What are the sources for these standard deviation calculations?

Sample and population formulas follow OpenStax Introductory Statistics 2e, section 2.7, Measures of the Spread of the Data: https://openstax.org/books/introductory-statistics-2e/pages/2-7-measures-of-the-spread-of-the-data The sample denominator is n − 1. The population denominator is n.

Standard Deviation Calculator

Enter a list of numbers for the sample and population standard deviation, plus both variances.

Smart Tips

• Sample standard deviation divides by n − 1. Population standard deviation divides by n.

• Variance is the square of the matching standard deviation. Both variances are on this page, not on a separate URL.

• With one number, the population standard deviation is 0. The sample figure is left blank because n − 1 would be 0.

• Order does not matter. Negatives are allowed.

The Formula

s = √[ Σ(x − mean)² / (n − 1) ] σ = √[ Σ(x − mean)² / n ]

s Sample standard deviation
Divide by n − 1

σ Population standard deviation
Divide by n

s² Sample variance
Square of s, before the root

σ² Population variance
Square of σ, before the root

The mean is the sum divided by n. Each deviation is the value minus that mean. Squaring keeps a negative deviation from cancelling a positive one.

Key Insights

n − 1 vs n

The sample divisor is the Bessel correction. It keeps a sample variance from running small as an estimate of a larger population.

Same page as variance

Variance is the average squared deviation. Take the square root and you have the standard deviation.

Units stay readable

Variance is in squared units. The standard deviation is back in the units of the data.

One number

A single value has no spread around itself. The population result is 0, and the sample result is not defined.

Frequently Asked Questions

The sample standard deviation divides the sum of squared deviations by n − 1. The population standard deviation divides by n. Each variance is the square of the matching standard deviation, and both are on this page.

Dividing by n − 1 is the Bessel correction. A sample variance that divided by n would tend to underestimate the variance of the larger population the sample came from.

The population standard deviation is 0, because that number is the mean. The sample standard deviation is not defined, because the divisor would be 0.

No. The mean, the variance, and the standard deviation do not depend on the order of the list.

Yes. A deviation can be negative. Squaring it makes that contribution positive, so the spread is still a positive number or zero.

Sample and population formulas follow OpenStax Introductory Statistics 2e, section 2.7, Measures of the Spread of the Data. The sample denominator is n − 1. The population denominator is n.

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Sample standard deviation

Enter values to calculate

Population standard deviation

Sample variance

Population variance

Note:The sample figure estimates spread in a larger population and divides by n − 1. The population figure describes the list itself and divides by n. This is not a t-test.