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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

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Lesson Transcript

Instructor:
*Kevin Newton*

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

If you're going to divide a data set into four groups in order to make a box and whisker plot, then it would likely be very helpful to know where to split the data. Quartiles tell you exactly where to make the divisions.

Imagine that you had a gigantic set of data to analyze. Okay, maybe not gigantic, but still big enough to want to split it up into four parts. Why four parts? Well, if you've used a box and whisker chart, you can see that it's incredibly useful for showing range and spread of data. But even if you haven't, four is still a pretty good number. After all, people can internalize what it is to be in the best or worst quarter, but ask them how they feel about being in the fifth-seventh and you might get some strange looks.

Of course, to be able to divide a set of data into four pieces, you have to know where to make the cuts. Just like a surgeon should, hopefully, know where to cut open a leg to help set a badly broken bone, so too must you know where to cut your data. Luckily, you don't have to deal with blood.

The name given to where to cut your data into four equal groups is known as a **quartile**. In this lesson, we'll learn how to find them, as well as go through an example of using quartiles in action.

So, how do you go about finding a quartile? Simply ask yourself how you would find the middle point of a collection of numbers. If you're not new to working with numbers as data, then you probably know that the middle point of a collection of numbers is the median. If you remember how to calculate the median, then you're already ahead of the game. If not, simply put the numbers in order and determine what the middle point is. If it's between two numbers, just take the average. In any event, that's the median.

It also happens to be one of the quartiles. We call this quartile **Q2**, since it is the second quartile. Keep in mind that there only three quartiles, since these three slices divide the data into four chunks. These groups are equal in that they all contain the same number of data points, but can span different ranges. At Q2, 50% of the data is on either side of the quartile while at Q1, 25% of the data is less than the quartile. Likewise at Q3, 25% of the data is greater than the quartile.

So that's one quartile, but what about the other two? Simple - do the same thing again, but this time replace the far end point with the median. As a result, if you are trying to find **Q1**, the first quartile that cuts between the lowest two quarters of a set, then you find the median between the median of the whole set and the lowest number.

Likewise, if you're trying to find **Q3**, the third quartile that cuts between the highest two quarters of the set, then find the median between the median for the whole set and the highest number. Remember to divide the two numbers on either side if the median falls in between numbers.

But how does all this factor into box and whisker plots? The quartiles tell us where to draw the points of separation in a box and whisker plot. The area between Q1 and Q3 represents the box, while Q2 represents the median line. Meanwhile, the whiskers of the plot are represented by the numbers that are furthest away from Q1 and Q3, the data set's minimum and maximum values. In short, Q1 and Q3 define the box, while Q2 defines the median.

Now that we've got that down, let's try an example. Say that you are finding the quartiles of a set of test scores for your teacher. Don't worry. It's for the other class, so you won't be seeing anyone's scores from your class. The numbers are as follows, in order of smallest to largest: 63, 65, 70, 74, 80, 88, 88, 89, 92, 93, 96, and 99. That's twelve numbers total.

So what's the first thing we do? Right, we find the median. The median happens to be between 88 and 88, so luckily, we don't have to do too much to find that average; it's 88. Therefore, Q2 is 88.

But what about the other two quartiles? Remember, just take the median of each half. For the first half, that means finding the average between 70 and 74. That happens to be 72, so Q1 is 72. Now we have divided this set into two quarters and a big half at the end.

Let's divide that last half. Here, the median falls between 92 and 93. Therefore, Q3 is equal to 92.5. There you have it: our quartiles are 72, 88, and 92.5.

In this lesson, we looked at how to find quartiles in a set of data. Finding **quartiles** allows us to use box and whisker charts, deal with smaller chunks of data at one time, and make it easier for people to understand exactly what point we are talking about of a data spread.

Medians are very important in helping us find the quartiles. To find the middle quartile, also known as **Q2**, simply take the median of the whole set. To find the smaller quartile, known as **Q1**, take the median from the smallest number to the median of the whole set. Finally, to find the largest quartile, known as **Q3**, take the median from the median of the whole set to the largest number.

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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

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