Standard deviation is a common summary statistic that describes the amount of variation or dispersion in a set of numerical values. Intuitively, it measures how far the typical observation lies from the central value of the distribution — usually the arithmetic mean or the theoretical expected value. A small standard deviation means values cluster closely around the centre; a large one means they are spread out more widely (close to the average vs far from it).
Definition and interpretation
Mathematically, the standard deviation is the square root of the variance. The variance is the average of the squared differences between each observation and the mean, so taking the square root returns a measure in the same units as the original data. Because of this, the standard deviation is often preferred when reporting variability: it conveys spread in familiar units (for example, dollars, centimeters, or test points).
How to compute it
- Compute the mean of the data set.
- For each value, calculate its deviation from the mean and square that deviation.
- Average those squared deviations (this gives the variance); then take the square root to obtain the standard deviation.
When all values in a population are available, the population standard deviation uses the divisor N. When only a sample is available, the sample standard deviation uses a slightly different formula that divides by (n−1) instead of n; this adjustment, known as Bessel's correction, gives an unbiased estimate of the population variance. The population standard deviation is conventionally denoted by the Greek letter σ (sigma) and the sample standard deviation by s.
Practical uses and examples
Standard deviation appears across science and everyday decision-making. Scientists report it to characterize experimental variability; differences larger than about two or three standard deviations are often treated as noteworthy. In finance, standard deviation is a basic measure of investment volatility: a higher value signals wider swings in returns. It also underlies many statistical tools: confidence intervals, hypothesis tests, control charts in manufacturing, and z-scores that standardize individual measurements.
Key properties and caveats
- Units: standard deviation has the same units as the data, unlike variance which is squared units.
- Normal distribution rule: for many bell-shaped (normal) distributions, roughly 68% of values lie within one standard deviation of the mean, about 95% within two, and 99.7% within three — a useful approximation, but it applies exactly only to the normal case.
- Sensitivity to outliers: because deviations are squared, large deviations have a disproportionate effect; alternative measures (median absolute deviation) are used when robustness is required.
- Relationship to the mean: standard deviation depends on the chosen center; using the mean minimizes the variance among choices of a central value.
History and notable facts
The idea of measuring spread has long roots in probability and statistics. The concrete symbol σ and the modern usage of "standard deviation" were popularized by statisticians in the late 19th and early 20th centuries as part of formalizing statistical inference. Because it links directly to the variance and to properties of the normal distribution, the standard deviation remains a cornerstone of descriptive and inferential statistics.
Further reading
For more detail on computation, interpretation and alternatives, consult introductory statistics texts or online resources that explain variance, standard error of the mean (σ/√n), and robust measures of spread. See also material on confidence intervals and the practical meaning of "two-sigma" or "three-sigma" thresholds in applied work. margin of error | mean | expected value | closeness to average | scientists | money | sigma symbol