The Midrange of a Sample, the Average of the Sample Maximum and Minimum, is a Good Estimator of the Population Mean

Consider the distribution of sample midrange, the average of the sample maximum and the sample minimum, for samples of a random variable uniformly distributed between -0.5 and +0.5. For n=1 the distribution of the sample midrange is just same as the distribution of the random variable..

As the sample size increases the distribution of the sample maximum approaches a spike function at 0.5 and the sample minimum approaches a spike at −0.5. The nature of the distribution of the midrange is not intuitive but the Central Limit Theorem establishes that it will approach a normal distribution for large sample size.