January 8, 2023 8, 5, x, 6

The median of the list of positive integers above is 5.5. Which of the following could be the average (arithmetic mean) of the list?

(A) 3

(B) 5.5 (THIS IS CORRECT)

(C) 6.25

(D) 7

(E) 7.5

The question asks you to find the average of the list for which you are given the median, 5.5. The median is the middle value in a list of values arranged in increasing order. When the list has an even number of terms (as in this problem), then find the average of the two middle terms in the list.

It’s easy to put 5, 6, and 8 in increasing order, but where does positive integer x fit in?

Is it: x, 5, 6, 8
5, x, 6, 8
5, 6, x, 8
5, 6, 8, x

Examine each case:

Case 1: {x, 5, 6, 8}: If x is less than or equal to 5, it would appear first on the list. In this case, the average of the two middle values 5 and 6 is 5.5, which is consistent with the information given in the problem. This is a valid possible case; keep it in.

Case 2: {5, x, 6, 8}: If x is located in this position, it could only be 5 or 6, since x must be an integer. If x is 5, the median would still be 5.5. That doesn’t provide any new information. Alternatively, if x is 6, the median would then be 6, which is inconsistent with what the problem said. So x can’t equal 6. Ignore this case.

Case 3: {5, 6, x, 8}: This would be true if x is 6, 7, or 8, but x can’t equal 6, as determined in Case 2. If x is 7, then the two middle values in the list would be 6 and 7, which average to 6.5. If x is 8, then the two middle values in the list would be 6 and 8, which average to 7. The median is actually 5.5, so x also can’t be 7 or 8. Ignore this case.

Case 4: {5, 6, 8, x}: If x is greater than 8, it would appear last in the list. In this case, the two middle values are 6 and 8, which average to 7. The median is actually 5.5, so cross off this case, as well.

The only possible list order is {x, 5, 6, 8}, so x has to be a positive integer equal to or less than 5. That is, x must be 1, 2, 3, 4, or 5. The question asks what could be the average, so the average will lie in the range of x = 1 to x = 5.

When x = 1, the average of the list is (1 + 5 + 6 + 8) / 4 = 5. That is, the least possible value for the average is 5. Eliminate answer choice (A).

When x = 5, the average of the list is (5 + 5 + 6 + 8) / 4 = 6. The greatest possible value for the average of the set is 6. Eliminate choices (C), (D), and (E).

If you feel really comfortable with how averages work, you don’t have to do that last calculation. The average is at least 5, but that is not an option in the answer choices. Some greater number will be the greatest possible average. The actual average must fall in the range between 5 and the greater number, inclusive. If the range is 5 to 6, then 5.5 is the correct answer. If the range is 5 to 6.2, then 5.5 is the correct answer. If the range is 5 to 7, then 5.5 would still be valid, but answers (B), (C), and (D) would also all be valid…and that’s impossible since there has to be just one correct answer! So the value 5.5 must be in the possible range of averages no matter what.