Variance

Compact study note.

Summary

Variance measures mean squared deviation from the expected value. It is nonnegative and has squared units.^[1]

Prerequisites

Expected Value

Notation and Assumptions

Variance identity:

Var (X) = E [(X - E [X])^{2}] = E [X^{2}] - (E [X])^{2} .

Use it when $E [X^{2}]$ is finite.

Essential Result

For constants $c, d$ ,

Var (c + d X) = d^{2} Var (X) .

If $X$ and $Y$ are independent,

Var (X + Y) = Var (X) + Var (Y) .

Small Example

If $X \sim Bernoulli (p)$ , then $E [X^{2}] = p$ and $Var (X) = p - p^{2} = p (1 - p)$ .

Common Mistakes

Squaring the added constant instead of the coefficient multiplying $X$ .
Adding variances without independence or zero covariance.

Connections

References

OpenStax, Introductory Statistics 2e, "Chapter 4: Discrete Random Variables", https://openstax.org/books/introductory-statistics-2e/pages/4-introduction ↩︎