Variance calculator

What is variance in statistics?

Variance measures how spread out a set of numbers is. It is the average of the squared distances between each value and the mean. If every value is the same, the variance is 0. The further the values sit from the mean, the larger it gets.

Because the distances are squared, variance is in squared units. Test scores in points give a variance in points squared, which is hard to picture. That is why reports usually quote the standard deviation, the square root of the variance, while statistical tests such as ANOVA work with the variance itself.

Variance formula

There is one formula for a sample and one for a population. They differ only in what you divide by.

Sample variance: s² = Σ(x − x̄)² / (n − 1)

Population variance: σ² = Σ(x − μ)² / N

x is each value, x̄ and μ are the sample and population means, and n and N are the number of values. The top of both fractions is the sum of squares, so once you have that, either version is one division away.

How to find variance

To find the variance of a data set by hand, work out the mean first, then measure how far each value is from it:

1. Add the values and divide by how many there are to get the mean.
2. Subtract the mean from each value.
3. Square each result.
4. Add the squares to get the sum of squares.
5. Divide by n − 1 for a sample or by N for a population. That n − 1 is the degrees of freedom.

Worked example

Take the five values 2, 7, 3, 12 and 9.

• Mean: 33 / 5 = 6.6
• Differences from the mean: −4.6, 0.4, −3.6, 5.4, 2.4
• Squared: 21.16, 0.16, 12.96, 29.16, 5.76
• Sum of squares: 69.2
• Population variance: 69.2 / 5 = 13.84
• Sample variance: 69.2 / 4 = 17.3

Statistics Canada’s guide to calculating variance uses the same five numbers and divides by 5, which is the population version, to get 13.84. Paste the numbers into the calculator above and switch between sample and population to see both results.

Sample variance vs population variance

Use the sample variance when your numbers are part of a bigger group and you want to estimate the spread of that group. Survey answers from 200 people, 12 weeks of sales or one class out of a school are all samples. Most homework and research data falls here.

Use the population variance only when you have every value you care about, such as the scores of all 30 students in a class you are describing and nothing more.

The sample formula divides by n − 1 instead of n. A sample tends to sit a little closer to its own mean than to the true mean of the whole group, so dividing by n would make the variance too small. With the five values above, the sample variance is 17.3 against 13.84 for the population, a 25% difference. With 100 values the gap drops to about 1%.

The “Data set is” dropdown turns the tool into a sample variance calculator or a population variance calculator, and the results always show the other version too.

Variance of a frequency table

When values repeat, enter each one once with its count in the Frequency table tab. The formula multiplies each squared difference by its count f:

s² = Σf(x − x̄)² / (n − 1), where n = Σf

For 60 dice rolls where 1 to 6 came up 8, 11, 9, 12, 10 and 10 times, the mean is 3.5833, the sum of squares is 164.5833 and the sample variance is 164.5833 / 59 = 2.7895. Click “Dice rolls” in the calculator to see each step.

Variance vs standard deviation

They measure the same thing. The standard deviation is the square root of the variance, and the variance is the standard deviation squared. For the five values above, the sample variance is 17.3 and the sample standard deviation is √17.3 = 4.1593.

Use the standard deviation when you want to describe spread in the same units as the data, such as “the average score was 80 with a standard deviation of 9 points.” It is also the only one of the two you can draw on the same axis as the values, which is what a standard deviation graph does. Use the variance when you add up spread from different sources, because variances of independent variables add together and standard deviations do not. Our standard deviation calculator uses the same engine with the standard deviation as the main result.

How to calculate variance in Excel and Google Sheets

=VAR.S(A2:A11) sample

=VAR.P(A2:A11) population

Microsoft’s help pages for VAR.S and VAR.P describe the first as an estimate from a sample and the second as the variance of an entire population. Google Sheets accepts the same two names as well as the older VAR and VARP.

If your answer does not match a textbook, check which version the book used. For 2, 7, 3, 12 and 9, VAR.S returns 17.3 and VAR.P returns 13.84, so a gap like that usually means one of you used the other formula.