_2-1}},\; v_i = \frac{s_i^2}{n_i}

Welch interval (default): no equal-variance assumption

$\xbar_i, s_i, n_i$: mean, SD ( $n - 1$ ) and size of sample $i$
$\nu$: Welch–Satterthwaite degrees of freedom (usually not a whole number)

(\xbar_1 - \xbar_2) \pm t_{1-\alpha/2,\,n_1+n_2-2}\; s_p\sqrt{\frac{1}{n_1} + \frac{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}

Pooled interval (assumes equal population variances)

$s_p$: pooled standard deviation

\bar d \pm t_{1-\alpha/2,\,n-1}\,\frac{s_d}{\sqrt{n}}

Paired interval: a one-sample t interval on the differences

$d_i = x_i - y_i$: difference within pair $i$
$\bar d, s_d$: mean and SD of the differences
$n$: number of pairs

When to use

Independent groups (two different sets of units): Welch. It is the engine's default because it stays valid when variances differ and loses almost nothing when they are equal.
Pooled: only when equal variances are justified by design or prior knowledge.
Paired (same units measured twice, or matched pairs): analyse the differences.

Assumptions

Random samples / random assignment; independence within (and, for two-sample, between) groups; approximately normal data or differences, or reasonably large samples.

How to interpret

The interval gives plausible values for the difference (group 1 − group 2). If it excludes 0, the matching two-sided test rejects "no difference" at that α. Judge the size: is the whole interval practically important?

Common mistakes

Treating paired data as independent: the SE ignores the pairing and is usually far too large.
Choosing pooled vs Welch after looking at which gives the smaller p-value.
Concluding "no difference" because two separate intervals overlap.

Example

Group A: 34, 29, 41, 38, 30, 36 ( $\xbar = 34.67$ , $s = 4.633$ ); group B: 27, 31, 25, 29, 24, 28, 26 ( $\xbar = 27.14$ , $s = 2.410$ ). Welch 95% interval: $(2.60, 12.45)$ with $\nu = 7.26$ ; pooled: $(3.12, 11.93)$ with df 11. Paired example: before 142, 150, 138, 147, 155, 141 and after 136, 146, 135, 140, 150, 139; $\bar d = 4.5$ , $s_d = 1.871$ , 95% interval $(2.54, 6.46)$ .