Law of total variance

\underset{total variance}{\underset{⏟}{var [Y]}} = \underset{within groups variance}{\underset{⏟}{E [var [Y | X]]}} + \underset{between groups variance}{\underset{⏟}{var [E [Y | X]]}} .

Where:

$Y$ is a vector of observations, e.g. trait values
$X$ is a vector of indicators telling which group each individual measurement comes from.
- Like how we used X = 0 or 1 in last week's example comparing two populations with linear regression
This only holds if the residual variance is independent from the variance in covariates - if there is correlation there are some problems, but we won't worry about this for now