- 1Enter your data points in the text area, separated by commas, spaces, or line breaks.
- 2The calculator displays two results side by side: the sample variance (s²) and the population variance (σ²).
- 3The mean and dataset size (N) are shown below for reference.
Variance measures how far each value in a dataset is from the mean, on average. It is calculated by averaging the squared differences from the mean. Because the differences are squared, variance is always zero or positive. A variance of zero means all values are identical. Variance is the foundation for standard deviation — the standard deviation is simply the square root of the variance.
Used when the data represents the entire population. Divides the sum of squared differences by N.
σ² = Σ(xᵢ − μ)² / NUsed when the data is a sample from a larger population. Divides by (N − 1) to correct for estimation bias (Bessel's correction).
s² = Σ(xᵢ − x̄)² / (N − 1)Mean = 5.2 → Squared differences: (4−5.2)²=1.44, (8−5.2)²=7.84, (6−5.2)²=0.64, (5−5.2)²=0.04, (3−5.2)²=4.84 → Sum = 14.8Mean = 8; Σ(x−8)² = 25 + 9 + 0 + 4 + 36 = 74; Population variance = 74 / 5Sample variance = 74 / (5 − 1)- Use Population Variance (σ²) when your dataset includes every member of the group you are studying.
- Use Sample Variance (s²) when your data is a sample from a larger population and you want to estimate the full population's variability.
- Variance is expressed in squared units (e.g., if your data is in metres, variance is in square metres). To return to the original units, take the square root to get the standard deviation.
- With only one data point, the sample variance is undefined because the denominator (N − 1) equals zero.
What is the difference between variance and standard deviation?
Standard deviation is the square root of variance. They both measure dispersion, but standard deviation is in the same units as the original data, while variance is in squared units.
Why does the sample variance divide by (N − 1) instead of N?
Dividing by (N − 1) is called Bessel's correction. It compensates for the fact that a sample tends to underestimate the true population variance. This correction makes the sample variance an unbiased estimator of the population variance.
Why is variance always zero or positive?
Each term is a squared difference from the mean, and squares cannot be negative. A variance of zero means every value is identical.
When should I use the sample version?
When your numbers are a sample drawn from a larger population you want to describe. Use the population version when your data already covers the entire group.