Standard Deviation

For a population of $N$ values $x_1, \ldots, x_N$ with mean $\mu$, the population standard deviation $\sigma$ is

$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2}.$

For a sample of $n$ values with sample mean $\bar{x}$, divide by $n - 1$ instead of $n$ — Bessel's correction, an unbiased estimator of population variance.

Standard deviation is in the same units as the original data (unlike variance, which is in squared units), making it directly interpretable. It is the natural "ruler" for normal distributions: roughly 68% of values fall within one standard deviation of the mean, 95% within two, 99.7% within three.