Variance

Updated 28 Sep 2026

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What it is
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  a measure of how far data points spread out from the mean
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Sample variance formula
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  $s^2 = \dfrac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}$
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Grouped sample-variance formula
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  with class midpoints $m_i$ and $n=\sum_i f_i$ :
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  $s^2\approx\dfrac{\sum_i f_i(m_i-\bar{x})^2}{n-1}$
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  equivalently, $s^2\approx\dfrac{n\sum_i f_i m_i^2-(\sum_i f_i m_i)^2}{n(n-1)}$
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Population formula
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  $\sigma^2=\dfrac{\sum_{i=1}^{N}(x_i-\mu)^2}{N}$
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Distinction
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  variance is expressed in squared measurement units; taking its square root gives Standard Deviation
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Caution
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  this shortcut subtracts two large, similar-magnitude sums — for large data values, this is vulnerable to Catastrophic Cancellation (see the Numerical Methods knowledge base), which can even produce a nonsensical negative variance under limited computational precision
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Related

Linked references 4