Median Absolute Deviation

Here again are the deviations or errors for Sets A and B. We want to find the typical deviation.

Set A			Set B
Data	Median	Deviation	Data	Median	Deviation

10	12	2	4	12	8
14	12	2	20	12	8
9	12	3	3	12	9
13	12	1	19	12	7
8	12	4	2	12	10
12	12	0	12	12	0
13	12	1	19	12	7
12	12	0	18	12	6
7	12	5	1	12	11
14	12	2	20	12	8
13	12	1	19	12	7
14	12	2	20	12	8
12	12	0	6	12	6
11	12	1	5	12	7
11	12	1	5	12	7

These graphs display the deviations for both sets of data. Find the typical deviation for each set by moving the model line until the sum of the errors of the deviations is as small as possible.

The typical or best representative of the deviations in each set is of course the median deviation. Listing the deviations in order:

Set A Deviations: 0 0 1 1 1 1 1 1 2 2 2 2 3 4 5

Set B Deviations: 0 6 6 7 7 7 7 7 8 8 8 8 9 10 11

The typical deviation in this case has the special name median absolute deviation or MAD. For Set A, the MAD = 1 and for Set B the MAD = 7. If the typical deviation or MAD is large, it tells us that the data values are spread out far away from the typical value or median. In this case, the spread for Set B is much greater than the spread for Set A, as reflected by the differences in their MADs.

A concise summary of the data in Set A (which, remember, are the real data for the Statistical Knowledge Quiz) is: Median = 12, MAD = 1. This tells us that the typical score was twelve questions correct and that all the scores were typically only one question more or one question fewer correct relative to the typical score. That is, there wasn't much spread around the typical value.

For Set B, the concise summary is: Median = 12, MAD = 7. This tells us that the typical score was twelve questions correct and that all the scores were typically seven questions more or seven questions fewer correct relative to the typical score. That is, there was a lot of spread around the typical value.

In summary, the median absolute deviation (i.e., the typical deviation or error from the median) is a useful description of the spread of the data values.