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

72 CHAPTER 3 | Central Tendency
EXAMPLE 3.3
Now suppose that the 6 boys from Example 3.2 decide to sell their baseball cards on eBay.
If they make an average of M = $5 per boy, what is the total amount of money for the
whole group? Although you do not know exactly how much money each boy has, the new
definition of the mean tells you that if they pool their money together and then distribute
the total equally, each boy will get $5. For each of n = 6 boys to get $5, the total must be
6($5) = $30. To check this answer, use the formula for the mean:
M 5 SX
n 5 $30
6 5 $5 ■
The Mean as a Balance Point The second alternative definition of the mean describes
the mean as a balance point for the distribution. Consider a population consisting of N = 5
scores (1, 2, 6, 6, 10). For this population, SX = 25 and μ = 25
5 = 5. Figure 3.3 shows
this population drawn as a histogram, with each score represented as a box that is sitting
on a seesaw. If the seesaw is positioned so that it pivots at a point equal to the mean, then
it will be balanced and will rest level.
The reason the seesaw is balanced over the mean becomes clear when we measures the
distance of each box (score) from the mean:
Score
X 5 1
X 5 2
X 5 6
X 5 6
X 5 10
Distance from the Mean
4 points below the mean
3 points below the mean
1 point above the mean
1 point above the mean
5 points above the mean
Notice that the mean balances the distances. That is, the total distance below the mean
is the same as the total distance above the mean:
below the mean: 4 + 3 = 7 points
above the mean: 1 + 1 + 5 = 7 points
Because the mean serves as a balance point, the value of the mean will always be located
somewhere between the highest score and the lowest score; that is, the mean can never be
outside the range of scores. If the lowest score in a distribution is X = 8 and the highest is
X = 15, then the mean must be between 8 and 15. If you calculate a value that is outside
this range, then you have made an error.
The image of a seesaw with the mean at the balance point is also useful for determining
how a distribution is affected if a new score is added or if an existing score is removed. For
FIGURE 3.3
The frequency distribution shown as a seesaw
balanced at the mean. Based on Weinberg,
G. H., Schumaker, J. A., and Oltman, D. (1981).
Statistics: An intuitive approach. Belmont, CA:
Wadsworth. (p. 14)
1 2 3 4 5 6 7 8 9 10
m