The GMAT's Central Tendency

October 1, 2010

The GMAT likes to test your knowledge of "measures of central tendency." While finding an average (arithmetic mean, or simply "mean") requires some calculation, the GMAT is often more concerned with the logical aspects.

It's one thing to know how to calculate the mean (add up the terms, and divide by the sum of terms) or the median (line up the terms in order, and take the middle term, or average of two middle terms). It's another to have an instinctual understand of how to compare the two.

For what types of sets are the mean and median equal? When is the mean greater than the median? When is the mean less than the median?

The mean and the median are the same in any set where the terms are consecutive or equally spaced. For instance, take the set {1, 2, 3, 4, 5}. The median is the middle term, 3. The average is (1 + 2 + 3 + 4 + 5)/5, which is equal to 15/5 = 3.

(I've written at length about this special case.)

What happens when you tweak that set? Say, you replace the 5 with a 10:

{1, 2, 3, 4, 10}

The median is still 3. The mean, however, is 4. It's clear what happened: Making one of the terms a whole lot bigger results in the sum being greater, and when the sum is greater, the average is greater.

Consider a different variation of the original {1, 2, 3, 4, 5}. This time, we're going to tweak the bottom end:

{-4, 2, 3, 4, 5}

Can you guess what will happen? The median remains the same, but by decreasing the lowest term, the sum of terms decreases, as does the average. The average is now 2.

Let's observe and summarize. When the terms in a set are equally spaced, the mean and median are the same. When they are not equally spaced, the mean and median are different.

The mean skews in the direction that the set spreads out. It may be helpful to imagine all of the numbers plotted on a number line. If the terms greater than the median are more spread out than those less than the median, the mean is greater than the median. The inverse is also true.

Some GMAT questions ask you to determine what happens to the mean or median of a set when some or all of the terms are altered. In those cases, the test-makers don't expect you to do all the calculations. Fundamentals like those described in this article help you get the answer faster and more intuitively.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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