Scaled Scores
(A worked Problem:  Procedure explained in more detail on Pages 137-142)

Suppose that you are the professor of a  Psychology 100 class and the mean of the first test is 70 and the standard deviation is 10.  What would happen to the mean and standard deviation if you decided to:

    I.     Add 15 points to each score in the distribution?

    II.     Divide each score by 2?

    III.     Subtract the mean from each score and then divide each score by the standard deviation?

I.  Add 15 points to each score

    A.    New Mean = Old Mean + 15

            New Mean = 70 + 15 = 85

    B.    New Standard Deviation = Old Standard Deviation

            New Standard Deviation = 10

Hint:    The standard deviation does not change if you add or subtract a constant from every score.

II.  Divide each score by 2

    A.    New Mean = Old Mean  ÷ 2

            New Mean = 70 ÷ 2 = 35

    B.    New Standard Deviation = Old Standard Deviation ÷ 2

            New Standard Deviation = 10 ÷ 2 = 5

Hint:  Both the mean and standard deviation change if you multiply or divide every score by a constant.

III.  Subtract the mean from each score and then divide by the standard deviation

    A.    First subtract the mean from each score

                New Mean = Old Mean - 70

                New Mean = 70 - 70 = 0

                New Standard Deviation = Old Standard Deviation

                New Standard Deviation = 10

            Then divide each of the new scores by the standard deviation

                New New Mean = New Mean ÷ 10

                New New Mean = 0 ÷ 10 = 0

                New New Standard Deviation = New Standard Deviation ÷ 10

                New New Standard Deviation = 10 ÷ 10 = 1

Hint:    Subtracting the mean from each score then dividing each score by the standard deviation
            creates Z scores that have a mean of 0 and a standard deviation of 1.

Copyright © 2004 by Mark W. Vernoy