Variance and Standard Deviation

Probability · Axiom Academy

LESSON Variance and Standard Deviation Measuring the spread and variability of random variables The variance of a random variable X measures the average squared deviation from the mean μ = E[X]. We square the deviations so positive and negative differences don't cancel out. The second formula (E[X²] - (E[X])²) is often easier to compute. The standard deviation σ is the square root of variance. It's measured in the same units as X, making it more interpretable. Standard deviation gives us a typical distance between a random value and the mean. Larger σ means more spread out; smaller σ means more concentrated around the mean. • Var(X) ≥ 0 always (variance is never negative) • Var(aX + b) = a²Var(X) (adding constants doesn't change spread; scaling by a multiplies variance by a²) • If X and Y are independent: Var(X + Y) = Var(X) + Var(Y) Computational Formula: Var(X) = Σ x² · p(x) - μ², which avoids computing all deviations individually.

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