Definition · Plain-language
Variance
Variance is the average of the squared deviations from the mean — a measure of how spread out a dataset is. It is the square of the standard deviation, so it is in squared units.
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How variance is calculated
To find the variance, you take each value’s deviation from the mean, square it, and average those squared deviations. Squaring serves two purposes: it removes negative signs, so deviations above and below the mean do not cancel out, and it gives extra weight to values far from the mean. The result is a single number summarising total dispersion. A variance of zero means every value equals the mean; the larger the variance, the more the data are scattered. Like the standard deviation, it has a population form (dividing by N) and a sample form (dividing by N − 1).
Why the units are squared
Because variance averages squared deviations, it is expressed in the square of the data’s units — if heights are measured in centimetres, the variance is in square centimetres. That makes variance hard to interpret directly: a "variance of 25 square centimetres" has no intuitive meaning. Taking the square root returns the figure to the original units and produces the standard deviation, which is why standard deviation is usually preferred for reporting and communication. Variance remains the more fundamental quantity in the underlying mathematics.
Where variance is used
Despite being harder to read than standard deviation, variance is central to statistical theory because it behaves neatly in calculations: variances of independent quantities add together, which standard deviations do not. This property makes variance the natural building block for techniques such as the analysis of variance (ANOVA), which partitions total variability into components to test for differences between groups. Variance also appears throughout regression, portfolio theory and experimental design wherever total dispersion needs to be decomposed or compared.
Key facts
At a glance
- Definition: the average of the squared deviations from the mean
- Relationship: equal to the standard deviation squared
- Units: the square of the data’s units
- Symbols: σ² (population) and s² (sample)
- Population: divides squared deviations by N
- Sample: divides by N − 1 for an unbiased estimate
Common misconceptions
What people often get wrong
Often heard: Variance and standard deviation are interchangeable.
Actually: They measure the same idea but differ in units. Variance is in squared units and standard deviation is its square root, in the original units. Standard deviation is easier to interpret; variance is more convenient mathematically.
Often heard: A higher variance means the data are wrong or unreliable.
Actually: No. High variance simply means the values are more spread out around the mean. Whether that is a problem depends entirely on context — some phenomena are naturally highly variable.
Often heard: Variance can be negative if many values fall below the mean.
Actually: Never. Because every deviation is squared before averaging, variance is always zero or positive. A variance of zero occurs only when all values are identical.
Going deeper
