State whether it is the mean or median that minimizes the mean absolute 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:
the point on which a distribution would balance
the value whose average absolute deviation from all the other values is minimized
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 \(\PageIndex{1}\) 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 \(\PageIndex{1}\): 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 \(\PageIndex{1}\) 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.
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.