Editing Law of total variance (section)

=== Example 1 (Exam Scores) ===
Suppose five students take an exam scored 0–100. Let {{mvar|Y}} = student’s score and {{mvar|X}} indicate whether the student is *international* or *domestic*:

{| class="wikitable"
! Student !! {{mvar|Y}} (Score) !! {{mvar|X}}
|-
| 1 || 20 || International
|-
| 2 || 30 || International
|-
| 3 || 100 || International
|-
| 4 || 40 || Domestic
|-
| 5 || 60 || Domestic
|}

* Mean and variance for international: <math>\operatorname{E}[Y\mid X=\text{Intl}] = 50,\; \operatorname{Var}(Y\mid X=\text{Intl}) \approx 1266.7.</math>
* Mean and variance for domestic: <math>\operatorname{E}[Y\mid X=\text{Dom}] = 50,\; \operatorname{Var}(Y\mid X=\text{Dom}) = 100.</math>

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