Variance Calculator
Paste a list of numbers to get the sample and population variance, plus the standard deviation, mean and count — with the n vs n−1 difference explained.
How it works
mean = Σx ÷ n · Sample variance s² = Σ(x − mean)² ÷ (n − 1) · Population variance σ² = Σ(x − mean)² ÷ n · s = √s²
Variance measures how far a set of numbers spreads out from its mean, in squared units. Sample variance divides the summed squared deviations by n − 1 (Bessel’s correction) because a sample tends to underestimate the true spread, while population variance divides by n when your numbers are the entire group. The standard deviation is simply the square root of the variance, which brings the figure back into the original units.
Worked example
For 4, 8, 15, 16, 23, 42 the mean is 108 ÷ 6 = 18. The squared deviations sum to 910, so the sample variance is 910 ÷ 5 = 182 and the population variance is 910 ÷ 6 ≈ 151.67. The sample standard deviation is √182 ≈ 13.49.
Sources: Variance — Wikipedia
Frequently asked questions
Should I use sample or population variance?
Use population variance (÷ n) only when your numbers are the entire group you care about. If they are a sample drawn to estimate a larger population, use sample variance (÷ n − 1), which corrects for the tendency of a sample to underestimate spread.
Why does variance use squared differences?
Squaring stops positive and negative deviations from cancelling out and gives extra weight to values far from the mean. The trade-off is that variance is in squared units, which is why the standard deviation — its square root — is usually reported alongside it.
What is Bessel’s correction?
It is the use of n − 1 instead of n in the denominator of the sample variance. Because the sample mean is closer to the sample data than the true population mean is, dividing by n − 1 removes the resulting downward bias in the estimate.
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