Calculate population and sample standard deviation, variance, mean, and more from a data set.
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Enter or paste your data separated by commas, spaces, semicolons or new lines (e.g. "2, 4, 4, 5, 9" or one value per line from a spreadsheet). Choose whether your data is a sample or the whole population; all statistics update instantly.
Population σ: √[ Σ(xᵢ − μ)² / N ]
Sample s: √[ Σ(xᵢ − x̄)² / (N − 1) ] — needs at least two values
Standard error of the mean: s / √N
Use population SD when you have complete data. Use sample SD when your data is a subset of a larger population — dividing by N−1 (Bessel's correction) gives a better estimate of the population variance.
The sample formula divides by N − 1, which is zero when you have a single value, so it is undefined. The population standard deviation of one value is 0.
Use population standard deviation (÷N) when you have data for the entire group you are studying. Use sample standard deviation (÷(N-1)) when your data is a subset of a larger population and you want to estimate the population's variability. In practice, most real-world data is a sample, so sample standard deviation is more commonly appropriate.
Did you know?
In a normal distribution, approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. This is known as the 68-95-99.7 rule or the empirical rule.
Standard deviation was introduced by Karl Pearson in 1894 as a measure of data spread. The concept builds on earlier work by Carl Friedrich Gauss on the normal distribution (bell curve) in 1809. Pearson's contributions to statistics were so fundamental that the correlation coefficient and chi-squared test also bear his name.
Population standard deviation (σ) divides by N, while sample standard deviation (s) divides by N-1. The N-1 correction (Bessel's correction) accounts for the fact that a sample underestimates population variability.
A standard deviation of 0 means all values in the dataset are identical.
Standard deviation is used in finance as a measure of investment risk — higher standard deviation means more volatile returns.
Z-scores express how many standard deviations a value is from the mean, enabling comparison across different datasets.