Variance Calculator — Sample & Population Variance and Std Dev

Formula

How this is calculated

Variance measures how spread out values are around their mean. For each observation the squared deviation from the mean is computed; those squared deviations are summed and divided by either n (population variance, σ²) or n − 1 (sample variance, s²). The n − 1 denominator is Bessel's correction — when you have a sample drawn from a larger population, dividing by n − 1 corrects for the fact that the sample mean is slightly closer to all sample points than the true population mean would be, giving an unbiased estimate of σ².

Standard deviation (SD) is the square root of variance and shares the units of the original data, making it far more interpretable. The coefficient of variation (CV = SD / |mean| × 100%) expresses spread as a proportion of the mean, enabling comparison across data sets with different units or very different mean magnitudes. The standard error of the mean (SE = s / √n) estimates how much the sample mean would vary across repeated samples of the same size — it shrinks as n grows, reflecting the precision gain from larger samples.

When to use which: use population variance (σ²) only when your data is the entire population with no sampling uncertainty. In practice — surveys, experiments, quality-control samples — use sample variance (s²). The distinction becomes negligible for n above ~30. The distribution curve shown uses the sample standard deviation and assumes normality purely for visualisation; your data may not follow a normal distribution.