Use this variance calculator to measure data dispersion with explicit sample and population options. Returns variance, standard deviation, and supporting descriptive statistics from raw input.
What is variance?
Variance is a measure of how spread out values are around their mean. A high variance means values are widely dispersed; a low variance means they cluster closely. It is one of the foundational statistics in quantitative analysis because it underpins standard deviation, confidence intervals, regression, and almost every inferential method that follows.
The intuition is straightforward: variance calculates the average squared distance of each value from the mean. Squaring the distances prevents positive and negative deviations from canceling each other out, and it gives more weight to large deviations than to small ones.
Variance formula
Sample variance: s² = Σ(xi − x̄)² / (n − 1)
Population variance: σ² = Σ(xi − μ)² / N
Where s² is the sample variance, σ² is the population variance, xi is each individual observation, x̄ is the sample mean, μ is the population mean, n is the number of observations in the sample, and N is the total population size. The Σ notation means squared deviations are summed across all observations before dividing.
The denominators differ. Sample variance uses n − 1 (Bessel’s correction) to produce an unbiased estimate of the population variance from a sample. Using n instead of n − 1 when working with a sample systematically underestimates the true dispersion, which is why the distinction is not just a technicality.
How the variance calculator works
The calculator keeps the sample versus population choice explicit rather than resolving it silently, which matters when the intent is to report or communicate the result with precision. Variance is returned alongside standard deviation, mean, and count so interpretation has the full context it needs.
Variance interpretation benefits from examining the raw values alongside the result. A single outlier can dominate the dispersion estimate, particularly in smaller samples. Seeing the mean and knowing the sample size makes it easier to judge whether the variance is reflecting genuine spread in the data or being driven by a small number of extreme observations.
How variance is used in statistics and finance
Variance appears in hypothesis testing (F-tests compare variances directly), quality control (control charts use it to flag process instability), and finance (portfolio variance incorporates correlations between assets, making it central to diversification analysis). In everyday data work, variance and standard deviation are the first checks for whether a distribution is tight or dispersed, and whether a single summary metric meaningfully represents the data or obscures important spread.