Posts tagged with: sets

I made a video today that I hope will help you understand something I get asked about an awful lot. The SAT loves asking a particular kind of Venn diagram question—not so common that it appears on every test, but common enough that high scorers need to know it. The test writers keep coming back to this kind of question because it’s so easy to trick students with it. Here’s an example:

  1. In a certain neighborhood, 11 children play baseball and 13 children play soccer. If 16 of the children play only one of the two sports, how many children play both sports?(A) 1
    (B) 3
    (C) 4
    (D) 8
    (E) 10

I solve this two ways in the video above (beginning at 4:05)—first I backsolve, then I do the algebra. My general preference is to backsolve a question like this because once you’ve practiced doing so it’s really fast and easy, but it’s good to know the algebra in case you ever come across such a question as a grid-in.

Think you’ve got this question type mastered? Try this short, 3-question quiz:

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How are people doing on this quiz?

56% got 3 right
23% got 2 right
7% got 1 right
14% got 0 right

Like the average (or, as some say, the arithmetic mean), the median and the mode are useful properties of a set of numbers and can give statisticians great at-a-glance insight into the nature of copious data. When the SAT gets its hands on them, though, they are usually stripped of any analytical utility and instead used as a framework in which to ask tricky reasoning questions. Also, the SAT doesn’t really test either of these concepts very often — doubly so for mode. So I’ll leave it to your college stats class to elucidate the myriad ways median and mode are useful in real life, and just show you what you need to know to PWN the rare median and/or mode question on the SAT.

Median

The median is the middle value in an ordered list of numbers. If the list of numbers you’re given isn’t in numerical order, you can’t find the median until you put it in numerical order. If there are an even number of values in your list, the median is the average of the two middle values. That’s it. That’s all you need to know about the median. Mouse over the following example sets to see their medians.

  • {4, 6, 7, 9, 12, 16, 30}
  • {9, 30, 16, 4, 7, 6, 12}
  • {2, 200, 300, 700}
  • {17, 22, 6, 110, 68, 52, 29, 8456}
Sample Median Question
  1. Which of the following CANNOT change the value of a median in a set of five numbers?
     
    (A) Adding 0 to the set
    (B) Multiplying each value by -1
    (C) Increasing the least value only
    (D) Increasing the greatest value only
    (E) Squaring each value

This isn’t the hardest question in the world, and if you’ve seen a similar one before you probably know the answer instantly. If you haven’t, well, now you have and you’ll nail a similar one if you see it on the SAT. You’re welcome.

Let’s plug in a set to make this easier to comprehend. Say our set is {2, 3, 4, 5, 6}. The original value of the median is 4.

If we add 0 to the set like it says to do in (A), the median becomes the average of 3 and 4, or 3.5. Cross off (A).

If we multiply each value by -1, the median becomes -4. That’s not the same as 4. Cross off (B).

If we increase the least value (by more than 2) we change the median as well. Say that 2 became a 10. Now the median is 5 instead of 4. Cross off (C). Note that since the questions says “CANNOT” it doesn’t matter that we wouldn’t change the median if we only increased the least value by 1. If we can come up with a way to change the median by increasing the least value, we can cross off (C).

Increase the greatest value as much as you like, you won’t change the order of the values at all. If we change 6 to 6,000,000, the median is still 4. (D) is our answer.

Obviously, squaring each value changes each value, and thus changes the median value. Cross off (E).

Not so bad, right?

Mode

The mode of a list of numbers is the number that appears most in that list. Be aware that it’s possible for a list to have multiple modes, but all modes will appear the same number of times, and no other number will appear more often. For example: 5 and 6 are modes, in {4, 4, 5, 5, 5, 6, 6, 6, 7, 8}. Mouse over the following example sets to see their modes:

  • {2, 4, 2, 7, 9, 7, 3, 2}
  • {2, 4, 2, 7, 9, 7, 3, 2, 7}
Sample Mode Question
{2, 3, 9, 4, 11, 4x – 8, 3y – 4}
  1. The modes of the set above are 2 and 11. What is one possible value of x + y?
(This is a grid-in.)

OK. In order for 2 and 11 to be the modes of the set above, each need to appear an equal number of times, and more often than any other value in the list. Which means one of two things must be true:

  1. 4x – 8 = 2 and 3y – 4 = 11
  2. 4x – 8 = 11 and 3y – 4 = 2

Let’s deal with possibility 1 first:

4x – 8 = 2
4x = 10
x = 2.5

3y – 4 = 11
3y = 15
y = 5

So one possible value of x + y is 2.5 + 5 = 7.5.

6.75.

 
Let’s try a few more!
#12 in the set below is a grid-in. Also, it’s possible that the SAT will throw median, mode, and average at you all at once. Whatcha gonna do when this comes for you (like in #20)?

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