Standard Deviation Calculator

Use the sample version (divide by n − 1) when your numbers are drawn from a larger group whose spread you want to estimate: survey answers, test scores from a few classes, measurements from one batch. Use the population version (divide by n) only when the numbers are every member of the group you care about. When in doubt it is a sample, which is also what the STDEV.S function in spreadsheets assumes.

Because the deviations are measured from the sample’s own mean, which sits closer to the sample’s values than the true population mean does. That makes the squared deviations slightly too small on average, and dividing by n − 1 instead of n (Bessel’s correction) removes that bias from the variance. With 6 values it raises the variance by a fifth; with 1,000 values the difference is 0.1%.

The variance is the average squared deviation, so it is in squared units: cm², kg², points². The standard deviation is its square root, back in the units of the data, which is why it is the figure people quote. For 4, 8, 15, 16, 23 and 42 the sample variance is 182 and the standard deviation about 13.49.

How many of your values sit within one or two standard deviations of the mean. For data that follows a bell-shaped normal distribution, about 68% of values fall within one and 95% within two. Skewed data or outliers can move your own counts well away from those figures, and that difference is itself worth knowing.

The standard error of the mean, s ÷ √n, says how precisely the sample mean estimates the population mean; it shrinks as the sample grows. The coefficient of variation divides the standard deviation by the mean, giving the spread as a percentage of the mean, so data on different scales can be compared. It is undefined when the mean is 0 and hard to read when values can be negative.