Standard deviation

Standard deviation is the square root of the variance. It measures how widely values in a dataset spread around their mean and is expressed in the same units as the original data, so it is directly interpretable.

Also known asSD · std dev

ByHoang TruongUpdated June 30, 2026

See it move

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A bell-shaped frequency distribution plots monthly sales in €'000 on the horizontal axis, with the mean at €70,000 and a sample standard deviation of approximately €7,900. The region spanning one standard deviation either side of the mean — roughly €62,100 to €77,900 — covers about 68% of stores. Because the standard deviation is expressed in the same currency units as the original data, it can be read alongside the mean directly, without further conversion.

Where it fits

TopicDescriptive StatisticsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n - 1}}

Variables

$s$: Sample standard deviation
$x_{i}$: i-th observation
$\overset{x}{ˉ}$: Sample mean
$n$: Number of observations in the sample

Denominator n − 1 (not n) corrects for using the sample mean as a stand-in for the unknown population mean.

LaTeX

\sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}}

Variables

$σ$: Population standard deviation
$μ$: Population mean
$N$: Population size

Population formula. Denominator is N because the true mean μ is known.

Check yourself

PracticeCORE

Two investment portfolios each report a mean monthly return of 2%. Portfolio A has a standard deviation of 0.5% and Portfolio B has a standard deviation of 3.2%. What does this tell an investor about the two portfolios?

Select an answer to check your understanding.