Total variance

01Variability within and between cases#

Overall variance includes spread inside each case and differences between case means. The law of total variance accounts for both:

The first term is the probability-weighted average of conditional variances. The second treats the conditional means as values of a new random variable and computes their variance.

02Compute both parts#

Suppose cases $a,b$ are equally likely, have means $1,5$, and variances $2,6$. The within-case term is $(2+6)/2=4$. The overall mean is $(1+5)/2=3$.

The case means deviate from $3$ by $-2$ and $2$, so the between-case variance is $4$. Adding gives total variance $8$. Averaging the conditional variances alone would miss half the variability.

03Split a deviation at its group mean#

Write $X-E[X]$ as the sum of $X-E[X\mid Y]$ and $E[X\mid Y]-E[X]$. The first piece fluctuates within a case; the second shifts the case mean relative to the overall mean.

After squaring, the cross term has expectation zero: within each case, the first piece averages to zero. The two remaining squared terms produce the stated decomposition. Finite variances are assumed.

If case means coincide, the between-case term is zero. If each case is internally constant, the within-case term is zero, yet the overall variance can remain positive. This distinction is useful when interpreting heterogeneous datasets.