Mean, Variance and Standard Deviation

Abstract

Recall that the range is the difference between the upper and lower limits of the data. While this is important, it does have one major disadvantage. It does not describe the variation among the variables. For instance, both of these sets of data have the same range, yet their values are definitely different. 90, 90, 90, 98, 90 Range = 8 1, 6, 8, 1, 9, 5 Range = 8 To better describe the variation, we will introduce two other measures of variation-variance and standard deviation (the variance is the square of the standard deviation). These measures tell us how much the actual values differ from the mean. The larger the standard deviation, the more spread out the values. The smaller the standard deviation, the less spread out the values. This measure is particularly helpful to teachers as they try to find whether their students' scores on a certain test are closely related to the class average. To find the standard deviation of a set of values: a. Find the mean of the data b. Find the difference (deviation) between each of the scores and the mean c. Square each deviation d. Sum the squares e. Dividing by one less than the number of values, find the "mean" of this sum (the variance*) f. Find the square root of the variance (the standard deviation) *Note: In some books, the variance is found by dividing by n. In statistics it is more useful to divide by n-1. EXAMPLE Find the variance and standard deviation of the following scores on an exam: 92, 95, 85, 80, 75, 50 SOLUTION First we find the mean of the data: Mean = 92+95+85+80+75+50 6 = 477 6 = 79.5 Then we find the difference between each score and the mean (deviation). Score Score-Mean Difference from mean 92 92-79.5 +12.5 95 95-79.5 +15.5 85 85-79.5 +5.5 80 80-79.5 +0.5 75 75-79.5-4.5 50 50-79.5-29.5 Next we square each of these differences and then sum them. Difference Difference Squared +12.5 156.25