# An X% trimmed mean means?

Rand Wilcox in Fundamentals Of Statistical Methods, 1st. edition, gives a formula which says that for a 20% trimmed mean, you would trim away 20% of one end of the ranked data, and 20% of the other end, making 40% trimmed away in total.

But spreadsheets such as the Calc of LibreOffice5, would for a 20% trimmed mean only trim away 10% from one end and another 10% from the other end, making 20% trimmed away in total.

Which one is right?

The author also writes that a 20% trimmed mean is best for mixture distributions. Is this correct?

• There can't be a universal prescription (use 20%!) for mixture distributions any more than there is for any other kind of data. Choice of trimming fraction is a dark art in which how much contamination or fraction of wild(er) observations you expect should be considered with how much protection you need. Trimming is insurance against being badly off because of wild values, but sometimes the wild values are genuine too. If in doubt, explore the sensitivity of results to trimming fraction. – Nick Cox Sep 27 '18 at 11:43
• See stats.stackexchange.com/questions/117950/… for one device. – Nick Cox Sep 27 '18 at 11:44

## 2 Answers

Neither is "right" or "wrong"; it's just that usage is not universal. However, I've seen Wilcox's definition used more than the other. Wikipedia agrees with him, as do several other sites I browsed to, and so do SAS, and R.

• (+1) I agree with this. I'll add that there are situations where trimming in one tail only is entirely reasonable. In that case the terminology would, or should, agree. – Nick Cox Sep 27 '18 at 11:39

As Peter correctly points out, the conventions on usage of this term differ, and the definition used by Wilcox seems (unfortunately) to be the more common. I disagree with the view that neither is right or wrong. The definition that removes X% from each side of the ordered data vector, but refers to this as an "X% trimmed mean" is a zombie definition --- it seems to be impossible to kill despite obvious and serious flaws:

• Under this definition you are actually removing twice as much of the data as the "headline" amount you refer to in your description of the statistic. In particular, a "50% trim" removes all the data! That is contrary to the basic meaning of language, and it is highly misleading to the reader, who would expect removal of all the data to be described as a "100% trim". Use of this term, without explicit elaboration on its idiosyncrasy, is highly misleading.

• This definition is also completely inconsistent with analogous usage of significance levels for hypothesis tests and confidence intervals in statistical discussion. In those contexts, if you have a significance level $$\alpha$$ and you create a two-sided test/interval, the value $$\alpha$$ refers to the total area on both sides. So, for example, an equal-tail $$1-\alpha$$ confidence interval excludes an area of $$\alpha/2$$ from either side, and a two-sided symmetric hypothesis test at $$\alpha$$ significance level constructs the rejection region by allocating a null rejection probability of $$\alpha/2$$ to each side. In both cases the terminology respects the fact that the significance level is fixed as a total.

• The definition fails on both counts: it is contrary to ordinary language and it is inconsistent with well-established (an linguistically appropriate) conventions for statistical description in other core areas of the subject.

If you are going to report trimmed means in your own analysis for any purpose, please to not feed the zombies. Please use this term in its more appropriate meaning, where an X% trimmed mean refers to the removal of X% of the data. If you are concerned about interpretation, leave a footnote explaining your usage of the term.

• You make a good case. although the zombie terminology over-states it. The wording "in each tail" can be enough to clarify. Often in statistics, original or even existing terminology appears poorly chosen. Many of us would like to start again with different terminology (and more consistent notation conventions), preferably our own. In this case I think that the definition you dislike, which I have followed in a paper and programs to do this without feeling strongly either way, is so strongly entrenched that explaining choice of a non-standard definition is essential, not optional. – Nick Cox Sep 28 '18 at 8:01