Standard Deviation Calculator

Population and sample divide by different numbers

Population standard deviation divides the sum of squared deviations by N. Sample standard deviation divides by N minus 1. The difference is not a convention — dividing by N when working from a sample systematically underestimates the true spread, because the deviations are measured from the sample's own mean, which sits closer to the sample points than the true mean does. Subtracting one, known as Bessel's correction, compensates. On small samples the gap is large: with 10 observations the two differ by about 5 percent, and with 4 by nearly 15.

Which one to use

Use the population formula only when your data genuinely is the entire group you are describing — every employee in a company, every item produced in one batch. Use the sample formula whenever the data is a subset from which you are inferring something about a wider group, which covers almost all real analysis. Spreadsheets expose both: STDEV.S and STDEV.P in Excel, and the default in most statistical software is the sample version.

What the number means

Standard deviation is in the same units as the data, which variance is not, and that is why it is the one usually reported. For roughly normal data, about 68 percent of values fall within one standard deviation of the mean, 95 percent within two and 99.7 percent within three. That rule is what makes the figure interpretable, and it fails badly for skewed or multi-modal data, where a standard deviation can be reported precisely and mean very little.

Comparing spread across different scales

A standard deviation of 5 is large for exam scores out of 20 and negligible for house prices. The coefficient of variation — standard deviation divided by the mean — makes such comparisons possible by expressing spread as a proportion. It breaks down when the mean is near zero or the data contains negatives.

Outliers dominate it

Because deviations are squared before averaging, a single distant value moves standard deviation far more than it moves the mean. One mistyped figure can double it. Where robustness matters, the interquartile range or median absolute deviation describes spread without that sensitivity — and a large gap between standard deviation and IQR is itself a signal to go looking for outliers.

It says nothing about accuracy

Low standard deviation means measurements agree with each other, not that they are correct. A systematically biased instrument produces tightly clustered wrong answers. Precision and accuracy are separate properties, and only one of them is what standard deviation measures.