Statistics 101: The Trimmed Mean and Median

In a previous article: Statistics 101: The mean www.associatedcontent.com/article/2110277/statistics_101_the_arithmetic_mean.html, I discussed the mean, or average. In this article, I go into the trimmed mean, including the most commonly known trimmed mean, the median. In the article on the mean, I used the example of heights in a psychology class. Those heights were

64 65 64 67 64 67 66 70 66 66

66 64 69 69 62 67 64 59 66 67

65 71 67 68 59 69 67 65 68 66

68 67 75 67 69 70 67 76 67 70

68 67 78 67 73 64 75 65 70 68.

And the arithmetic mean of the above is 67.36 inches.

For the trimmed mean and median, it is useful to first sort the numbers from smallest to largest:

59 59 62 64 64 64 64 64 64 65

65 65 65 66 66 66 66 66 66 67

67 67 67 67 67 67 67 67 67 67

67 68 68 68 68 68 69 69 69 69

70 70 70 70 71 73 75 75 76 78.

The median is the number that splits the data into two equal halves, with half being higher, and half lower (there are slightly more technical definitions, to deal with things like ties, and sparse data, but this will do for our purposes). The median height in the psychology class is 67 inches.

A less commonly used measure is the trimmed mean. The trimmed mean is the mean after you throw out some extreme values (typically the highest 10% and the lowest 10%).

For the 10% trimmed mean, we delete the 5 smallest (59,59, 62, 64, 64) and 5 largest values (73, 75, 75, 76, 78(, and take the mean of the remaining 40. Here, the trimmed mean is the same as the regular mean - 67.36 inches.

Sometimes, though, the trimmed mean and median can be very different from the mean. Take income; suppose you sample 50 American adults and get household incomes of

4416
11280
7339
7882
3821

14367
11223
11197
5152
6169

28058
33362
26730
23546
32838

27679
25582
31776
26288
20113

45847
44699
39966
35535
52081

41582
52301
41308
36916
44841

76424
55663
64971
58316
55778

65888
70922
70174
76397
81837

111359
114360
153072
141380
135553
97504
136559
119445
160962
405354

Then the mean is $60,916, but the median is $43,140, and the 10% trimmed mean is $50,540.

While many people have no problem with the median, sometimes the trimmed mean is looked on with suspicion. But the median is just the 50% trimmed mean - it ignores all the data except the central point.

In a later article, I will describe when to use which measure, and why.

Sources: http://en.wikipedia.org/wiki/Household_income_in_the_United_States.