Paste a list of numbers (comma or newline separated) — pick population or sample depending on whether your data is the whole group or just part of it.
Population standard deviation divides the sum of squared deviations by N, used when your data represents the entire group you care about. Sample standard deviation divides by N−1 instead — a correction (Bessel's correction) that gives an unbiased estimate of the population's spread when your data is only a sample drawn from a larger population.
It measures spread, or variability, around the mean. A small standard deviation means most values sit close to the average; a large one means values are spread out over a wider range.
Squaring each deviation makes it positive, so values above and below the mean don't cancel each other out when summed, and it weights larger deviations more heavily than small ones — which is exactly the behavior we want from a measure of spread.
The Statistics Calculator covers mean, median, mode, range, variance, and standard deviation all together. This tool is a focused, standalone version for when you just need standard deviation specifically, with an explicit population-vs-sample toggle front and center.
Worked example: for the data set 2, 4, 4, 4, 5, 5, 7, 9 (n=8), the mean is 5. The population variance is 4, giving a population standard deviation of exactly 2. The sample variance (dividing by n−1=7 instead) is 32/7 ≈ 4.57, giving a sample standard deviation of about 2.14 — slightly larger, as Bessel's correction always produces.
The two formulas differ in one place: the population version divides by n, the sample version by n−1. That adjustment (Bessel's correction) exists because a sample's own mean sits closer to its own data points than the true population mean would, which systematically understates spread. Dividing by a smaller number corrects the bias.
Take the dataset 2, 4, 4, 4, 5, 5, 7, 9. The mean is 5. Treated as an entire population the standard deviation is exactly 2; treated as a sample it is 2.1381. Same numbers, different question, different answer — and with only eight data points the gap is about 7%.
Use population when you have measured everything. Use sample — the safer default — when you are generalising.
Standard deviation is in the same units as your data, which makes it directly interpretable in a way variance is not. A standard deviation of 2 on exam marks means 2 marks. For a roughly normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
To compare spread across different scales, divide the standard deviation by the mean to get the coefficient of variation. A standard deviation of 5 means something very different on data averaging 10 than on data averaging 10,000.
Standard deviation assumes spread is symmetric around the mean, so it describes skewed data poorly and is strongly influenced by outliers — squaring the deviations gives extreme values disproportionate weight. One mistyped figure can dominate the result.
Before trusting it, check whether the mean and median are close. If they are far apart the distribution is skewed, and the median plus an interquartile range describes the data more honestly.
Related statistics tools: the Z-Score Calculator converts a value into standard deviations from the mean, the Mean, Median and Mode Calculator covers the other measures of central tendency, and the Confidence Interval Calculator turns a sample into a range.