The three key ways of describing a dataset in descriptive statistics are the mean, median and mode. (All three can be referred to as types of “average,” though in the U.S., it is common to use the term for the mean alone.) The mean of a list of n numbers is the sum of the numbers divided by n. The median is the middle number when the list is arranged in size order. If there is an even amount of numbers on the list, the median is the mean of the two middle numbers. The mode is the most frequently occurring number on the list. A data series can have multiple modes.
How many positive integers on a list with a unique mode are needed, at minimum, so that the list’s mode < mean < median?
If all n numbers are equal, then the mode, median and mean are also equal, which is not allowed. Because the mode must be unique, at least one number must appear more than once. For n = 3 and a < b, there are therefore only two possibilities: (a, a, b) and (a, b, b). Because the mode and median are then both equal, however, these cases don’t work.
With a < b < c for n = 4, there are five possibilities: (a, a, a, b), (a, b, b, b), (a, a, b, c) , (a, b, b, c) and (a, b, c, c).
The first, second and fourth possibilities are eliminated because the mode and median would be equal. The fifth possibility is also unsuitable because the mode, c, would be greater than both the median and the mean. In the third possibility, the mode = a, the mean = (2a + b + c) / 4, and the median = (a + b) / 2. Because the mean must be smaller than the median, (2a + b + c) / 4 < (a + b) / 2, which simplifies to c < b. This is a contradiction. Therefore, n must be >4.
With a < b < c < d for n = 5, the following possibilities are possible: (a, a, a, a, b), (a, a, a, b, b), (a, a, b, b, b), (a, b, b, b, b), (a, a, a, b, c), (a, b, b, b, c), (a, b, c, c, c), (a, a, b, c, d), (a, b, b, c, d), (a, b, c, c, d) and (a, b, c, d, d).
For (a, a, b, c, d), the mode = a, the median = b, and the mean = (2a + b + c + d) / 5, which is a valid possible outcome. For example, the dataset (1, 1, 5, 6, 7) meets the required conditions: mode = 1, mean = 4, and median = 5.
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This puzzle originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.