stdev

Stdev, short for standard deviation, is a measure of the dispersion or spread of a set of values. It indicates how much the data deviate from the mean. A small stdev means observations are close to the mean; a large stdev indicates wider variation.

There are two common forms: the population standard deviation σ and the sample standard deviation s. Population: σ = sqrt( (1/N) ∑ (xi − μ)² ). Sample: s = sqrt( (1/(n−1)) ∑ (xi − x̄)² ). Here μ is the population mean, x̄ is the sample mean, N is the population size, and n is the sample size. Variance is the square of stdev.

Stdev shares the same units as the data. It relates to the standard error of the mean as an estimate of the variability of sample means: SE ≈ σ/√n (population) or s/√n (sample).

Common interpretation: for normally distributed data, about 68% lie within μ ± σ, about 95% within μ ± 2σ, and about 99.7% within μ ± 3σ.

Stdev is sensitive to outliers and non-normal distributions. It can be misleading if data are skewed or contain extreme values; robust alternatives include the median absolute deviation.

In practice, use s for a sample and σ for a population analysis. Many software packages implement stdev with differing ddof settings, e.g., Excel STDEV.S (n−1) and STDEV.P (n).

Example: for data 2,4,4,4,5,5,7,9, the sample stdev is about 2.14, while the population stdev is 2.0.