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What Is Sample Variance?

Sample variance, written s², is what you calculate when your numbers are a sample from a bigger group and you want to estimate how spread out that bigger group is. A survey of 200 customers, ten bolts measured from a day's production, or eight volunteers in a pilot study are all samples. The calculation is the same as for population variance except for the last step: you divide the sum of squared deviations by n − 1 instead of n. That small change, called Bessel's correction, stops the result from systematically underestimating the true spread. This page explains when to use it, why it works and how to calculate it by hand.

Worked Example: Task Completion Times

A manager times 8 employees, chosen at random from a team of 200, on the same task. The 8 times are a sample, so the variance of the whole team is estimated with s².

Data (minutes):12, 15, 11, 14, 13, 16, 9, 14

1. Mean. x̄ = 104 / 8 = 13
2. Deviations and squares. Subtract the mean from each value and square the result:

3. Sum of squares. Σ(xᵢ − x̄)² = 36
4. Sample variance. s² = 36 / (8 − 1) = 5.1429; sample standard deviation s = √5.1429 = 2.2678

Sample variance s² = 36 / 7 ≈ 5.143 square minutes, so the sample standard deviation is s ≈ 2.268 minutes. Dividing by 8 instead would give 4.5, which would understate the spread of the whole team.

Formula

Sample variance uses the sample mean x̄ and divides by n − 1:

How to Calculate Sample Variance Step by Step

1. Check that you have a sample.Your data should be a subset of the group you want to describe. If it is the whole group, use population variance instead.
2. Find the sample mean.x̄ = Σxᵢ / n.
3. Subtract the mean from each value.These deviations always add up to zero.
4. Square and add the deviations.This is the sum of squares, Σ(xᵢ − x̄)².
5. Divide by n − 1.For 8 values divide by 7, for 30 values divide by 29, and so on.
6. Take the square root if you need s.The sample standard deviation is in the same units as the data.

More Sample Variance Examples

How to Use This Sample Variance Calculator

Make sure the calculator is set to Sample, then:

1. Enter your sample values, separated by commas, spaces or line breaks.
2. Check that the result card says Sample Variance (s²). You need at least two values.
3. Read s², the sample standard deviation s and the sample mean.
4. Open the calculation steps to see the n − 1 division written out with your numbers.

What the Calculator Shows You

• The sample variance and sample standard deviation with the n − 1 divisor applied.
• The population figure for the same data, so you can see how much the correction changes the answer.
• Every deviation and squared deviation, useful for homework or checking a spreadsheet.
• Calculations run in your browser; your numbers are not sent anywhere.

When to Use Sample Variance