How this calculator works
Population variance averages the squared distances of all included observations from their population mean. It uses N rather than the sample-estimation divisor N - 1. The square gives distant observations extra influence and produces squared units; taking a square root gives population SD in the original unit. A one-member population is valid and has zero variance.
Formula
mu = Σx/N; population variance sigma² = Σ(x_i - mu)²/N; population SD sigma = sqrt(sigma²).
Worked example
For 1, 2, 3, 4, 5, the population mean is 3. Squared distances sum to 10, so variance is 10/5 = 2 and SD is sqrt(2).
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Frequently asked questions
When should I use the sample divisor instead?
When using observations to estimate a broader population variance, the usual sample estimator divides by N - 1.
Can variance be zero?
Yes. It is zero exactly when all entered observations have the same numeric value, subject to floating-point precision.
Why are the units squared?
Every deviation is squared before averaging. For measurements in metres, variance is in square metres and SD is in metres.