How to Calculate Variance

Calculating variance allows you to measure how far a set of numbers is spread out. Variance is one of the descriptors of probability distribution, and it describes how far numbers lie from the mean. Variance is often used in conjunction with standard deviation, which is the square root of the variance. If you want to know how to calculate the variance of a set of data points, just follow these steps.

Steps

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Variance Cheat Sheet

Calculating Variance

1. Write down the formula for calculating variance. The formula for measuring an unbiased estimate of the population variance from a fixed sample of n observations is the following:(s²) = Σ [(x_i - x̅)²]/n - 1. The formula for calculating the variance in an entire population is the same as this one except the numerator is n, not n - 1, but it should not be used any time you are working with a finite sample of observations. Here's what the parts of the formula for calculating variance mean:
   
   
   
   
   
   
   • s² = Variance
   
   
   • Σ = Summation, which means the sum of every term in the equation after the summation sign.
   
   
   • x_i = Sample observation. This represents every term in the set.
   
   
   • x̅ = The mean. This represents the average of all the numbers in the set.
   
   
   • n = The sample size. You can think of this as the number of terms in the set.


2. Calculate the sum of the terms. First, create a chart that has a column for observations (terms), the mean (x̅), the mean subtracted from the terms (x_i - x̅) and then the square of these terms [(x_i - x̅)²)]. After you've made the chart and placed all of the terms in the first column, simply add up all of the numbers in the set. Let's say you're working with the following numbers: 17, 15, 23, 7, 9, 13. Just add them up: 17 + 15 + 23 + 7 + 9 + 13 = 84.
   
   
   


3. Calculate the mean of the terms. To find the mean of any set of terms, simply add up the terms and divide the result by the number of terms. In this case, you already know that the sum of the terms is 84. Since there are 6 terms, just divide 84 by 6 to find the mean. 84/6 = 14. Write "14" all the way down the column for the mean.
   
   
   


4. Subtract the mean from each term. To fill the third column, simply take each term from the sample observations and subtract it from 14, the sample mean. You can check your work by adding up all of the results and confirming that they add up to zero. Here's how to subtract each sample observation from the average:
   
   
   
   
   
   
   • 17 - 14 = 3
   
   
   • 15 - 14 = 1
   
   
   • 23 - 14 = 9
   
   
   • 7 - 14 = -7
   
   
   • 9 - 14 = -5
   
   
   • 13 - 14 = -1


5. Square each result. Now that you've subtracted the average from each sample observation, simply square each result and write the answer in the fourth column. Remember that all of your results will be positive. Here's how to do it:
   
   
   
   
   
   
   • 3² = 9
   
   
   • 1² = 1
   
   
   • 9² = 81
   
   
   • (-7)² = 49
   
   
   • (-5)² = 25
   
   
   • (-1)² = 1


6. Calculate the sum of the squared terms. Now simply add up all of the new terms. 9 + 1 + 81 + 49 + 25 + 1 = 166
   
   
   


7. Substitute the values into the original equation. Just plug in the values into the original equation, remembering that "n" represents the number of data points.
   
   
   
   
   
   
   • s² = 166/(6-1)


8. Solve. Simply divide 166 by 5. The result is 33.2 If you'd like to find the standard deviation, simply find the square root of 33.2. √33.2 = 5.76. Now you can interpret this data in a larger context. Usually, the variance between two sets of data are compared, and the lower number indicates less variation within that data set.
   
   
   

Video

Tips

• Since it is difficult to interpret the variance, this value is usually only calculated as a start in calculating the standard deviation.

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