variancecalculation

Variance calculation refers to the process of measuring how data values differ from the mean of the data. It is a fundamental concept in statistics used to quantify dispersion and is defined for both populations and samples.

For a population with values x1,...,xn and mean μ, the population variance is σ^2 = (1/n) Σ (x_i − μ)^2. For a sample, the variance is s^2 = (1/(n−1)) Σ (x_i − x̄)^2, where x̄ is the sample mean.

An equivalent computation uses sums of squares: s^2 = (Σ x_i^2 − n x̄^2) / (n−1) and σ^2 = (Σ x_i^2 − n μ^2) / n. This form avoids repeated subtraction of means.

Calculation steps typically include: collect data, compute the mean, subtract the mean from each value, square the results, sum the squares, and divide by n or n−1. The resulting value is the variance; the square root gives the standard deviation.

Variance is a measure of dispersion with units squared, which can be less interpretable than the standard deviation. It is widely used in descriptive statistics and as an input in inferential methods such as hypothesis testing, ANOVA, and regression analysis.

When data carry weights, or when sampling design affects probabilities, weighted variances or population variances require adjusting the formulas. Software tools provide variants, often distinguished by degrees of freedom (ddof) in calculations such as VAR.P, VAR.S, or Python's numpy.var.