What is the mean of the mode, median and range of the data given below? 17, 8,…

Concept: For an ungrouped data set the mode is the observation with the highest frequency; the median is the middle observation once the data are arranged in ascending order, and for an even count n it is the mean of the (n/2)th and the (n/2 + 1)th ordered observations; the range is the largest observation minus the smallest. The mean of k numbers is their sum divided by k.

Application: apply each definition to the twelve given observations.

1. Arrange the twelve observations in ascending order: 7, 8, 9, 10, 11, 11, 13, 14, 15, 16, 17, 23.

2. Mode: the value 11 occurs twice while every other value occurs once, so the mode is 11.

3. Median: n = 12 is even, so the median is the mean of the 6th and the 7th ordered observations, that is (11 + 13) / 2 = 12.

4. Range: largest observation minus smallest observation = 23 − 7 = 16.

5. Mean of the three statistics: (11 + 12 + 16) / 3 = 39 / 3 = 13.

Cross-check: the three statistics 11, 12 and 16 total 39, and 39 / 3 = 13, while multiplying back gives 13 × 3 = 39, the same total. Note also that with an even number of observations the median 12 need not itself belong to the data set, and a repeated value such as 11 is counted once as the mode, not once per occurrence.