Midrange vs Median

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.

Sort the values from smallest to largest and the median is whatever sits at position $\frac{n+1}{2}$ (n plus 1, over 2), where $n$ is how many values there are. When $n$ is odd that position lands on an actual observation. When $n$ is even it lands between two of them and the median is their average, so the median of an even-sized set need not be a number that appears in the data at all. It is commonly written $M$.

Take the seven quiz scores 12, 15, 15, 18, 22, 24, 91. They are already sorted and $n = 7$, so the median is the value in position 4: $M = 18$. The mean of the same seven scores is 28.14. Now change the 91 to 27. The median is still 18, while the mean drops to 19. That is resistance: the median only counts how many values sit on each side of it and ignores how far away they are, so moving one extreme value cannot shift it by more than one position.

With an even count the averaging step kicks in. For 2, 6, 8, 10 the median is $(6 + 8) / 2 = 7$, a number that appears nowhere in the list.

The most common error is reporting the position instead of the value. For those seven scores the answer is 18, not 4; position 4 only tells you where to look. Two related slips are taking the middle of the list before sorting it, and averaging the smallest and largest values, which gives the midrange rather than the median.

The median is also the 50th percentile and the center of the five-number summary. The quartiles are the medians of the halves on either side of it, with the median itself left out of each half under the TI-84 convention this site uses: for the seven scores, 12, 15, 15 gives $Q_1 = 15$ and 22, 24, 91 gives $Q_3 = 24$. Comparing the two centers is also a quick read on shape: here the mean of 28.14 sits far above the median of 18, which usually signals either a right skew or a single high value dragging the mean up. In this set it is the second of those, since the 1.5 IQR rule puts the upper fence at $24 + 1.5(9) = 37.5$ and flags 91 as an outlier.