Variance Calculator — Population and Sample Variance

Sample variance: s² = Σ(x⊂i; − x̅)² / (n−1)
Population variance: σ² = Σ(x⊂i; − μ)² / n

Standard deviation = √variance

Variance measures the average squared distance from the mean. Standard deviation is its square root, returning results to the original units. Both measure spread — variance is used in statistical formulas; standard deviation in reporting.

Sample variance

=VAR(A1:A20)

Population variance

=VARP(A1:A20)

Standard deviation

=STDEV(A1:A20)

Variance ExplainedSearch YouTube tutorials →

Variance vs Standard Deviation

Variance and standard deviation both measure how spread out a dataset is, but variance is expressed in squared units while standard deviation is in the original units. If you're measuring heights in centimeters, variance is in cm² — a unit with no intuitive meaning. Standard deviation is in centimeters, which makes sense and is therefore more commonly reported.

Despite being less interpretable on its own, variance is mathematically convenient. In statistics, when you combine independent random variables, their variances add directly. You cannot simply add standard deviations. This is why variance appears in ANOVA, regression, and portfolio theory, even though standard deviation is the metric reported to audiences.

The 68-95-99.7 rule (that 68% of data falls within one standard deviation of the mean, 95% within two, 99.7% within three) is stated in standard deviations rather than variance — a reflection of how much more interpretable standard deviation is for communicating results.

Sample vs Population: Why It Matters

Population variance divides by N (the count of all values). Sample variance divides by N-1. The N-1 correction, called Bessel's correction, makes sample variance an unbiased estimator of the true population variance. Without it, sample variance would systematically underestimate the true spread of the population.

In almost all real-world applications, you are working with a sample (a subset of data) rather than an entire population. Unless you have literally measured every member of the population you're studying — every employee at a company, every student in a school — use sample variance.

Frequently Asked Questions

Squaring serves two purposes: it makes all deviations positive (preventing positive and negative deviations from canceling), and it weights larger deviations more heavily than small ones. An outlier 10 units away contributes 100 to the sum, not 10 — making variance (and standard deviation) sensitive to extreme values.

CV = Standard Deviation / Mean × 100%. It normalizes variability by the mean, enabling comparison of spread across datasets with different units or scales. A CV of 15% means the standard deviation is 15% of the mean. Useful when comparing variability across measures in different units.

In statistical formulas that combine multiple variance sources: portfolio variance in finance (sum of weighted variances plus covariance terms), ANOVA (Analysis of Variance) tests, regression analysis, and any situation where you're adding or comparing variance from independent sources. For reporting and data exploration, standard deviation is almost always preferred.

A variance of zero means every value in the dataset is identical — there is no spread at all. Any non-zero variance indicates variability. Variance cannot be negative — squaring the deviations ensures all contributions are positive or zero.