What Is Population Variance?
Population variance measures the average squared deviation of each data point from the population mean. Unlike sample variance, it divides by N (the total population size) rather than N-1. This is the true measure of spread for a complete population.
Variance is fundamental to statistics, forming the basis for standard deviation, confidence intervals, hypothesis tests, and many other statistical methods. A larger variance indicates data points are more spread out from the mean, while a variance of zero means all values are identical.
Formula
Where μ is the population mean, x_i is each individual value, and N is the population size.
Population vs Sample Variance
| Feature | Population Variance | Sample Variance |
|---|---|---|
| Symbol | σ² | s² |
| Denominator | N | n - 1 |
| When to use | Complete data set | Subset of population |
| Bias | Exact (no bias) | Unbiased estimator |
Example
For data {4, 8, 6, 5, 3}: Mean = 5.2. Sum of squared deviations = (4-5.2)^2 + (8-5.2)^2 + (6-5.2)^2 + (5-5.2)^2 + (3-5.2)^2 = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8. Population variance = 14.8 / 5 = 2.96.
Frequently Asked Questions
When should I use population variance vs sample variance?
Use population variance when your data set includes every member of the population. Use sample variance when your data is a sample drawn from a larger population. In practice, most data is a sample, so sample variance (dividing by n-1) is more common.
Can variance be negative?
No. Variance is always non-negative because it is the average of squared deviations. Squared numbers are always zero or positive. A variance of zero means all values in the data set are identical.
What are the units of variance?
Variance is expressed in squared units of the original data. If your data is in meters, the variance is in meters squared. This is why standard deviation (the square root of variance) is often preferred for interpretation, as it shares the original units.