Standard deviation

Standard deviation measures how much individual values in a dataset typically differ from the average. A small standard deviation means values cluster tightly around the mean; a large one means they are spread widely, so two datasets with the same average can look very different once spread is accounted for.

It is calculated as the square root of the variance: take each value's difference from the mean, square it, average those squared differences, then take the square root, sd = sqrt(sum((x - mean)^2) / n). Taking the square root returns the measure to the original units (dollars, seconds, visits), which is why standard deviation is reported far more often than variance itself. Population and sample versions differ slightly: sample standard deviation divides by n - 1 rather than n, correcting for the bias of estimating from a subset.

Analysts use standard deviation to judge whether a difference between two numbers is meaningful, to compute a z-score, and to build control limits and confidence bands. In a roughly normal distribution, about 68% of values fall within one standard deviation of the mean and about 95% within two. The common pitfall is applying that rule to skewed or heavy-tailed data, where it does not hold, or comparing standard deviations across datasets with very different means, where the interquartile range is often a more honest comparison of spread.