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Standard Deviation Calculator

Calculate sample or population standard deviation, variance, mean and standard error with a deviation table.

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Enter up to 1,000 finite numbers separated by commas, spaces or newlines. Choose whether the list is a sample or the whole population.

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Formula and variables

x̄ = Σx/n; s² = Σ(x−x̄)²/(n−1); σ² = Σ(x−μ)²/N; SD = √variance; SE = SD/√n

n (or N) is the count. x̄ is the sample mean, μ the population mean, s the sample SD and σ the population SD. The standard error is for a mean of independent draws.

Worked example

For 2, 4, 4, 4, 5, 5, 7, 9: n = 8, sum = 40, mean = 5. Squared deviations sum to 32. Population variance = 32/8 = 4 and SD = 2. Sample variance = 32/7 ≈ 4.57143 and SD ≈ 2.13809.

When to use this formula

Use SD to describe spread in the original measurement units. Choose sample mode when observations are used to estimate a larger population’s variability. An independent-observation SE is not appropriate for dependent or complex-survey data.

Common mistakes

Population and sample formulas have different denominators. SD describes spread of observations; SE describes sampling variability of a mean. An enumerated whole population has no sampling uncertainty about its own mean.

Why can’t I calculate sample SD from one value?

The denominator n−1 would be zero. Population SD for a one-value population is zero; these are different questions.

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