VARIANCE AND STANDARD DEVIATION
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 variationโvariance 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 studentsโ 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 โmeanโ 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 โ 79.5
+12.5
95
95 โ 79.5
+15.5
85
85 โ 79.5
+5.5
80
80 โ 79.5
+0.5
75
75 โ 79.5
-4.5
50
50 โ 79.5
-29.5
Next we square each of these differences and then sum them.
Difference
Difference Squared
+12.5
156.25
