Midrange

The midrange is the minimum plus the maximum, divided by 2. It is a measure of center built from exactly two observations, so a data set of 500 values hands it no more information than a data set of 5. It sits exactly halfway between the extremes, so the midrange and the range together rebuild the endpoints: the minimum is the midrange minus half the range, the maximum is the midrange plus half the range.

Seven daily high temperatures: 41, 55, 58, 60, 62, 64, 77 degrees. The midrange is $(41 + 77)/2 = 59$, the mean is $417/7 \approx 59.57$, and the median is 60. All three agree because the data are fairly balanced. Now push the hottest day to 105. The midrange becomes $(41 + 105)/2 = 73$, the mean becomes $445/7 \approx 63.57$, and the median stays at 60. One value moved 28 degrees, and the midrange moved half of that, 14 degrees, against 4 degrees for the mean and nothing at all for the median.

The name suggests the middle of the data, and it is not. It is the middle of the range. After that hot day, 6 of the 7 temperatures sit below the midrange of 73, so a statistic sold as a center has nearly the whole data set on one side of it. The value with half the data on each side is the median, a different number and a different idea.

The midrange is honest exactly when the data are symmetric. If a data set is a mirror image about some value, the minimum and maximum are equally far from it, and the midrange, the median, and the mean all land on it together. The more skewed the data, the further the midrange drifts toward the long tail, since the tail holds the extreme that defines it.

The pattern is worth memorising. The median does not move when a value shifts without crossing it. The mean moves by the change divided by $n$. The midrange moves by half the change whenever the value that moved was an extreme and stayed one.