What Is a Population Variance Calculator?
A population variance calculator measures how spread out a complete set of data is around its mean. Variance, written σ² (sigma squared), is the average of the squared differences from the mean — a higher variance means the data is more dispersed. Enter your data set and the calculator returns the population variance, the mean, and the standard deviation, with the steps shown.
How to Use the Calculator
- Enter your data values separated by commas or spaces.
- Calculate — see the mean, variance (σ²), and standard deviation.
The Population Variance Formula
σ² = Σ(xᵢ − μ)² ÷ N
where xᵢ are the data values, μ is the population mean, and N is the number of values. You subtract the mean from each value, square the result, add them up, and divide by N.
Step-by-Step Example
For the data 4, 8, 6, 5, 3: the mean is 26 ÷ 5 = 5.2. The squared differences are 1.44, 7.84, 0.64, 0.04, and 4.84, summing to 14.8. Divide by N = 5 to get a population variance of 2.96. The standard deviation is √2.96 ≈ 1.72.
Population vs Sample Variance
| Type | Divides By | Use When |
|---|---|---|
| Population variance | N | You have data for the entire group |
| Sample variance | N − 1 | You have a sample of a larger group |
Using N − 1 for samples (Bessel's correction) gives an unbiased estimate of the larger population's variance.
Why Variance Matters
- Measuring spread: quantifies consistency or volatility in data.
- Standard deviation: its square root, in the same units as the data.
- Statistics and finance: risk and variability are often measured by variance.
Frequently Asked Questions
What is population variance?
Population variance is the average of the squared differences between each data value and the mean, for a complete population. It measures how spread out the data is.
What is the formula for population variance?
σ² = Σ(xᵢ − μ)² ÷ N — subtract the mean from each value, square it, sum them, and divide by the number of values.
What is the difference between population and sample variance?
Population variance divides by N and is used when you have the entire group, while sample variance divides by N − 1 to give an unbiased estimate from a sample.
How is variance related to standard deviation?
Standard deviation is the square root of the variance, expressed in the same units as the original data, which makes it easier to interpret.
Is this population variance calculator free?
Yes — it is completely free, requires no signup, and shows the mean, variance, and standard deviation.