Standard deviation calculator

What standard deviation describes

Standard deviation summarizes how far values typically lie from their arithmetic mean. A value of zero means every entered observation is identical. A larger result indicates more spread in the same measurement units as the original data. Variance is the square of the standard deviation and therefore uses squared units.

Sample versus population

Select population when the entered values are every member of the group you intend to describe. Population variance divides the sum of squared deviations by n. Select sample when the values are observations used to estimate a broader population. The sample calculation divides by n − 1, commonly called Bessel’s correction, and requires at least two values.

How the calculation is performed

The calculator finds the mean and accumulates squared differences from it. It then divides by the selected denominator and takes the square root. Inputs are parsed as finite decimal numbers. Results also include the count, sum, mean, and variance so you can check the data and distinguish squared variance from standard deviation.

Trace a small dataset

For 2, 4, and 6, the mean is 4 and the squared deviations are 4, 0, and 4, totaling 8. Treating these three values as the full population gives variance 8 ÷ 3 ≈ 2.6667 and standard deviation ≈ 1.633. Treating them as a sample gives variance 8 ÷ 2 = 4 and standard deviation 2. The data did not change; the denominator changed because the inferential question changed. Recalculate the sum and mean first if the result looks wrong, and preserve full precision until the final display.

A single summary cannot reveal outliers, clusters, skew, data-entry errors, dependence, or whether a sample is representative. Review the underlying distribution before drawing statistical conclusions.