When the Mean is the Median

Take a group of 50 numbers, say the integers between 1 and 50, inclusive. What's the average (arithmetic mean)?

More importantly, how long would it take you to figure it out? You could probably approximate very quickly–it's about 25. Being more exact, however, is demanding. Technically, to find the average of a set of numbers, you need to add them up and divide by the number of terms. Do you really want to do that for all 50 numbers?

There is a decent shortcut for that sort of thing: take the outer two numbers (50 and 1), add them up (51), then figure out how many similar pairs there are. 50 and 1, 49 and 2, 48 and 3, etc., all add up to 51. So if there are 25 pairs like that, the sum is 25 times 51, or 1275. Then you'd have to divide 1275 by 50, resulting in 25.5.

That's better, but still not very fast. When the numbers in the set are consecutive, we can do better.

The Mean is the Median

It is easy to find the median of that set. The median is defined as the "middle term" in a set. In the set {0, 3, 5}, the median is 3. If the set has an even number of terms, the median is the average of the middle two terms. For example, in the set {10, 12, 15, 20}, the median is the average of 12 and 15: 13.5.

So, in the set of all integers between 1 and 50, inclusive, the median is the average of the 25th and 26th term. The 25th term is 25 and the 26th term is 26, so the median is 25.5. Since the numbers are consecutive, the mean and the median are the same.

Why Does This Work?

When you're calculating an average, think of it as a kind of tug of war. If two people, equally strong, pulled on either end of a rope, the middle of the rope would be right between them. Similarly, the average of 0 and 10 is 5–right in between 0 and 10.

However, if another equally strong person got on one side, the middle of the rope would end up much closer to the side with two people on. That extension of the analogy represents a set like {0, 10, 10}, the mean of which is 6.67.

When the integers are consecutive, there's always an equal amount of "pull" on both sides. Using the original example: 25 and 26 average to 25.5. 27 and 24 average to 25.5. 2 and 49 average fo 25.5. 1 and 50 average to 25.5. Any one of those pairs is pulling on the rope enough so that the middle of the rope remains at 25.5.

When Does It Work?

This trick works whenever a set consists of consecutive numbers. It doesn't have to be consecutive integers, as in the initial example. It could also be consecutive evens (such as {2, 4, 6, 8, 10}), or consecutive multiples of 7 (such as {21, 28, 35, 42}). The GMAT loves these kinds of sequences, so this is a shortcut you'll have plenty of opportunities to take advantage of.

In fact, we can do even better. Initially, I said that you just need to find the median of the set. In the case of the integers from 1 to 50, that's easy. But what if you needed the mean or median of the set of numbers between 79 and 130? It would take quite a bit of time to figure out where exactly the middle of that set is.

Because the numbers in a set of consecutive numbers are equally spaced, the median of such a set is the median of the two outer numbers. So, instead of locating the two middle numbers in the set from 1 to 50, simply average 1 and 50. 79 and 130? Find the average of those two numbers.

Summary

When you have a set of consecutive numbers (integers, evens, odds, multiples), the mean is equal to the median. To locate the median, find the average of the endpoints.

To extend the usefulness of this tip a bit further: this technique can also be used to find the sum of large sequences of numbers. Say that, instead of looking for the mean of the set of integers from 1 to 50, you wanted to the sum of that set of integers.

The sum of a set of numbers is equal to the average times the number of terms. The number of terms is usually easy to come up with: in this case it's 50. The average, as we now know, is also easy: we know it's 25.5. The sum of the numbers in the set, then, is 25.5 times 50, or 1275. That's much easier than adding up all of the numbers, or even finding all the pairs of numbers, as I did earlier in this article.

As usual, when it looks like the GMAT is asking you to perform a lengthy calculation, there's a way around it. This is just one more example of how the GMAT isn't an arithmetic test, it's a thinking skills test.

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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