Standard Deviation Calculator — Sample & Population

What standard deviation measures

Standard deviation describes how spread out a set of numbers is around its mean. A small value means the numbers cluster tightly together; a large value means they are widely scattered. Two classes can both average 70 % on a test, yet one has scores between 65 and 75 and the other between 40 and 100. The mean cannot tell them apart, but the standard deviation can. Because it is expressed in the same units as the data, it is easy to interpret, which is why it appears in science, finance, quality control and everyday statistics.

Sample versus population

The only difference between the two versions is the divisor. The population standard deviation σ divides the sum of squared deviations by n and applies when you have every value in the group. The sample standard deviation s divides by n − 1 and is used when your numbers are a sample drawn from a bigger group. This calculator shows both at once and highlights the one you select. A sample needs at least two values, because with a single number there is no spread to measure.

The deviation table lists each value, its distance from the mean and that distance squared, so you can follow the working by hand or copy it into a report. Alongside the standard deviation you get the variance, the sum of squares, the standard error of the mean and the coefficient of variation.

Reading the result

For data that follows a bell curve, roughly 68 % of values fall within one standard deviation of the mean, 95 % within two and 99.7 % within three. Values far outside that range are candidates for outliers worth checking. Remember that standard deviation is sensitive to extreme values, because the differences are squared. All the maths runs locally in your browser, so your data is never uploaded.