Remember that the mean absolute deviation is the average of the absolute values of the differences between the mean and each value in the data set. Let's start by calculating the mean of the given set of numbers.
Mean
First, we can find the sum of the given values.
1.3 + 3.8 + 1.5 + 8.4 +
0.9 + 1.4 + 2.3 + 1.3 =
20.9
Because there are 8 values in our set, we need to divide the sum by 8.
Mean: 20.9/8 = 2.6125
Mean Absolute Deviation
Now, the MAD of a set of data is the average of the absolute values of the differences between the mean and each value in the data set. Let's use a table to find the sum of the absolute values of the differences.
Data Value
Data Value - Mean
|Data Value - Mean|
1.3
1.3-2.6125=-1.3125
|-1.3125|=1.3125
3.8
3.8-2.6125=1.1875
|1.1875|=1.1875
1.5
1.5-2.6125=-1.1125
|-1.1125|=1.1125
8.4
8.4-2.6125=5.7875
|5.7875|=5.7875
0.9
0.9-2.6125=-1.7125
|-1.7125|=1.7125
1.4
1.4-2.6125=-1.2125
|-1.2125|=1.2125
2.3
2.3-2.6125=-0.3125
|-0.3125|=0.3125
1.3
1.3-2.6125=-1.3125
|-1.3125|=1.3125
Sum of Values
13.95
Finally, we need to divide this sum by 8.
Mean Absolute Deviation (MAD)
13.95/8=1.74375≈ 1.74
A MAD of about 1.74 indicates that the data, on average, are about 1.74 units away from the mean. In the context of the problem, this means that the populations of largest U.S. cities, on average, are about 1.74 millions away from the mean of about 2.61 millions.
