# When the Median and the Mean are Equal

Consider the following Quantitative Comparison question, which you might find on the GRE:

Set Q consists of all multiples of 7 from 42 to 210 inclusive

Quantity A           Quantity B

Median of set Q       Mean of set Q

One option here is to tediously calculate the mean and median of set Q, but given how much time it would take to do so, you might be better off guessing and moving on. Fortunately, we can apply a nice rule here. It says:

If the numbers in a set are equally spaced, then the median and mean of that set are equal.

Now what exactly does “equally spaced” mean? If the numbers in a set are arranged in ascending order, then the numbers are “equally spaced” if the difference between any two adjacent numbers is always be the same. For example, the numbers in the set {-24, -14, -4, 6, 16, 26} are equally spaced since any two adjacent numbers differ by 10.

Since set Q in the original question looks like this {42, 49, 56, 63, . . . , 203, 210}, we can see that the numbers are, indeed, equally spaced. As such, the mean of set Q must equal the median, which means the answer to the original question is C.

Notice that our handy rule enables us to answer the question without making any calculations whatsoever.

Now see if you can use the rule to answer this one:

Set A: {-29, -22, -15, -8, -1, 6, 13, 20, 27}

Set B: {-19, -14, -9, -4, 1, 6, 11, 16}

Quantity A           Quantity B

Mean of set A       Mean of set B

Once we recognize that the numbers in set A and in set B are equally spaced, then we know that the mean will equal the median in each set.

How does this help us?

Well, finding the mean of a set of numbers can be a lot of work, but finding the median is much easier. So, rather that find the mean of each set, we’ll find the median.

Set A: {-29, -22, -15, -8, -1, 6, 13, 20, 27}

There are 9 elements in set A. Since there is an odd number of elements, the median will equal the middlemost element. The middlemost value here is -1. So, the median of set A is -1, which means the mean of set A is also -1.

Set B: {-19, -14, -9, -4, 1, 6, 11, 16}

There are 8 elements in set B. Since there is an even number of elements, the median will equal the average of the two middlemost elements. As such, the median of set B equals the average of -4 and 1, which is -1.5. So, the median of set B is -1.5, which means the mean of set B is also -1.5.

Since Quantity A = -1 and Quantity B = -1.5, the answer to the question is A.