In statistics a population is the complete set of things you care about – every employee of one company, every game a team played this season, every pixel in an image. When your data really are the whole population, you do not estimate the spread, you calculate it exactly, and the right measure is the population variance σ²: the plain average of the squared deviations from the mean.
Population variance divides the sum of squares by N, the number of values. The sample variance, by contrast, divides by n − 1 (Bessel’s correction) because a sample’s values sit slightly closer to their own mean than to the unknown population mean. Using the sample formula on a full population would overstate the spread; using the population formula on a sample would understate it. The calculator shows both so you can see the difference, which shrinks as N grows.
Variance is in squared units, so its square root, the population standard deviation σ, is usually easier to interpret: it is in the same units as the data.
How to use the Population Variance Calculator
- Population data (all values)
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How it works
Enter all values of the population. The calculator finds the mean μ, subtracts it from every value, squares the differences, adds them to get the sum of squares and divides by N. The table shows every step.
Worked example
The eight values 2, 4, 4, 4, 5, 5, 7, 9 add to 40, so μ = 5. The deviations are −3, −1, −1, −1, 0, 0, 2 and 4; their squares 9, 1, 1, 1, 0, 0, 4 and 16 add to 32. The population variance is 32 ÷ 8 = 4 and the population standard deviation √4 = 2. Treated as a sample, the same data would give s² = 32 ÷ 7 = 4.571 and s = 2.138.
Population or sample?
| Population | Sample | |
|---|---|---|
| Symbol | σ² | s² |
| Divide by | N | n − 1 |
| Use when | You have every value | Data are a subset of a larger group |
| Excel | VAR.P | VAR.S |
Formula
σ² = Σ(xᵢ − μ)² ÷ N · μ = Σxᵢ ÷ N · σ = √σ² · shortcut: σ² = Σxᵢ² ÷ N − μ²
Worked example
Inputs — Population data (all values): 2, 4, 4, 4, 5, 5, 7, 9.
Results — Population variance σ² = 4; Population standard deviation σ = 2; Population mean μ = 5; Number of values N = 8; Sum of squares Σ(x − μ)² = 32; Sum of squared values Σx² = 232.
Key terms
- Population
- The complete set of individuals or values of interest.
- Population variance (σ²)
- The average squared deviation from the population mean.
- Sum of squares
- Σ(xᵢ − μ)², the total squared deviation from the mean.
Frequently asked questions
What is the formula for population variance?
σ² = Σ(xᵢ − μ)² ÷ N: subtract the mean from each value, square, add up and divide by the number of values.
When do I use population variance instead of sample variance?
When the data include every member of the group you want to describe, e.g. the test scores of all 28 pupils in one class. If the class is a sample of a whole school, use the sample variance.
What is the population variance of 2, 4, 4, 4, 5, 5, 7, 9?
The mean is 5, the squared deviations add to 32, so σ² = 32 ÷ 8 = 4 and σ = 2.
Why is the sample variance larger?
It divides the same sum of squares by N − 1 instead of N. For 8 values that is 32 ÷ 7 = 4.571 instead of 4.
Can population variance be zero?
Yes, when all values are identical. It can never be negative.
What is the shortcut formula?
σ² = Σx² ÷ N − μ². For the example 232 ÷ 8 − 25 = 4. It is quick by hand but can lose precision on a computer with large numbers.
What are the units of variance?
The square of the data's units – cm² for heights in cm – which is why the standard deviation is usually reported.