Probability

Law of total variance

Splits variability into the noise you would still face knowing YY, plus the variability that knowing YY would remove.

Var⁡(X)=E[Var⁡(X∣Y)]+Var⁡(E[X∣Y])\operatorname{Var}(X) = \mathbb{E}\big[\operatorname{Var}(X \mid Y)\big] + \operatorname{Var}\big(\mathbb{E}[X \mid Y]\big)

Where

E[Var⁡(X∣Y)]\mathbb{E}[\operatorname{Var}(X\mid Y)]
Within-group: irreducible noise.
Var⁡(E[X∣Y])\operatorname{Var}(\mathbb{E}[X\mid Y])
Between-group: the value of the information in YY.

Assumptions

  • Both terms are non-negative, so the total is never below either one.

Sanity check. A total variance smaller than one of your conditional variances means you dropped a term.

Where this is taught