Let Y be a random variable and X another random variable on the same probability space. The law of total variance can be understood by noting:

1. ${\displaystyle \operatorname {Var} (Y\mid X)}$  measures how much Y varies around its conditional mean ${\displaystyle \operatorname {E} [Y\mid X].}$ 
2. Taking the expectation of this conditional variance across all values of X gives ${\displaystyle \operatorname {E} [\operatorname {Var} (Y\mid X)]}$ , often termed the “unexplained” or within-group part.
3. The variance of the conditional mean, ${\displaystyle \operatorname {Var} (\operatorname {E} [Y\mid X])}$ , measures how much these conditional means differ (i.e. the “explained” or between-group part).

Adding these components yields the total variance ${\displaystyle \operatorname {Var} (Y)}$ , mirroring how analysis of variance partitions variation.

Suppose five students take an exam scored 0–100. Let Y = student’s score and X indicate whether the student is *international* or *domestic*:

• Mean and variance for international: ${\displaystyle \operatorname {E} [Y\mid X={\text{Intl}}]=50,\;\operatorname {Var} (Y\mid X={\text{Intl}})\approx 1266.7.}$ 
• Mean and variance for domestic: ${\displaystyle \operatorname {E} [Y\mid X={\text{Dom}}]=50,\;\operatorname {Var} (Y\mid X={\text{Dom}})=100.}$ 

Both groups share the same mean (50), so the explained variance ${\displaystyle \operatorname {Var} (\operatorname {E} [Y\mid X])}$  is 0, and the total variance equals the average of the within-group variances (weighted by group size), i.e. 800.

Let X be a coin flip taking values Heads with probability h and Tails with probability 1−h. Given Heads, Y ~ Normal(${\displaystyle \mu _{h},\sigma _{h}^{2}}$ ); given Tails, Y ~ Normal(${\displaystyle \mu _{t},\sigma _{t}^{2}}$ ). Then ${\displaystyle \operatorname {E} [\operatorname {Var} (Y\mid X)]=h\,\sigma _{h}^{2}+(1-h)\,\sigma _{t}^{2},}$  ${\displaystyle \operatorname {Var} (\operatorname {E} [Y\mid X])=h\,(1-h)\,(\mu _{h}-\mu _{t})^{2},}$  so ${\displaystyle \operatorname {Var} (Y)=h\,\sigma _{h}^{2}+(1-h)\,\sigma _{t}^{2}\;+\;h\,(1-h)\,(\mu _{h}-\mu _{t})^{2}.}$ 

Consider a two-stage experiment:

1. Roll a fair die (values 1–6) to choose one of six biased coins.
2. Flip that chosen coin; let Y=1 if Heads, 0 if Tails.

Then ${\displaystyle \operatorname {E} [Y\mid X=i]=p_{i},\;\operatorname {Var} (Y\mid X=i)=p_{i}(1-p_{i}).}$  The overall variance of Y becomes ${\displaystyle \operatorname {Var} (Y)=\operatorname {E} {\bigl [}p_{X}(1-p_{X}){\bigr ]}+\operatorname {Var} {\bigl (}p_{X}{\bigr )},}$  with ${\displaystyle p_{X}}$  uniform on ${\displaystyle \{p_{1},\dots ,p_{6}\}.}$ 

Let ${\displaystyle (X_{i},Y_{i})}$ , ${\displaystyle i=1,\ldots ,n}$ , be observed pairs. Define ${\displaystyle {\overline {Y}}=\operatorname {E} [Y].}$  Then ${\displaystyle \operatorname {Var} (Y)={\frac {1}{n}}\sum _{i=1}^{n}{\bigl (}Y_{i}-{\overline {Y}}{\bigr )}^{2}={\frac {1}{n}}\sum _{i=1}^{n}{\Bigl [}(Y_{i}-{\overline {Y}}_{X_{i}})+({\overline {Y}}_{X_{i}}-{\overline {Y}}){\Bigr ]}^{2},}$  where ${\displaystyle {\overline {Y}}_{X_{i}}=\operatorname {E} [Y\mid X=X_{i}].}$  Expanding the square and noting the cross term cancels in summation yields: ${\displaystyle \operatorname {Var} (Y)=\operatorname {E} {\bigl [}\operatorname {Var} (Y\mid X){\bigr ]}\;+\;\operatorname {Var} \!{\bigl (}\operatorname {E} [Y\mid X]{\bigr )}.\!}$ 

Using ${\displaystyle \operatorname {Var} (Y)=\operatorname {E} [Y^{2}]-\operatorname {E} [Y]^{2}}$  and the law of total expectation: ${\displaystyle \operatorname {E} [Y^{2}]=\operatorname {E} {\bigl [}\operatorname {E} (Y^{2}\mid X){\bigr ]}=\operatorname {E} {\bigl [}\operatorname {Var} (Y\mid X)+\operatorname {E} [Y\mid X]^{2}{\bigr ]}.}$  Subtract ${\displaystyle \operatorname {E} [Y]^{2}={\bigl (}\operatorname {E} [\operatorname {E} (Y\mid X)]{\bigr )}^{2}}$  and regroup to arrive at ${\displaystyle \operatorname {Var} (Y)=\operatorname {E} {\bigl [}\operatorname {Var} (Y\mid X){\bigr ]}+\operatorname {Var} \!{\bigl (}\operatorname {E} [Y\mid X]{\bigr )}.\!}$ 

In a one-way analysis of variance, the total sum of squares (proportional to ${\displaystyle \operatorname {Var} (Y)}$ ) is split into a “between-group” sum of squares (${\displaystyle \operatorname {Var} (\operatorname {E} [Y\mid X])}$ ) plus a “within-group” sum of squares (${\displaystyle \operatorname {E} [\operatorname {Var} (Y\mid X)]}$ ). The F-test examines whether the explained component is sufficiently large to indicate X has a significant effect on Y.^[3]

In linear regression and related models, if ${\displaystyle {\hat {Y}}=\operatorname {E} [Y\mid X],}$  the fraction of variance explained is ${\displaystyle R^{2}={\frac {\operatorname {Var} ({\hat {Y}})}{\operatorname {Var} (Y)}}={\frac {\operatorname {Var} (\operatorname {E} [Y\mid X])}{\operatorname {Var} (Y)}}=1-{\frac {\operatorname {E} [\operatorname {Var} (Y\mid X)]}{\operatorname {Var} (Y)}}.}$  In the simple linear case (one predictor), ${\displaystyle R^{2}}$  also equals the square of the Pearson correlation coefficient between X and Y.

In many Bayesian and ensemble methods, one decomposes prediction uncertainty via the law of total variance. For a Bayesian neural network with random parameters ${\displaystyle \theta }$ : ${\displaystyle \operatorname {Var} (Y)=\operatorname {E} {\bigl [}\operatorname {Var} (Y\mid \theta ){\bigr ]}+\operatorname {Var} {\bigl (}\operatorname {E} [Y\mid \theta ]{\bigr )},}$  often referred to as “aleatoric” (within-model) vs. “epistemic” (between-model) uncertainty.^[4]

Credibility theory uses the same partitioning: the expected value of process variance (EVPV), ${\displaystyle \operatorname {E} [\operatorname {Var} (Y\mid X)],}$  and the variance of hypothetical means (VHM), ${\displaystyle \operatorname {Var} (\operatorname {E} [Y\mid X]).}$  The ratio of explained to total variance determines how much “credibility” to give to individual risk classifications.^[2]

For jointly Gaussian ${\displaystyle (X,Y)}$ , the fraction ${\displaystyle \operatorname {Var} (\operatorname {E} [Y\mid X])/\operatorname {Var} (Y)}$  relates directly to the mutual information ${\displaystyle I(Y;X).}$ ^[5] In non-Gaussian settings, a high explained-variance ratio still indicates significant information about Y contained in X.

The law of total variance generalizes to multiple or nested conditionings. For example, with two conditioning variables ${\displaystyle X_{1}}$  and ${\displaystyle X_{2}}$ : ${\displaystyle \operatorname {Var} (Y)=\operatorname {E} {\bigl [}\operatorname {Var} (Y\mid X_{1},X_{2}){\bigr ]}+\operatorname {E} {\bigl [}\operatorname {Var} (\operatorname {E} [Y\mid X_{1},X_{2}]\mid X_{1}){\bigr ]}+\operatorname {Var} (\operatorname {E} [Y\mid X_{1}]).}$  More generally, the law of total cumulance extends this approach to higher moments.

• Law of total expectation (Adam’s law)
• Law of total covariance
• Law of total cumulance
• Analysis of variance
• Conditional expectation
• R-squared
• Fraction of variance unexplained
• Variance decomposition

• Blitzstein, Joe. "Stat 110 Final Review (Eve's Law)" (PDF). stat110.net. Harvard University, Department of Statistics. Retrieved 9 July 2014.
• "Law of total variance". The Book of Statistical Proofs.
• Billingsley, Patrick (1995). "Problem 34.10(b)". Probability and Measure. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-00710-2.
• Weiss, Neil A. (2005). A Course in Probability. Addison–Wesley. pp. 380–386. ISBN 0-201-77471-2.
• Bowsher, C.G.; Swain, P.S. (2012). "Identifying sources of variation and the flow of information in biochemical networks". PNAS. 109 (20): E1320 – E1328. doi:10.1073/pnas.1118365109.