When is the average median and mode the same?

Answer 1

The average, median and mode of a list of numbers will be the same when the middle (or mean of the two middle numbers) is equal to the most common number in the list, and that number is also the mean. This assumes that the list has only a single mode.

If you arrange such a list of numbers from least to greatest, the mode value will be grouped at the middle of the list, thus becoming the median as well. The average of the non-median values will be equal to the median/mode.

Given a target mean/median/mode and a list length, you can construct an infinite number of lists that qualify.

Here are some examples:

10 10 10 10 10 (or any list of only one number)

11 12 12 12 12 13

1 2 3 4 5 5 5 6 7 8 9