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

What it is

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.

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.