# Name for the special estimate of the mean

During my masters studies, I heard about the following estimate of the mean: We take the minimal and the maximal value from sample and simply average them out.

Does this estimate have any name? And when is it useful? I have never seen it in practice.

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Looks like a disastrous way to estimate a mean. – Stéphane Laurent Aug 5 '14 at 8:05
Yes, it seems to be very nonrobust. However, the expected value is the true mean. But what about variance. – Miroslav Sabo Aug 5 '14 at 8:07
@MiroslavSabo The expected value is NOT the true mean (in general). Easiest case to consider is a binary outcome with the probability for 0 being 0.9 and the probability for 1 being 0.1. The midrange will quickly be (and stay at) 0.5 but the true mean is just 0.1. – Erik Aug 5 '14 at 9:40
Good remark, I considered only symmetric distributions. – Miroslav Sabo Aug 5 '14 at 10:08
It happens to be the best estimate of the mean of the uniform distribution, though. so it may perform well with platycurtic dist. – Germaniawerks Sep 9 '14 at 10:24

It's called the midrange.

It's a good way to estimate the population mean of a $\text{Unif}(\mu-\theta,\mu+\theta)$.

It may be quite good in a variety of other circumstances; they'll generally be ones where the density is both symmetric and 'cuts off' relatively quickly at the bounds, rather than ones that very smoothly tail off.

So it should do fairly well as an estimator for the center of say a Beta(1.5,1.5), even though it's not ML (indeed, it looks like it's more efficient than the sample mean even at a Beta(2,2), at least in moderately small samples.)

(It will not in general be suitable as an estimator of the population mean for a non-symmetric distribution, even if it's distinctly platykurtic. So for example, it wouldn't be suitable for estimating the mean of a Beta(0.45,1.8), say.)

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I think the answer would be improved by explicitly correcting the misperception that the midrange is always the mean. – Erik Aug 5 '14 at 9:42
I've made an edit to explicitly discuss the case of non-symmetry; I deliberately didn't consider the case where the mean doesn't exist (such as the Cauchy), since it's far from cases where one would contemplate the midrange. – Glen_b Aug 5 '14 at 10:05