What Is How to Find Population Variance?
Population variance is the average squared distance of each value from the mean, when your data include every member of the group. The formula divides the sum of squared deviations by N.
If you only have a sample, the estimated population variance divides by n − 1 instead. That small change removes the bias that comes from measuring spread around a sample mean rather than the true mean.
Choose your case, then enter a list of values or just n, Σx and Σx² if that is what your problem gives you. The steps show the mean, the sum of squares and the division.
How to Use the How to Find Population Variance Calculator
- Choose whether your data are the whole population or a sample.
- Enter the values, or enter n, Σx and Σx² if you already have the sums.
- Follow the steps from the mean to the final variance.
How to Find Population Variance Formula
Population variance
Estimated population variance
From sums
How to Find Population Variance Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| N or n | Number of values | N for a population, n for a sample |
| μ or x̄ | The mean | |
| Σ(x − mean)² | Sum of squared deviations | |
| σ² and s² | Population variance and its sample estimate |
How to Find Population Variance Worked Example
With the example values loaded in the calculator above, the steps are:
- Mean = 33 ÷ 6 = 5.5.
- Square each deviation from the mean and add: Σ(x − mean)² = 17.5.
- Population variance σ² = 17.5 ÷ 6 = 2.916667.
Result: Population variance (σ²) = 2.916667; Population standard deviation (σ) = 1.70783.
Worked example: 4, 8, 6, 5, 3, 7
The mean is 33 ÷ 6 = 5.5. The squared deviations are 2.25, 6.25, 0.25, 0.25, 6.25 and 2.25, which add to 17.5.
Population variance = 17.5 ÷ 6 = 2.9167. Estimated population variance from a sample = 17.5 ÷ 5 = 3.5. The standard deviations are 1.7078 and 1.8708.
Population or estimated population variance?
If your values are every member of the group, divide by N. If they are a sample used to learn about a larger group, divide by n − 1. The estimate is slightly larger, which corrects for the fact that a sample sits closer to its own mean than to the true mean.
Where How to Find Population Variance Is Used
Statistics coursework
Practise the difference between σ² and the unbiased estimate.
From a summary table
Find variance when only Σx and Σx² are given.
Quality checks
Measure the spread of a full batch rather than a sample.
Common How to Find Population Variance Mistakes to Avoid
- If your textbook says estimated or unbiased variance, choose the sample option.
- The sums shortcut is handy when you only have Σx and Σx² from a summary table.
How to Find Population Variance Terms and Definitions
- Population variance σ²
- The mean squared deviation of all values in a population.
- Unbiased estimator
- A statistic whose average over many samples equals the true value.
How to Find Population Variance FAQs
How do you find population variance?
Find the mean, subtract it from each value, square the results, add them and divide by N, the number of values in the population.
What is the estimated population variance?
It is the sample variance, which divides by n − 1 so that, on average, it equals the true population variance.
How do I find variance from Σx and Σx²?
Compute Σx² − (Σx)² ÷ n for the sum of squares, then divide by N for a population or n − 1 for a sample.
Why is population variance smaller than sample variance?
Dividing by n instead of n − 1 gives a smaller number for the same data.
Is population variance ever negative?
No. It is an average of squares, so it is zero or positive.
How do I get the standard deviation?
Take the square root of the variance.