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What Is Population Variance?

Population variance, written σ², describes the spread of a group when you have measured every member of it. There is nothing to estimate, so you take the average of the squared deviations from the mean exactly as it is: divide by N, the number of values. Typical examples are the marks of every student in one class, the weight of every parcel in a single shipment you weighed in full, or one company's monthly sales for a completed year. The key question is not how big your data set is, but whether it is the whole group you want to describe.

Worked Example: A Complete Tutorial Group

A tutor has exactly six students and wants to describe how spread out their test scores are. These six scores are the entire group, so population variance applies.

Data (marks):70, 85, 75, 90, 65, 95

1. Mean. x̄ = 480 / 6 = 80
2. Deviations and squares. Subtract the mean from each value and square the result:

3. Sum of squares. Σ(xᵢ − x̄)² = 700
4. Population variance. σ² = 700 / 6 = 116.6667; population standard deviation σ = √116.6667 = 10.8012

Population variance σ² = 700 / 6 ≈ 116.67, so the population standard deviation is σ ≈ 10.80 marks. If the same six scores were treated as a sample, s² = 700 / 5 = 140; the gap between the two answers is largest for small groups like this one.

Formula

Population variance uses the population mean μ and divides by N:

How to Calculate Population Variance Step by Step

1. Confirm you have the entire population.Every member of the group you are describing must be in the data.
2. Find the population mean (μ).Add all values and divide by N.
3. Compute deviations.Subtract μ from each value.
4. Square each deviation.Calculate (xᵢ − μ)².
5. Add the squared deviations.Σ(xᵢ − μ)² is the sum of squares.
6. Divide by N.σ² = Σ(xᵢ − μ)² / N.

More Population Variance Examples

How to Use This Population Variance Calculator

Switch the calculator to Population, then:

1. Enter every value in the population, separated by commas, spaces or line breaks.
2. Check that the result card says Population Variance (σ²).
3. Read σ², the population standard deviation σ and the mean μ.
4. Open the calculation steps to see the division by N written out with your numbers.

What the Calculator Shows You

• The exact population variance and standard deviation, with no correction applied.
• The sample figure for the same data, so you can see the difference if you are unsure which applies.
• Every deviation and squared deviation.
• Calculations run in your browser; your numbers are not sent anywhere.

When to Use Population Variance