Standard Deviation Calculator
Calculate sample & population standard deviation, variance, and data metrics.
Population, sample, and the n−1 you keep seeing
Standard deviation measures typical distance from the mean. Compute the mean, square each deviation from it, average those squares to get the variance, then take the square root to return to the original units. The squaring exists so that deviations above and below the mean cannot cancel to zero.
The division step differs depending on what your data is. Population standard deviation divides by n and applies when you have every member of the group.Sample standard deviation divides by n − 1 and applies when you have a subset and are estimating the wider population. The reduced denominator — Bessel's correction — exists because a sample's own mean sits closer to its own points than the true mean does, which biases the estimate downward. On small samples the difference is substantial; past a few hundred points it barely matters.
For roughly bell-shaped data the empirical rule gives a quick read: about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. Note that standard deviation is not robust — a single extreme outlier inflates it sharply, because the deviation is squared before averaging. For skewed data the interquartile range often describes spread more honestly.
Frequently Asked Questions
What is the standard error of the mean?
The standard error (SE) is the sample standard deviation divided by the square root of the count: SE = s / √n. Standard deviation describes how spread out the individual values are; standard error describes how precisely the sample mean estimates the population mean. It shrinks as the sample grows, so quadrupling the sample size halves it.
How do I enter numbers into the calculator?
Enter numbers separated by commas, spaces, or newlines (e.g. 10, 20, 30, 40).
Is my data private?
Yes. All statistical calculations run 100% locally in your browser.
What is the difference between population and sample standard deviation?
Population standard deviation divides by N; sample divides by n-1. That correction, Bessel's correction, compensates for the fact that a sample's own mean sits closer to its data than the true population mean does, which would otherwise systematically underestimate the spread. Use the sample version unless you genuinely have every member of the population.