Sample Variance

The sample variance s² = SS/(n − 1) is used when your data are a subset of a larger population, and you want to estimate that population's variance. The population variance σ² = SS/n is used when your data include every member of the population. The two differ only in the denominator: n − 1 versus n. For large samples the difference is negligible — for n = 1,000, dividing by 999 instead of 1,000 changes the result by 0.1 %. But for small samples the difference matters: for n = 5, s² is 25 % larger than σ². In statistical inference the sample variance is almost always what you want, because you rarely have the entire population.

Because each deviation is squared before being summed. A point at 2 standard deviations from the mean contributes 4 to SS; a point at 10 standard deviations contributes 100 — 25 times more. This squaring amplifies the effect of extreme values disproportionately, which is a feature in some applications (variance is the natural measure of squared error) and a bug in others. If your dataset may contain outliers or heavy tails, use robust alternatives: the interquartile range (IQR = Q₃ − Q₁), the median absolute deviation (MAD), or a trimmed variance that removes the highest and lowest k percent of the data before computing SS.

Because the deviations (xᵢ − x̄) are themselves measured in the data's original units, and squaring them produces squared units. If x is in metres, then (xᵢ − x̄)² is in m². If x is in dollars, then s² is in dollars². This is why the standard deviation s = √s² is more intuitive — it restores the original units and can be compared directly to the mean. But the variance is mathematically cleaner: it is additive for independent variables, it appears naturally in the central limit theorem, and it is the parameter that characterises the normal distribution. In practice, both s² and s are reported, and the choice depends on the application.

There is no sharp threshold, but rules of thumb exist. For exploratory work, n ≥ 10 gives a rough estimate. For routine analysis, n ≥ 30 is common. For precise estimation of σ², n ≥ 100 is preferred, especially when the underlying distribution is not normal. The relative standard error of s² is approximately √(2/(n − 1)) for normal data: for n = 10 it is 47 %, for n = 30 it is 26 %, for n = 100 it is 14 %, and for n = 1,000 it is 4.5 %. So with n = 10, your sample variance could easily be off by half or double from the true population variance. With n = 100 the estimate is usually within about 15 % of the truth.

The standard error of the mean SE = s/√n measures how precisely the sample mean estimates the population mean μ. It is directly related to the sample standard deviation through the factor 1/√n. A larger sample gives a smaller standard error — the sample mean becomes a more precise estimate. For the sample variance itself, the standard error (for normal data) is SE(s²) = s²·√(2/(n − 1)). The 95 % confidence interval for the mean is approximately x̄ ± 1.96·SE for large n, or x̄ ± t_{0.025,n−1}·SE using the t-distribution for smaller samples. All of these depend on s², which is why the sample variance is such a central quantity in statistics.

Five steps. (1) Compute the mean x̄ = (Σxᵢ)/n. (2) For each value, subtract the mean to get the deviation dᵢ = xᵢ − x̄. (3) Square each deviation: dᵢ². (4) Sum the squares: SS = Σdᵢ². (5) Divide by n − 1: s² = SS/(n − 1). For example, given the dataset {4, 7, 8, 10, 11}: mean = 8; deviations = {−4, −1, 0, 2, 3}; squares = {16, 1, 0, 4, 9}; SS = 30; s² = 30/4 = 7.5. The dataset analyzer above performs all five steps automatically for arbitrary datasets — just paste your numbers and click Analyze.