State whether it is the mean or median that minimizes the mean absolute deviation

State whether it is the mean or median that minimizes the mean squared deviation

State whether it is the mean or median that is the balance point on a balance scale

In the section "What is central tendency,"
we saw that the center of a distribution could be defined three
ways: (1) the point on which a distribution would balance, (2)
the value whose average absolute
deviation from all the other values is minimized, and (3)
the value whose average squared difference from all the other values is
minimized. From the simulation in this chapter, you discovered
(we hope) that the mean is the point on which a distribution would
balance, the median is the value that minimizes the sum of absolute
deviations, and the mean is the value that minimizes the sum of
the squared deviations.

Table 1 shows the absolute and squared deviations
of the numbers 2, 3, 4, 9, and 16 from their median of 4 and their
mean of 6.8. You can see that the sum of absolute deviations from
the median (20) is smaller than the sum of absolute deviations
from the mean (22.8). On the other hand, the sum of squared deviations
from the median (174) is larger than the sum of squared deviations
from the mean (134.8).

Table 1. Absolute and squared deviations
from the median of 4 and the mean of 6.8.

Value

Absolute Deviation from Median

Absolute Deviation from Mean

Squared Deviation from Median

Squared Deviation from Mean

2

2

4.8

4

23.04

3

1

3.8

1

14.44

4

0

2.8

0

7.84

9

5

2.2

25

4.84

16

12

9.2

144

84.64

Total

20

22.8

174

134.8

Figure 1 shows that the distribution balances
at the mean of 6.8 and not at the median of 4. The relative advantages
and disadvantages of the mean and median are discussed in the
section "Comparing Measures"
later in this chapter.

Figure 1. The distribution balances at the mean of 6.8 and not at the median of 4.0.

When a distribution is symmetric, then the mean and the median
are the same. Consider the following distribution: 1, 3, 4, 5,
6, 7, 9. The mean and median are both 5. The mean, median, and
mode are identical in the bell-shaped normal distribution.