Variance

Variance measures how widely values are scattered around their mean. Deviations from the mean are squared before averaging, so large departures carry more weight.

ByHoang TruongUpdated June 30, 2026

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A formula card presents the sample variance formula: s² = Σ(xᵢ − x̄)² ÷ (n − 1), where xᵢ − x̄ is each observation's deviation from the mean and n − 1 is the degrees of freedom. A worked example uses a bakery with a mean daily sales figure of €440: five squared deviations sum to 9,400, giving a variance of 9,400 ÷ 4 = 2,350 €² and a standard deviation of approximately €48. Dividing by n − 1 rather than n yields an unbiased estimate of the population variance.

Where it fits

TopicDescriptive StatisticsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

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

Variables

$s^{2}$: Sample variance
$x_{i}$: Value of observation i
$\overset{x}{ˉ}$: Sample mean
$n$: Number of observations in the sample (observations)

Sample variance; denominator (n − 1) rather than n corrects for bias when estimating the unknown population variance σ².

LaTeX

\sigma^2 = \frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}

Variables

$σ^{2}$: Population variance
$μ$: Population mean
$N$: Population size (observations)

Population variance; divides by N because all values are known and no estimation is involved.

LaTeX

s = \sqrt{s^2}

Taking the square root of the variance returns the spread to the original unit of measurement, making it directly interpretable.

Check yourself

PracticeCORE

A sample of five daily output figures has a mean of 200 units. The sum of squared deviations from the mean is 2,000. What is the sample variance?

Select an answer to check your understanding.