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

SECTION 3.2 | The Mean 75
TABLE 3.1
Statistics quiz scores for a
section of n = 8 students.
Quiz Score (X ) f fX
10 1 10
9 2 18
8 4 32
7 0 0
6 1 6
■ Characteristics of the Mean
The mean has many characteristics that will be important in future discussions. In general,
these characteristics result from the fact that every score in the distribution contributes
to the value of the mean. Specifically, every score adds to the total (SX) and every
score contributes one point to the number of scores (n). These two values (SX and n)
determine the value of the mean. We now discuss four of the more important characteristics
of the mean.
Changing a Score Changing the value of any score will change the mean. For example,
a sample of quiz scores for a psychology lab section consists of 9, 8, 7, 5, and 1. Note
that the sample consists of n = 5 scores with SX = 30. The mean for this sample is
M 5 SX
n 5 30 5 5 6.00
Now suppose that the score of X = 1 is changed to X = 8. Note that we have added
7 points to this individual’s score, which will also add 7 points to the total (SX). After
changing the score, the new distribution consists of
9, 8, 7, 5, 8
There are still n = 5 scores, but now the total is SX = 37. Thus, the new mean is
M 5 SX
n 5 37 5 5 7.40
Notice that changing a single score in the sample has produced a new mean. You should
recognize that changing any score also changes the value of SX (the sum of the scores), and
thus always changes the value of the mean.
Introducing a New Score or Removing a Score Adding a new score to a distribution,
or removing an existing score, will usually change the mean. The exception is when
the new score (or the removed score) is exactly equal to the mean. It is easy to visualize
the effect of adding or removing a score if you remember that the mean is defined as the
balance point for the distribution. Figure 3.4 shows a distribution of scores represented as
boxes on a seesaw that is balanced at the mean, μ = 7. Imagine what would happen if we
added a new score (a new box) at X = 10. Clearly, the seesaw would tip to the right and
we would need to move the pivot point (the mean) to the right to restore balance.
Now imagine what would happen if we removed the score (the box) at X = 9. This time
the seesaw would tip to the left and, once again, we would need to change the mean to
restore balance.
Finally, consider what would happen if we added a new score of X = 7, exactly equal to
the mean. It should be clear that the seesaw would not tilt in either direction, so the mean
would stay in exactly the same place. Also note that if we remove the new score at X = 7,
the seesaw will remain balanced and the mean will not change. In general, adding a new