Expected Value and Variance: Explained with Examples

It is the centre of mass of the distribution. Its square root is the standard deviation σ.

These two numbers matter because they summarise any distribution the way people actually think: what to expect, and how much it wobbles. A gambler compares games by expected value; an investor compares portfolios by expected return and variance as risk. The mean gives the fair price of a bet, and the standard deviation tells you how wild the ride will be. Almost every statistical method, from confidence intervals to regression, is built on means and variances.

Key Concepts

• Expected value E(X) = μ: the probability-weighted average of all values; the balance point of the distribution.
• Variance Var(X) = σ²: average squared deviation from the mean; always non-negative; zero only for a constant.
• Standard deviation σ: the square root of variance, in the same units as X, so it is the interpretable spread.
• Linearity: E(aX + b) = a·E(X) + b and E(X + Y) = E(X) + E(Y), true with no independence needed.
• Variance rules: Var(aX + b) = a²·Var(X); for independent X and Y, Var(X + Y) = Var(X) + Var(Y); adding a constant never changes variance.
• Units: if X is in metres, E(X) is in metres but Var(X) is in square metres; take the square root to return to metres.

Formulas

Mean, discrete: E(X) = Σ x·p(x). Mean, continuous: E(X) = ∫ x·f(x) dx. Variance: Var(X) = E[(X − μ)²] = E(X²) − μ². Standard deviation: σ = √Var(X). Linearity: E(aX + b) = aμ + b, Var(aX + b) = a²σ². For independent variables: E(XY) = E(X)·E(Y) and Var(X ± Y) = Var(X) + Var(Y). All assume the sums and integrals converge.

Worked Example

Roll one fair die, X = number shown, p(k) = 1/6 for k = 1 … 6. Mean: E(X) = (1 + 2 + 3 + 4 + 5 + 6)/6 = 21/6 = 7/2 = 3.5. Next E(X²) = (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6. Variance: Var(X) = 91/6 − (7/2)² = 91/6 − 49/4 = 182/12 − 147/12 = 35/12, about 2.9167. Standard deviation: σ = √(35/12) ≈ 1.708. The definition agrees: averaging (k − 3.5)² over k = 1 … 6 gives (6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25)/6 = 17.5/6 = 35/12.

Common Misconceptions

• The expected value must be a possible value: Not so. The die's mean is 3.5, which no single roll can show; the mean is an average, not a prediction.
• Var(X + Y) always equals Var(X) + Var(Y): Only for independent or uncorrelated variables. In general add 2·Cov(X, Y), which can be positive or negative.
• Variance and standard deviation are interchangeable: They differ by a square root and by units. Report σ when you want the spread in the original units.

Real-World Applications

Expected value prices insurance premiums, lottery tickets and casino games: the premium must exceed expected claims for the business to survive. Variance quantifies risk in finance, where investors trade expected return against volatility, and in manufacturing, where process variance decides whether parts meet tolerance. Machine learning minimises expected loss, which is an expected value over the data distribution.

Linearity of expectation needs no independence, while variance adds only for independent variables. Together they are the two-number summary behind pricing, risk, quality control and nearly all of statistics.