Because the mean was estimated from the same data. Using it uses up one degree of freedom, and dividing by n would systematically underestimate the spread. That correction is named after Bessel.

Sample, almost always. You use the population divisor only when your data genuinely is every member of the group rather than a draw from it.

Math & Statisticsstatistics

Variance Calculator

Variance and standard deviation on both conventions. Dividing by n-1 rather than n is Bessel correction, and on small samples the two answers differ substantially.

Also called: standard deviation calculator, sample variance.

Variance

182

Variance 182 and standard deviation 13.49, treating the data as a sample. On the other convention it would be 151.67, a difference of 20%.

Standard deviation

13.49

On the other convention

151.67

Mean

How many values

Difference between the conventions

20%

Basis used

a sample

How this is calculated

Method and background

How this is calculated

Variance is the average squared distance from the mean. The divisor is the whole question: a sample uses n-1 because the sample mean is itself estimated from the data, which uses up a degree of freedom and would otherwise bias the estimate downwards. A population uses n because the mean is known rather than estimated. On five values the two differ by twenty-five percent, which is why both are shown.

sample variance divides by n-1; population variance divides by n

x: Each value
n: How many values

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

six values as a sample

Values: 4, 8, 15, 16, 23, 42
Treat the data as: A sample

Variance182

Sum of squares 910 over 5, computed independently

Open this example

the population divisor gives a smaller figure

Values: 4, 8, 15, 16, 23, 42
Treat the data as: The whole population

Variance151.67

boundary: 910 over 6, the same data on the other convention

Open this example

Method and limits

What it assumes

Numeric values, with non-numeric entries ignored.

What it deliberately does not model

Variance is in squared units, which is why standard deviation is usually reported instead.

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Frequently asked questions

Why divide by n-1?: Because the mean was estimated from the same data. Using it uses up one degree of freedom, and dividing by n would systematically underestimate the spread. That correction is named after Bessel.
Which should I use?: Sample, almost always. You use the population divisor only when your data genuinely is every member of the group rather than a draw from it.