Variance decomposition using ANOVA

Question

Say you have a random variable $X$ (e.g., kilometers driven). Getting its variance is straightforward. But what if you want to say, $A$ percent of the variance in $X$ is due to $\text{Var}(X)$ for female drivers and $B$ percent is the rest, that is, $\text{Var}(X)$ for male drivers? $A + B$ should be 100 percent.

Is this possible? Are there assumptions to be made to simplify things? Independence of female and male drivers?

[Question was also asked here https://mathoverflow.net/questions/88185/variance-decomposition-anova but I realized it's best to do it at Stats instead.]

Answer 1

This is pretty much what analysis of variance (ANOVA) does. Except that there is an additional source of variance in the response variable which is variation between individuals that is not explained by sex.

A model is fit of the form:

$y_i=\beta_0+\beta_1x_i+\epsilon_i$

where $x_i$ is 1 if the individual is male, 0 otherwise; and $\epsilon_i$ has a normal distribution. It is then possible to divide the variance into an element explained by the difference between sexes (the structural part of the model above) and an element explained by the difference between individuals (the $\epsilon_i$ part).

It's not possible to say what variance is explained by men and what by women - only a total amount explained by the difference between the two.

Answer 2

Variance is one of true distribution characteristics indicating how widely values of given random variable are distributed. In a sense, it is a similar concept to width of range (difference between min and max) So in almost all cases, you can't say "A percent of the variance in X is due to the Var(X) for female drivers and B percent is the rest. A + B should be 100 percent" whether that variable is independent of gender.

If you want to run conventional ANOVA, then Peter Ellis's answer is the right approach. Otherwise, describe what you want to achieve in detail.