Sample and population standard deviation, variance and mean — with worked steps
Paste or type your numbers below to get the sample and population standard deviation instantly, along with the mean, variance and a full breakdown of how the answer is reached.
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Standard deviation measures how spread out a set of numbers is around their mean. A small standard deviation means the values cluster tightly around the average; a large one means they are widely scattered. Because it is expressed in the same units as the original data, it is usually easier to interpret than variance, which is in squared units.
The two versions differ only in their denominator. The population standard deviation divides the sum of squared deviations by n and is used when your data represents the entire group you care about. The sample standard deviation divides by n − 1 (Bessel's correction) and is used when your data is a sample drawn from a larger population — the slightly larger result corrects the tendency of a sample to underestimate the true spread. When in doubt for statistics coursework, the sample version is the usual default.
The procedure is: find the mean, subtract it from each value to get the deviations, square those deviations, add them up, divide by n (population) or n − 1 (sample) to get the variance, then take the square root. The step-by-step panel above shows the mean and both standard deviations for your specific numbers.
The population version divides by n and describes a complete data set; the sample version divides by n − 1 and estimates the spread of a wider population from a sample. This calculator reports both.
Neither on its own — it simply means more variability. Whether that is desirable depends on context: low variability is good for a manufacturing process, but high variability might be expected for, say, household incomes.
No. It is the square root of an average of squared numbers, so it is always zero or positive. It equals zero only when every value is identical.