Definition
The variance of a random variable X, written Var(X) or sigma squared, is the expected value of the squared deviation from the mean: Var(X) = E((X - mu)²) where mu = E(X). Its square root, the standard deviation, has the same units as X and is often easier to interpret.
Variance quantifies how 'spread out' the random variable is. A small variance means values cluster tightly around the mean; a large variance means they scatter widely.
Why it matters
How it works
To compute variance, find the mean mu, subtract it from each value, square the result, weight by probability, and sum (or integrate). For a fair six-sided die with mean 3.5, the variance is the average of (1 - 3.5)², (2 - 3.5)², ..., (6 - 3.5)², which equals 35/12 ≈ 2.92, giving a standard deviation of about 1.71.
In statistics, the sample variance estimates the population variance from data, with a small bias correction (dividing by n - 1 instead of n) for unbiasedness. Variance computations are everywhere in regression, control charts, portfolio optimisation, and signal processing.