Consider the following question:
Set T consists of all multiples of 5 from 30 to 225 inclusive
| Column A | Column B |
|---|---|
| Mean of Set T | Median of Set T |
A. The quantity in Column A is greater
B. The quantity in Column B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
As you can imagine, we really don’t want to calculate the mean and median of set T. Fortunately, we can apply a nice rule that says:
If the numbers in a set are equally spaced, then the mean and median of that set are equal.
What does it mean to have “equally spaced” numbers?
“Equally spaced” means that, if the numbers in a set are arranged in ascending order, then the difference between any two adjacent numbers will always be the same.
So, in the set {30, 35, 40, . . . , 215, 220, 225}, the difference between any two adjacent numbers will always equal 5. As such, the mean of this set will equal the median of this set (which means the answer to the above question is C). Notice that we can answer this question without making any calculations whatsoever. We can answer it by recognizing that the numbers in the set are equally spaced.
Similarly, the numbers in the set {5, 8, 11, 14, 17, 20, 23} are equally spaced. As such, the mean of this set will equal the median of this set.
As you might imagine, this little rule as a shortcut can come in handy. It will help you quickly answer the original question, and it will help you answer this one:
Set X: {-27, -20, -13, -6, 1, 8, 15}
Set Y: {-23, -19, -15, -11, -7, -3, 1, 5, 9, 13}
| Column A | Column B |
|---|---|
| Mean of set X | Mean of set Y |
A. The quantity in Column A is greater
B. The quantity in Column B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
Column A: Since the numbers in set X are equally spaced, the mean will equal the median. We can quickly find the median by locating the middlemost element: {-27, -20, -13, -6, 1, 8, 15}. Set X has 7 elements and the middlemost element is -6. So, the median of set X is -6, which means the mean of set X is -6 as well.
Column B: The numbers in set Y are also equally spaced, so the mean will equal the median. Set Y has 10 elements, so there are two middlemost elements: {-23, -19, -15, -11, -7, -3, 1, 5, 9, 13}. As such, the median of set Y will equal the average (mean) of -7 and -3, which is -5. So, if the median of set Y is -5, then the mean of set Y is -5 as well
So, using our rule, we found that Column A = -6, and Column B= -5. As such, the answer to the question is B.
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