Question

What exactly does the pooled variance in an independent samples t-test tell us?Why is it necessary to modify this formula for unequal values of n?

Answer 1

For an independent sample t test, if we have two samples of sizes n₁ and n₂. Let S₁² and S₂² be the respective variances of the two samples. Then, the pooled variance of the two samples is given by,

Pooled variance is a method for estimating variance of several different populations when the mean of each population may be different, but the variance of each population may be assumed to be the same.

Under the assumption of equal population variances, the pooled sample variance provides a higher precision estimate of variance than the individual sample variances. This helps in providing statistical tests of higher power as in the case of t tests.

For more that two samples,

Now, if the value of n_i, are equal for all k, the above formula reduces to

Hence the two formula are not identical