Posts tagged with: mode

Hi Mike. Can you please explain Practise test 1, section 4, question 14? I figured out that the range was the largest change (but only by physically calculating the mean and median etc.)

Is there a better way to do this?

Also, I’d just like to say THANK YOU THANK YOU THANK YOU for this book it really is amazing!!

Well, you have to consider every answer choice, but there are a few mental shortcuts that I think are helpful.

The main thing to recognize right away is that the data is in order, so calculating the median is very easy. The middle value is 12, and if you remove that outlier 24, the median will STILL be 12. (It’s useful to remember that, in general, outliers have little to no effect on the median—that’s why you usually hear about median household income instead of mean household income, for example.) So I can cross of median without calculating it, and then I can cross off choice D as well.

The data being neatly sorted also helps one to think about the mean. Notice how nicely clustered the data is: how the 13s, 14s and 15s pretty nicely balance out the 11s, 10s, and 9s. That tells you that the mean might not be exactly the same as the median, but it should still be very close to 12. So without adding up all the values, I know the average of 20 of the values should be about 12, which means their sum should be about 240 (average times # of numbers equals sum). Adding the erroneous 24 value in there won’t change the average much: \dfrac{\approx 240+24}{21}\approx 12.57. (Note that while this estimate is not completely accurate, it doesn’t matter because you’re just looking for the biggest change, not the exact change.)

The range, on the other hand, will change a lot! The range with the 24 measure in there is 24-8 = 16. Take that 24 out and the range becomes 16 - 8 = 8. That’s by far the biggest change.

PS: Glad you like the Math Guide! 🙂

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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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