Variance Calculator

Enter a data set for a population or sample to calculate variance using the calculator below.

Separate numbers using a comma (,)
Separate numbers using a comma (,)

Results:

Variance (σ²):
 
Standard Deviation (σ):
 
Sum of Squares (SS):
 
Population Size (N):
 
Mean (μ):
 

Steps to Solve

Variance Formula

graphic showing the variance formula where sigma squared is equal to the sum of squares divided by N

Step One: Find the Mean

Mean (μ) = sumpopulation size (N)

Step Two: Find the Sum of Squares

Sum of Squares (SS) = ∑(xi - μ)²

Step Three: Calculate the Variance

Variance (σ²) = SSN



How to Find Variance

Variance is a statistical measure of the variability from the mean in a data set. It is essentially a way to measure the spread between numbers and their center in the data set.

The symbol for variance is σ², pronounced sigma-squared. Variance can also be used to measure variability in a sample data set, and in this case, it is represented as .

Variance is often used to find the standard deviation, which is equal to its square root.

Variance Formula

You can use the following formula to find the variance.

graphic showing the variance formula where sigma squared is equal to the sum of squares divided by N

There are three main components to the formula, and you can find the variance of a data set in a few simple steps.

Find the Mean

Because variance is a measurement of the variability from the mean, it makes sense that the first step to finding the variance is to calculate the mean of the data set.

To calculate the mean, simply sum each value in the data set, then divide the result by the number of elements in the set.

mean = sumcount

The mean formula looks more like this:

μ = (x1 + x2 + … + xi) ÷ N
μ = ∑xi ÷ N

Find the Sum of Squares

Once you have the mean, the next step is to calculate the sum of squares. You can use the following formula to calculate the sum of squares.

SS = ∑(xi – μ)²

The sum of squares SS is equal to the sum of each value xi minus the mean μ, squared.

Find the Variance

Finally, using the sum of squares, you can find the variance.

σ² = SSN

Thus, the variance σ² is equal to the sum of squares SS minus the number of elements in the data set N.

How to Find Sample Variance

You can use the formula above to calculate the variance for a data set that represents a population, but the formula to find the variance of a sample is slightly different.

graphic showing the variance formula for a sample where s squared is equal to the sum of squares divided by n minus 1

The variance of a sample is equal to the sum of squares divided by the number of elements in the sample n minus 1.