Standard Deviation Calculator

Standard deviation measures dispersion around the mean. The population formula divides by n; the sample formula divides by n−1 (Bessel's correction) to give an unbiased estimate when working from a sample. Variance is the squared standard deviation.

The calculator accepts a comma- or newline-separated dataset and returns mean, median, mode, range, variance, and both population and sample standard deviations.

Standard deviation, denoted σ for population and s for sample, was named by Karl Pearson in 1893 building on earlier work by Gauss. The population formula σ = √(Σ(x − μ)² / n) measures dispersion when you have the entire population. The sample formula s = √(Σ(x − x̄)² / (n − 1)) divides by n − 1 because using the sample mean (rather than the unknown population mean) introduces a degree-of-freedom constraint — the sum of deviations from the sample mean is always zero, so the n − 1 deviations determine the nth. Without Bessel's correction, the sample variance underestimates the population variance by a factor of (n − 1)/n.

A worked example: dataset [2, 4, 4, 4, 5, 5, 7, 9]. Mean = 5. Deviations from mean: −3, −1, −1, −1, 0, 0, 2, 4. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16. Sum of squared deviations = 32. Population variance = 32/8 = 4, population SD = 2. Sample variance = 32/7 ≈ 4.571, sample SD ≈ 2.138. The two diverge by about 7% — a meaningful difference for small samples that shrinks as n grows. For 1000 data points, the population and sample formulas differ by 0.05%; for 10 points, by 5%.

Limitations: standard deviation assumes data approximates a normal distribution to be most useful — for skewed or heavy-tailed distributions, interquartile range and median absolute deviation are more robust descriptors. The 68/95/99.7 rule (68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD) only holds approximately for normal data; t-distributions, log-normal distributions, and bimodal data give different coverage. Outliers heavily influence SD because deviations are squared — a single extreme value can double the standard deviation of a small sample. For outlier-resistant dispersion, use the IQR or median absolute deviation. The calculator handles standard SD; robust alternatives are computed in different tools.