Stddev

Standard deviation, often abbreviated as stddev, is a statistic that measures the amount of variation or dispersion in a dataset. It indicates how far individual observations tend to be from the mean, providing a sense of the data’s spread.

There are two common forms. The population standard deviation, σ, is defined as sqrt( (1/N) Σ (xi − μ)² ), where N is the number of observations and μ is the population mean. The sample standard deviation, s, is defined as sqrt( (1/(n−1)) Σ (xi − x̄)² ), where n is the sample size and x̄ is the sample mean. The distinction matters because s is an unbiased estimator of the population variance under normal assumptions, while σ uses the true population parameters.

Stddev relates directly to variance, which is the average of the squared deviations from the mean. The standard deviation is simply the square root of the variance. In a normal distribution, about 68% of observations lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three, though this empirical rule depends on normality.

Calculation can be done by hand or with software. In practice, software packages may use different denominators (N vs. N−1) depending on whether they compute a population or sample stddev; this is often exposed as a degrees-of-freedom parameter (ddof). Standard deviation is a widely used measure of dispersion in statistics, quality control, finance, and scientific data analysis, but it is sensitive to outliers and depends on the scale of the data.