What is Population Variance?
Population variance (σ²) measures how much the values in a full population differ from the population mean. It reflects the average squared deviation of each value from the mean and gives a sense of how spread out the data are. Unlike sample variance, the population value is calculated using all members of the population and does not require Bessel’s correction.
Population variance is a key concept in descriptive statistics and is used when you have access to data for the entire population rather than just a sample. It provides an exact measure of variability without needing to estimate. Variance values are not as interpretable as the standard deviation.
Population Variance Formula
To calculate population variance (σ²), use the following formula:
Where:
- N is the total number of values in the population
- xᵢ is each individual value in the population
- μ is the population mean
- (xᵢ − μ)² is the squared deviation of each value from the mean
This formula works by calculating the difference between each value and the population mean, squaring those differences, adding them together, and dividing by N (the number of values). Because it uses the actual population mean, there’s no need to adjust for bias as you do with sample variance.
Example Calculations Using the Population Formula
Let’s use the formula to find the population variance.
A manager wants to understand the variation in weekly sales across all five stores in a small regional chain. Because data from every store in the chain is included, this is a full population—not a sample.
The weekly sales totals (in $1000s) are: 52, 56, 60, 62, and 70.
The population mean is:
(52 + 56 + 60 + 62 + 70) ÷ 5 = 60
Next, calculate the squared deviations from the mean:
(52 − 60)² = 64
(56 − 60)² = 16
(60 − 60)² = 0
(62 − 60)² = 4
(70 − 60)² = 100
Sum = 64 + 16 + 0 + 4 + 100 = 184
Now divide by N = 5:
Population variance = 184 ÷ 5 = 36.8
This tells us that the average squared deviation from the mean sales total is 36.8 (in $1000s squared). Because all units in the population were measured, no correction is needed.
« Back to Glossary Index