Standard Deviation

For a discrete random variable: first determine the expected value μ = Σxᵢ·pᵢ. Then for each value form the deviation xᵢ − μ, square it and weight it with pᵢ; the sum is the variance Var(X) = Σ(xᵢ − μ)²·pᵢ. Finally take the square root: σ = √Var(X). Die example: μ = 3.5, E(X²) = 91/6 ≈ 15.17, so Var = 15.17 − 3.5² = 2.92 and σ ≈ 1.71. Often faster is the shortcut formula Var(X) = E(X²) − μ², where you only average the squared values. For binomially distributed variables you skip everything: σ = √(n·p·(1−p)). Important: take the root only at the very end, never of the individual deviations.

σ is the theoretical standard deviation of a distribution, computed from the probability model. s is the empirical standard deviation of a sample, computed from data, and there you divide by n − 1 instead of n: s = √(Σ(xᵢ − x̄)²/(n − 1)). The reason for the Bessel correction: the sample mean x̄ automatically lies "in the middle of" its own data, so deviations from x̄ come out systematically smaller than those from the true μ. Dividing by n − 1 compensates this underestimation and makes s² an unbiased estimator of the variance. For large n the difference is tiny. Calculators and spreadsheets offer both variants (σₙ and sₙ₋₁, or STDEV.P and STDEV.S); choose by context: whole population or sample.

For a Bernoulli chain with n trials and success probability p the short formula σ = √(n·p·(1−p)) holds; q stands for 1 − p. Example: n = 400 applications with success rate p = 0.25 give μ = 100 and σ = √(400·0.25·0.75) = √75 ≈ 8.66. With this you build prediction intervals: about 95% of outcomes lie in [μ − 2σ; μ + 2σ] ≈ [83; 117]. The sigma rules require the Laplace condition σ > 3, clearly met here. Two pitfalls: the formula holds ONLY for binomially distributed variables, not for arbitrary distributions; and under the root stands the product n·p·q, not (n·p·q)² or n·p alone. Incidentally σ is maximal at p = 0.5, where the chain spreads most.

It measures how far outcomes typically lie from the expected value: small σ means tightly clustered, large σ widely spread values. Two distributions can have the same mean and completely different spread; only μ and σ together characterize the situation. σ becomes tangible via the sigma rules (for approximately bell-shaped distributions): about 68% of outcomes lie within μ ± σ, 95% within μ ± 2σ, 99.7% within μ ± 3σ. An outcome beyond 2σ is thus already remarkable, beyond 3σ very unusual; hypothesis tests and industrial quality control build exactly on this. Important for interpretation: σ carries the same unit as the measured quantity itself, unlike the variance with its squared unit.

Because simply averaging the deviations always gives zero: positive and negative distances from the expected value cancel exactly, Σ(xᵢ − μ)·pᵢ = 0 is a defining property of μ. So a trick is needed to remove the signs. Squaring does this and is mathematically well-behaved: it is differentiable, penalizes large outliers overproportionally and leads to elegant rules like the additivity of variance for independent variables, from which the √(npq) formula follows. The alternative of averaging absolute values gives the mean absolute deviation; it is intuitive but computationally unwieldy and plays no role in school stochastics. To restore the unit, you take the square root after squaring: that is exactly σ.