Median and mode

Of 11 salaries in an OpenStax example, the 6th in order is $54,000: five are below it and five above, so it is the median.

Is the median of 3, 9, 4, 7, 5 the number in the middle?

Look at 3, 9, 4, 7, 5. The number written in the middle is 4, and it is tempting to call that the median. It is not. Put the list in order and you get 3, 4, 5, 7, 9, with 5 in the middle. The order in which someone happened to write the numbers says nothing about their size. Sorting comes first, every time.

The mean is one way to describe the middle of a data set. Two others are just as common, and each answers a different question. The median is the middle value once the data are in order: half the values are at or below it, half at or above it. The mode is the value that turns up most often. Together with the mean, these are the measures of center.

Five values have one middle; four values have two

Sort first. With an odd number of values n, the median sits in position (n + 1) ÷ 2. For 3, 9, 4, 7, 5, sorted to 3, 4, 5, 7, 9, n = 5 and the median is the 3rd value, 5. An even count has no single middle. Take the mean of the two middle values instead. For 2, 4, 6, 8 those are 4 and 6, so the median is 5, a number that is not in the list at all.

OpenStax lists 11 salaries, in thousands of dollars. Sorted, they read 28, 33, 40.5, 42, 54, 54, 64.5, 68.5, 69, 72 and 120. The 6th value is 54, so the median salary is $54,000. Now change the top salary from 120 to 1,200. Nothing happens to the median. The 6th value is still 54. That stubbornness is the median's main strength, and the next lesson leans on it.

Frequency tables work for the median too. In the symmetric set of 16 values the median falls between positions 8 and 9. Count down the table. Positions 1 to 5 hold 4, 5, 6, 6 and 6; positions 6 to 11 are the six 7s. Both the 8th and the 9th values are 7, so the median is 7. A running count of the frequencies is enough, and you never have to write out all 16 numbers.

Seven appears six times, so the mode is 7

The mode is found by counting. In OpenStax's symmetric set 4, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10, the value 7 appears six times, more than any other. A set can have two modes, as in 1, 2, 2, 3, 4, 4, where 2 and 4 each appear twice; such a set is called bimodal, and a set can also have no mode at all, which happens when every value appears exactly once. The mode is the only measure of center that works for categories. You cannot take the mean of "red, blue, blue, green". You can say that blue is the most common answer.

In the symmetric set the mean, median and mode are all 7. The left-skewed set 4, 5, 6, 6, 6, 7, 7, 7, 7, 8 is different. OpenStax gives a mean of 6.3, a median of 6.5 and a mode of 7, because the long tail on the left pulls the mean down the most. In the right-skewed set the order flips: mode 7, median 7.5, mean 7.7. So the gap between mean and median is a quick clue to the shape of the data.