How to use this calculator
Paste numbers separated by commas, spaces, semicolons or new lines. Use decimal points, without thousands separators. Choose Population when the list is the whole group of interest, or Sample when it is a subset used to estimate that group.
The math behind it
Mean = sum of values / n Population variance = Σ(x − mean)² / n Sample variance = Σ(x − mean)² / (n − 1) Standard deviation = √variance
A worked example
For 2, 4, 4, 4, 5, 5, 7 and 9, the mean is 5. Population variance is 4 and population standard deviation is 2. Sample variance is 32/7 and sample standard deviation is about 2.13809.
Assumptions & precision
Unweighted data; up to 10,000 values. Sample mode requires at least two observations. Values must be finite and within ±10¹²; variance above 10¹⁵ is rejected. Welford’s algorithm reduces cancellation error. Displays round to six decimals, with scientific notation for tiny nonzero results.
A common question
Why divide by n − 1 for a sample?
Estimating the mean from the same sample leaves n − 1 degrees of freedom. That denominator corrects the bias of the sample variance estimator; it does not make the square-root standard deviation exactly unbiased.