Estimating the group means

We now assume a normal model with the same standard deviation in each group,

Group i:   Y   ~   normal, σ)

The sample means provide estimates of the {µi}:

Estimating σ2

The sample standard deviation in any single group, si, is a valid estimate of σ, but we need to combine these g separate estimates in some way.

It is easier to describe estimation of σ2 rather than σ. If the sample sizes are the same in all groups, a pooled estimate of σ2 is the average of the group variances,

If the sample sizes are not equal in all groups, this is generalised by adding the numerators and denominators of the formulae for the g separate group variances,

More mathematically, yij denotes the j 'th of the ni values in group i , for i  = 1 to g . The pooled estimate of σ2 can then be written as

The pooled variance is influenced most by the sample variances in the groups with biggest sample sizes.