Pictures are not only worth a thousand words, but they’re worth a lot of points on the SAT. Here’s how to use pictures (diagrams) on questions involving the median:

A quick review: the median is the number that lies in the middle when you put a list of numbers in order. So the median of {1,2,3,4,5} is 3, and the median of {1,2,3,4,4,4,4,5,6} is 4 (because one of the 4’s is in the middle).

If there is an even number of elements in the set, the median is the average (arithmetic mean) if the two middlemost numbers. So the median of {1,2,3,4} = the average of 2 and 3, which is 2.5.

Now let’s look at the same thing with simple diagrams:

This type of diagram will make easy problems even easier. But where it really helps is on harder problems, where some of the elements in the set will be unknown. Consider the following question:

Q. The table above shows the number of rockets sold by the BlastOff company for the years 2006 through 2010. If the median sales for those five years is 66, and no two years had the same sales, what is the least possible value for x?

First, just draw 5 blank spaces for the elements, since you’re not sure where they go yet:

You know that 66 is the median, so place it into the middle space:

Now 33, 35, 87, and x remain. Since 33 and 35 are lower than 66, they have to go on the left:

You don’t know exactly where x and 87 go, because you don’t know which one is bigger. But the question doesn’t ask where those values go; it only asks what’s the smallest possible x. And since x must go somewhere to the right of 66, it must be larger – at least 67.

Note that if the words “no two years had the same sales” weren’t included, the correct answer would be 66 (33,35,66,66,87 still has a median of 66). Read carefully!

Let’s look at another problem:

Q. If the median height of 9 basketball players is 6 feet, 5 inches, what is the largest number of players who could be under 6 feet tall?

You only know the height of one player, so put it into a diagram:

You can see from the diagram that there are 4 unknowns that must be less than or equal to 6’5″, and 4 that must be greater or equal. Since the smaller ones could possibly all be under 6 feet tall, the answer is 4.

Both of those questions are considered difficult. Let’s finish up with a killer (for any hot-shots who are still reading):

Q. The quiz grades of 5 students were 3, 3, 3, 5, and x, and the average (arithmetic mean) of the 5 students’ scores was twice the median. What is the value of x?

This would actually be the last question of an SAT section – what I like to call “The Harvard Question.” Even here, the diagram should help.

You don’t know where to put the given values, because you don’t know what x is. But if you monkey around with the numbers, you can see that there must be a 3 in the middle, because there will be 3 3’s in a row. Now it’s easy! If the median is 3, then the average is 6.

The average formula is sum/count = avg. So sum/5 = 6, and the sum is 30. Since 3 + 3 + 3 + 5 = 14, the missing number is 16 (sum = 3 + 3 + 3 + 5 + x).

See ya’ in Harvard!

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