Why Are Variances Squared?

Defining the Average

Because the variance measures how spread out numbers are from the middle of a data set, the middle of the data set must first be calculated. The average of a data set is one number that describes middle. The average can be several different numbers, including the mean, mode or median. To calculate the variance, your data must be continuous. Continuous data is made up of counting numbers such as 1, 2, 3 and 4. When calculating the middle of a continuous data set, the mean is the appropriate statistic. To calculate the mean, add up all of the numbers in the data set and divide by the total number of observations. If you have 10 observations and the sum is 1,000, the mean is 100.

Get the distance from the average for each observation in the data set by subtracting it from the mean. If your first data point was 101 and the mean is 100, the first data point differs from the mean by 1. If a number is less than the mean, its difference from the mean will be negative. For example, a data point of 99 is less than the mean, so its difference from the mean would be a negative number; in this example, 99 - 100 is (-1). The distances from the mean are squared because squaring eliminates the negative sign. Doing the exact same thing to every number in a data set is called adding a constant. Constants are added to make working with numbers easier but do not change the meaning of a data set.

On a number line, negative numbers fall to the left of the neutral zero point while positive numbers fall to the right. If you did not square the differences from the mean, some of the differences would fall to the left of the zero and some would fall to the right. When calculating variance, a statistician is concerned with how far numbers vary from the mean. If one point in the data set differs (-3) and one point differs by 3, they each differ an equal number of increments from the mean, in this example, 3. By eliminating the positive sign through squaring the number, the difference of 3 is just easier to read.

Squaring each of the differences from the mean when calculating variance also makes the differences larger so it is easier to observe trends. Because each number in the data set has been made larger by the same amount, the meaning of the data has not been altered.