Applies to:
StatTools 6.x/7.x

Equal vs. Unequal Variances in Two-Sample t Tests (Pooling)

When comparing two sample means, the sampling distribution of the difference follows a

Student’s t distribution.

Each t distribution is defined by its degrees of freedom (df).

• One-sample t test: df = n − 1

• Two-sample t test (general case): df is computed using the Welch–Satterthwaite formula (see image at right), which does not assume equal variances.

If you know that the population variances are equal, you can instead use:

df = n₁ + n₂ − 2

This is called pooling the variances, because the two samples are treated as coming from one population. Pooling usually gives slightly higher df, which can result in:

• a slightly smaller p-value

• a narrower confidence interval

• a slightly higher chance of rejecting the null hypothesis

However, in practice you almost never know that population variances are truly equal.

You can perform an F test, but:

• A large p-value only means you failed to reject equality; it does not prove it

• The F test requires normal populations, a stronger assumption than the t test

• Population normality is rarely known with certainty

For these reasons, pooling is controversial, and many textbooks recommend avoiding it.

Recommendation (StatTools):
Use the Unequal Variances (Welch) results unless you have strong theoretical justification for equal population variances. This is the conservative choice and usually makes little difference in practice.

 

 

 

 

Degrees of freedom formula for unequal variances (Welch–Satterthwaite).