Pooled Variance

When two independent samples are drawn from populations sharing the same (unknown) variance, we can combine their sample variances into a single, more reliable estimate called the pooled variance.

Given samples of sizes $n_1$ and $n_2$ with sample variances $s_1^2$ and $s_2^2,$ the pooled variance is

$$s_p^2 = \dfrac{(n_1-1)\,s_1^2 + (n_2-1)\,s_2^2}{n_1+n_2-2}.$$

Each sample variance is weighted by its degrees of freedom, $n_i-1.$ The total degrees of freedom is $n_1+n_2-2.$

For example, if $n_1 = 4,\, s_1^2 = 6$ and $n_2 = 6,\, s_2^2 = 10,$ then

$$s_p^2 = \dfrac{(4-1)(6) + (6-1)(10)}{4+6-2} = \dfrac{18 + 50}{8} = \dfrac{68}{8} = 8.5.$$