Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

120 CHAPTER 4 | Variability
(a)
(b)
M 5 16
s 5 8
3
s 5 2 s 5 2
f
2
1
m 5 80
13 14
15 16 17 18 19
x
F I G U R E 4.6
Showing means and standard deviations in frequency distribution graphs. (a) A population distribution
with a mean of µ 5 80 and a standard deviation of s 5 8. (b) A sample with a mean of
M 5 16 and a standard deviation of s 5 2.
to the left, or we could have drawn two lines (or arrows), with one pointing to the right
and one pointing to the left, as in Figure 4.6(b). In each case, the goal is to show the
standard distance from the mean.]
■ Transformations of Scale
Occasionally a set of scores is transformed by adding a constant to each score or by multiplying
each score by a constant value. This happens, for example, when exposure to a
treatment adds a fixed amount to each participant’s score or when you want to change the
unit of measurement (to convert from minutes to seconds, multiply each score by 60). What
happens to the standard deviation when the scores are transformed in this manner?
The easiest way to determine the effect of a transformation is to remember that the
standard deviation is a measure of distance. If you select any two scores and see what
happens to the distance between them, you also will find out what happens to the standard
deviation.
1. Adding a constant to each score does not change the standard deviation. If
you begin with a distribution that has m 5 40 and s 5 10, what happens to the
standard deviation if you add 5 points to every score? Consider any two scores in
this distribution: Suppose, for example, that these are exam scores and that you
had a score of X 5 41 and your friend had X 5 43. The distance between these two
scores is 43 2 41 5 2 points. After adding the constant, 5 points, to each score,
your score would be X 5 46, and your friend would have X 5 48. The distance
between scores is still 2 points. Adding a constant to every score does not affect
any of the distances and, therefore, does not change the standard deviation. This
fact can be seen clearly if you imagine a frequency distribution graph. If, for
example, you add 10 points to each score, then every score in the graph is moved
10 points to the right. The result is that the entire distribution is shifted to a new
position 10 points up the scale. Note that the mean moves along with the scores
and is increased by 10 points. However, the variability does not change because
each of the deviation scores (X 2 m) does not change.
2. Multiplying each score by a constant causes the standard deviation to be
multiplied by the same constant. Consider the same distribution of exam scores
we looked at earlier. If m 5 40 and s 5 10, what would happen to the standard
deviation if each score were multiplied by 2? Again, we will look at two scores,
X 5 41 and X 5 43, with a distance between them equal to 2 points. After all the