# Probability and Statistics

# Quartiles

**Quartiles** are numbers that break a list of data up into quarters ("quarters,'' "quartiles''... makes sense, right?). We are totally stuffed full of cookies at the moment, so let's turn our attention to, um, brownies.

To cut a pan of brownies into quarters we need to make three cuts:

Similarly, to break a list of data into quarters we need to find three quartiles.

To cut a pan of brownies into quarters, first we cut the pan down the middle:

Then we cut the left half down the middle:

...and finally cut the right half down the middle:

It appears we're using an exceptionally long pan. We're amazed that thing even fit in the oven.

To find quartiles, we do basically the same thing as we did with the brownies. We just don't eat them afterward, unless we want to have a severe bellyache.

### Sample Problem

Find the quartiles of the list of numbers

1, 4, 5, 7, 8, 10, 23, 25, 28, 32, 40.

To find the quartiles, first we cut the list in half down the middle; that is, we find the median:

1, 4, 5, 7, 8, **10**, 23, 25, 28, 32, 40.

Now we look at the left half of the list (not including the median):

1, 4, 5, 7, 8.

and find *its* median:

1, 4, **5**, 7, 8

Finally, we look at the right half of the list (not including the median):

23, 25, 28, 32, 40

and find *its* median:

23, 25, **28**, 32, 40.

The quartiles of this list are

5, 10, 28.

The quartiles are often called *Q*_{1}, *Q*_{2}, and *Q*_{3}. If you ever see *Q*_{4}, he is an impostor, and should be escorted from the premises immediately. Remember, there may be 4 quarters, but there are only three cuts. In the example above, we had

In summary, to find the quartiles of an **ordered list**, there are four steps:

- Find the median, which is quartile
*Q*_{2}.

- Find the median of the "left'' half of the list (not including
*Q*_{2}), which is quartile*Q*_{1}.

- Find the median of the "right'' half of the list (not including
*Q*_{2}), which is quartile*Q*_{3}.

- Draw seven new quartiles from the bag and wait for your opponent to take their next turn. Wait a second...

### Sample Problem

Find the quartiles of the list

2, 4, 6, 6, 7, 10, 14.

First we find the median, also known as *Q*_{2}:

2, 4, 6, **6**, 7, 10, 14

We know *Q*_{2 }= 6.

Now we find the median of the left half of the list, not including *Q*_{2}. The median of

2, **4**, 6

is 4, so

*Q*_{1 }= 4.

Finally, we find the median of the right half of the list, not including *Q*_{2}. The median of

7, **10**, 14

is 10, so

*Q*_{3 }= 10.

A couple of warnings are in order here. First of all, don't venture past the "No Trespassing" sign. That should go without saying. Second, and more to the point, we've said the quartiles are the numbers that divide the data set into four pieces, which is true. However, to make things confusing, "quartile'' can also refer to one of those four pieces. In the list

1, 2, 3, 4, 5, 6, 7, 8

we would say the quartiles are

*Q*_{1 }= 2.5, *Q*_{2 }= 4.5, *Q*_{3 }= 6.5.

We could also say the quartiles are the four sets into which the data has been divided:

As if that discrepancy wasn't bad enough, there is no global consensus on how to find the quartiles, so some calculators and computer programs may come up with different quartiles than you do. It doesn't mean you're doing something wrong; it just means you're following a different recipe. Your calculator may be making chicken parmesan and you might be making a skirt steak, but don't sweat it. It's all delicious.

Dr. Math talks here about some of the different ways quartiles can be calculated. We've gone with the most common way, but be sure it's in agreement with your teacher and textbook. We're honored you want to do it the Shmoop way, but we're not the ones grading your pop quizzes.