Standard deviation

Standard deviation is the most common measure of spread. It answers the question "how far is a typical value from the mean?" and is reported in the same units as the data itself, which makes it easier to interpret than the variance. In fact standard deviation is simply the square root of the variance.

The calculation follows four steps: find the mean, subtract the mean from each value and square the result, average those squared differences, and take the square root. Consider the set 2, 4, 4, 4, 5, 5, 7, 9. The mean is 5, the squared differences are 9, 1, 1, 1, 0, 0, 4, 16, and their average is 4, so the standard deviation is 2. A population standard deviation divides by n, while a sample standard deviation divides by n − 1 — a correction that keeps the estimate from understating the true spread.

Standard deviation is most useful as a yardstick for individual values. In a normal distribution, roughly 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three — the empirical rule. That framework is what lets you convert a raw score into a z-score, compare results measured on different scales, and judge whether an observation is unusual. In finance, the same statistic is used as a measure of volatility and therefore of risk.

Standard deviation shows up across the exam catalog. The SAT tests it conceptually, asking you to compare the spread of two data sets rather than compute a value. The GRE pairs it with percentiles and box plots. AP Statistics goes furthest, requiring you to compute standard deviation for random variables, apply the empirical rule to normal distributions, and distinguish population parameters from sample statistics.