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EDIT There are other discussions on CV. I've filtered out some mentions that add little, but many helpful details and references are given in

Approximate mean from .25, .5, .75 percentiles

How can I interpret a plot of trimming percentage vs. trimmed mean?

What location parameter is modelled by robust regression?

The context for Tukey, his students and his readers around 1970 was identifying median and quartiles by hand in a batch of numbers (usually as individual quantiles or order statistics or averages of adjacent quantiles) and then -- in the case of trimeans -- combining them with weights. Although Tukey was involved with computers from the late 1940s (suggesting the word bit and being an early use of the word software and suggesting hardware tweaks too), his exploratory style around 1970 put very heavy emphasis on mental arithmetic with paper, pens and pencils, both for himself (he did a lot of exploratory work on planes and in committee meetings) and for others. He was keen on as far as possible just ordering, counting and averaging pairs of numbers. The technological context was that, if I recall correctly, cheap portable calculators were not routine until the late 1970s. Use of computers was a long way from the obvious everyday and universal practice it later became.

The context for Tukey, his students and his readers around 1970 was identifying median and quartiles by hand in a batch of numbers (usually as individual quantiles or order statistics or averages of adjacent quantiles) and then -- in the case of trimeans -- combining them with weights. Although Tukey was involved with computers from the late 1940s (suggesting the word bit and being an early user of the word software and suggesting hardware tweaks too), his exploratory style around 1970 put very heavy emphasis on mental arithmetic with paper, pens and pencils, both for himself (he did a lot of exploratory work on planes and in committee meetings) and for others. He was keen as far as possible on just ordering, counting and averaging pairs of numbers. The technological context was that, if I recall correctly, cheap portable calculators were not routine until the late 1970s. Use of computers was a long way from the obvious everyday and universal practice it later became.

All that is history. A detail that may still bite occasionally is that if all you see in a report -- especially of other people's work -- is say median and quartiles, then a trimean combines the information in all three, uses more information than the median alone, and remains more robust than the mean. The weights 1/4, 1/2, 1/4 are arbitrary other than summing to 1, unless someone can suggest a rationale. In Tukey's case cutting down on arithmetic would have been a strong incentive for those weights. There is also an echo of weights (1/4, 1/2, 1/4) for moving averages, which Tukey called Hanning, although Julius von Hann surely didn't invent it.

The paper cited in the original question talks about truncated means. I would say that the term trimmed mean is more common in statistical literature, but that is at most a detail, and they are one and the same.

So also with the trimean: the major point is to use a bit more information about the distribution than the median provides.

That is mostly history. A detail that may still bite occasionally is that if all you see in a report -- especially of other people's work -- is say median and quartiles, then a trimean combines the information in all three, uses more information than the median alone, and remains more robust than the mean. The weights 1/4, 1/2, 1/4 are arbitrary other than summing to 1, unless someone can suggest a rationale. In Tukey's case cutting down on arithmetic would have been a strong incentive for those weights. There is also an echo of weights (1/4, 1/2, 1/4) for moving averages, which Tukey called Hanning, although Julius von Hann surely didn't invent it.

The paper cited in the original question talks about truncated means. I would say that the term trimmed mean is more common in statistical literature, but that is at most a detail, and they are one and the same.

I will put in a plug for geometric means as well as medians for any distribution that is right-skewed and always positive. The recipe exp(mean(log()) can provide another fair trade-off between using all the information and dampening outliers and/or long tails. They are more or less natural for anything close to lognormal.