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Population vs Sample Variance Calculator

See population variance (÷N) and sample variance (÷N−1) side by side from one list of numbers, with both standard deviations.

Values

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How it works

σ² = Σ(x−μ)² ÷ N · s² = Σ(x−x̄)² ÷ (N−1)

Use population variance when your numbers are the whole group you care about. Use sample variance when they are a sample standing in for a bigger population — dividing by N−1 (Bessel’s correction) removes the downward bias that comes from measuring spread around the sample’s own mean.

Worked example

For 4, 8, 15, 16, 23, 42 the mean is 18 and Σ(x−x̄)² = 940. Population variance = 940/6 = 156.67; sample variance = 940/5 = 188.

Frequently asked questions

Which one does Excel use?

VAR.S and STDEV.S divide by N−1 (sample); VAR.P and STDEV.P divide by N (population).

Does the difference matter for big samples?

Barely — with n = 1,000 the two differ by 0.1%. It matters most for small samples.

Related calculators

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.

Standard Error Calculator

Find the standard error of the mean from a sample standard deviation and size — how much the sample mean is expected to vary.

Coefficient of Variation Calculator

Compute the coefficient of variation (CV, or relative standard deviation) from a mean and standard deviation, or paste raw data. Get sample and population CV, the small-sample unbiased CV, a confidence interval for the CV, a two-set comparison and the full working.

For education — check assumptions (sample size, distribution) before using statistics in real decisions.