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

- 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}

- 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:

- 4
*x*– 8 = 2 and 3*y*– 4 = 11 - 4
*x*– 8 = 11 and 3*y*– 4 = 2

Let’s deal with possibility 1 first:

4*x* – 8 = 2

4*x* = 10

*x* = 2.5

3*y* – 4 = 11

3*y* = 15

*y* = 5

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

##### 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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It would really help if you would EXPLAIN the answers, both to these extra questions and those on the tests!!

I know. Eventually I will post explanations to everything, but in the meantime I’m happy to answer specific questions. Is there a specific question you have about this post?

You can also post questions at qa.pwnthesat.com/ask, which I try to respond to as quickly as possible.

Can you please explain #20?

I explained this one a while back on the Q&A site here.

Hi, can you explain #10?

Sure. If there are 4 positive integers with a mode of 3, that means at least 2 of them have to be 3. You want their sum to be as small as possible. You can’t do {1, 1, 3, 3} because then you’d have two modes. The next smallest choice is {1, 2, 3, 3}, which has a sum of 9.

I would like to add that we may be more careful by not ignoring the other cases possible here: {3,3,3,?} and {3,3,3,3},which we then calculate the sums of each 3+3+3+1=10, 3+3+3+3=12. Compare altogether 9, 10, 12 and we pick 9. I think it would be dangerous if SAT set traps in cases we sometimes ignore. Like in the question about the circle touching 2 axis in your Drill, I was wrong for not caring all the 4 quadrants of the xy-plane, ending up for m=n.

P/S: Mike, I would say your blog is great! some prep books are boring and not like the SAT in kinds of problems. Hope you will create more Math Drills, btw 😀

Thank you, Tuán!

What about something like {0, 1, 3, 3}?

0 isn’t a positive integer.

If 51 is the median of the values of 4f,f and 3f, what is the value of f?

Well of those three values, 3f must be the middle one. So 3f = 51. 51/3 = 17, so f = 17.

could you please explain #12

Sure! The key word in the question is

consecutive. If the median is –1, then there will be 3 consecutive integers after –1, and 3 consecutive integers before –1 to make 7 total consecutive integers. Like so:–4, –3, –2,

–1, 0, 1, 2The greatest integer in that set is 2.