Population SD divides by N; sample SD (s) divides by N − 1 (Bessel's correction) to avoid underestimating spread when you only have a sample. Here μ is the mean and xᵢ each value. Use sample SD when your data is a subset of a larger group.
Measuring spread
Standard deviation quantifies how much a data set varies around its mean. The steps are: find the mean, sum the squared differences from the mean, divide to get the variance, then take the square root.
Use population standard deviation when your numbers represent an entire population, and sample standard deviation (with Bessel's N − 1 correction) when they are a sample. This tool reports both, plus the mean, variance, count, and sum.
Worked example
Data set: 4, 8, 6, 5, 12. The mean is 7. Squared deviations: 9, 1, 1, 4, 25 — summing to 40. Population variance = 40 ÷ 5 = 8, so σ ≈ 2.83; sample variance = 40 ÷ 4 = 10, so s ≈ 3.16. The sample figure is always a bit larger — that's Bessel's correction compensating for estimating the mean from the same data.
Interpreting the number
Standard deviation shares the data's own units, which makes it intuitive: test scores averaging 75 with an SD of 3 are tightly clustered; with an SD of 15 they're all over the map. For roughly bell-shaped data, the 68–95–99.7 rule applies: about 68% of values fall within 1 SD of the mean, 95% within 2, and 99.7% within 3.
Population vs sample — how to choose
Ask: do my numbers include every member of the group I care about (all 30 students in the class → population), or are they a subset standing in for something bigger (30 shoppers surveyed out of thousands → sample)? When unsure, sample SD is the safer, slightly conservative choice — and it's what spreadsheets' STDEV defaults to. For solving equations rather than describing data, see the quadratic formula calculator.