variance
Two games can have the same expected payoff yet feel completely different: one pays a steady 5 dollars every time, the other pays 0 or 10 by a coin flip. The averages match, but the second is far riskier. Variance is the number that captures this — how spread out a random variable is around its mean, how much it 'wiggles'.
The definition takes the deviation from the mean, X - mu where mu = E[X], and asks for its average squared size: Var(X) = E[(X - mu)^2]. Why square? If you just averaged X - mu the positives and negatives would cancel to exactly zero, telling you nothing. Squaring makes every deviation positive and, by punishing big deviations more than small ones, makes variance sensitive to outliers. The result is a 'mean squared distance from the centre'. Because of the squaring, variance lives in squared units (if X is in metres, Var(X) is in metres squared), which is why we often take its square root, the standard deviation, to get back to natural units.
Variance is the headline measure of risk, noise, and uncertainty across all of probability and statistics. A small variance means outcomes cluster tightly around the mean and the mean is a reliable summary; a large variance means the mean alone tells you little. Note variance can be zero only for a constant (no randomness at all), and it can also be infinite or undefined for heavy-tailed distributions — and when variance is infinite, central tools like the central limit theorem can fail.
Steady game pays 5 always: Var = 0. Coin game pays 0 or 10, mean 5: deviations are ±5, so Var = (1/2)(25) + (1/2)(25) = 25, and standard deviation = 5. Same mean (5), but the second carries real spread.
Variance = average squared distance from the mean; zero only for a constant.
Variance is in squared units and overweights large deviations because of the square — so it is more sensitive to outliers than, say, the average absolute deviation.