# Outliers and the mean

I would like to know what the following example is called in mathematics: In a gymnastics competition the judges scored a competitor as 10, 8, 3, 7, 7, 9, and 8. I recall that the ending score was always calculated after they discarded the highest and lowest numbers in the group and used what was remaining to get the resulting score.

What is this kind of "math" called? Would this be an example of discarding the outliers? And is the result a mean of the remaining numbers?

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I believe this is there way of controlling for overly optimistic judges and overly pessimistic judges. – Dan Sep 3 '14 at 19:13

In statistics, this is usually now called trimming. You trim zero or more values in each tail of a distribution after ordering values, and then work on the rest, usually by taking the mean. (The question doesn't actually specify that taking the mean is what is done.)

So, in your example,

10, 8, 3, 7, 7, 9, 8


sorts to

3, 7, 7, 8, 8, 9, 10


and

trimming the outermost values would give us 7, 7, 8, 8, 9 which average 39/5 = 7.8

trimming two values in each tail would give 7, 8, 8 which average 23/3 = 7.67 to d.p.

trimming three values in each tail would leave 8.

Choosing the amount of trimming is a trade-off between discounting values in the tail which may be less reliable and using as much as possible of the information in the data.

The family of trimmed means includes as limiting cases the usual mean (trim no values) and the median (trim to leave only one value if the number of values is odd, or two values if even). Other trimmed means aren't usually assigned names; rather they are defined by the fraction of values trimmed. However, trimming 25% of values in each tail is quite often called taking the midmean. The name can be remembered by thinking of taking the mean of the middle half of the data. Fans of boxplots will see a connection here.

There is no absolute rule that trimming must be equal or symmetric in each tail, and it can make sense to trim differently in each tail (including not at all in one tail and quite hard in another).

A fairly recent tutorial review with discursive comments, including historical perspective, was given in

Cox, N.J. 2013. Trimming to taste. Stata Journal 13: 640-66. This article is accessible only to Stata Journal subscribers before fall (autumn) 2016. URL for Abstract is http://www.stata-journal.com/article.html?article=st0313

In addition to the historical references there, see

Verne, J. 1998. Twenty Thousand Leagues under the Seas. Trans. W. Butcher. Oxford: Oxford University Press. Original: Verne, J. 1870. Vingt Mille Lieues sous les Mers. Paris: Hetzel.

"Taking the average of the observations made at the various junctures---rejecting both the timid evaluations assigning the object a length of 200 feet and the exaggerated opinions making it three miles long by a mile wide---it could be affirmed that this phenomenal being greatly exceeded all the dimensions the ichthyologists had admitted until then---if indeed it existed at all." (p.5)

Whether this is thought of as discarding outliers is a subtle point. First off, if it is done after inspection shows that the extreme values are, or seem, outliers, it is not trimming in the strict sense. Rather, trimming implies that you set aside largest and/or smallest values as a matter of principle, not after inspection of the data. Second, it is as or more valid to think that outliers remain in the data; it is just that their literal (numeric) values are not fed into calculations.

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I don't know that it's called anything in Mathematics. In Statistics, it's an example of a trimmed mean, and particularly a 14% trimmed mean.

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