How to calculate interquartile range IQR | Data and statistics | 6th grade | Khan Academy

Let's get some practice calculating interquartile ranges. I've taken some exercises from the Khan Academy exercises here, and I'm going to solve it on my scratch pad. The following data points represent the number of animal crackers in each kid's lunchbox. Sort the data from least to greatest, and then find the interquartile range of the data set. I encourage you to do this before I take a shot at it.

All right, so let's first sort it. If we were actually doing this on the Khan Academy exercise, you could just drag these; you could just click and drag these numbers around to sort them, but I'll just do it by hand.

So let's see. The lowest number here looks like it's a four. So I've had that four. Then I have another four, and then I have another four. Let's see, are there any fives? No fives, but there is a six. So then there's a six, and then there's a seven. There doesn't seem to be an eight or a nine, but then we get to two tens. Then we get to 11, 12, no 13, but then we get 14, and then finally we have a 15.

So the first thing we want to do is figure out the median here. The median is the middle number. I have one, two, three, four, five, six, seven, eight, nine numbers, so there is going to be just one middle number. I have an odd number of numbers here; it's going to be the number that has four to the left and four to the right. That middle number, the median, is going to be 10. Notice I have 4 to the left and 4 to the right.

The interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half. It's a measure of spread—how far apart all of these data points are. So let's figure out the middle of the first half. We're going to ignore the median here and just look at these first four numbers. Since I have an even number of numbers, I'm going to calculate the median using the middle two numbers.

So, I'm gonna look at the middle two numbers here. I'm gonna take their average. The average of four and six, halfway between four and six, is five. Or you can say four plus six is equal to ten, but then I wanna divide that by two, so this is going to be equal to 5. So the middle of the first half is 5. You can imagine it right over there.

In the middle of the second half, I'm going to do the same thing. I have four numbers, and I'm going to look at the middle two numbers. The middle two numbers are 12 and 14. The average of 12 and 14 is going to be 13. If you took 12 plus 14 over 2, that's going to be 26 over 2, which is equal to 13. But an easier way for numbers like this—you say, "Hey, 13 is right exactly halfway between 12 and 14."

So there you have it. I have the middle of the first half: this five; I have the middle of the second half: 13. To calculate the interquartile range, I just have to find the difference between these two things. So the interquartile range for this first example is going to be 13 minus 5, the middle of the second half minus the middle of the first half, which is going to be equal to 8.

Let's do some more of these. This is strangely fun! Find the interquartile range of the data in the dot plot below: songs on each album in Shane's collection. So let's see what's going on here, and then, like always, I encourage you to take a shot at it.

So this is just representing the data in a different way, but we could write this again as an ordered list. So let's do that. We have one— we have one album with seven songs, I guess you could say. So we have a seven. We have two albums with nine songs, so we have two nines. Let me write those; we have two nines. Then we have three tens. Cross those out: 10, 10, 10. Then we have an 11, we have an 11, we have two 12s, and then finally, we have an album with 14 songs—14.

So all I did here is I wrote this data like this so we could see, okay, this album has seven songs, this album has nine, this album has nine, and the way I wrote it, it's already in order. So I can immediately start calculating the median.

Let's see, I have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 numbers. I have an even number of numbers, so to calculate the median, I'm going to look at the middle two numbers. The middle two numbers look like it's these two tens here because I have four to the left of them and then four to the right of them. Since I'm calculating the median using two numbers, it's going to be halfway between them.

It's going to be the average of these two numbers. Well, the average of 10 and 10 is just going to be 10, so the median is going to be 10. In a case like this, where I calculated the median using the middle two numbers, I can now include this left hand in the first half and I can include this right 10 in the second half.

So let's do that. The first half is going to be those five numbers and then the second half is going to be these five numbers. It makes sense because I'm literally looking at the first half—it's going to be five numbers and the second half is going to be five numbers. If I had a true middle number like the previous example, then we ignore that when we look at the first and second half. Or at least that's the way that we're doing it in these examples.

But what's the median of this first half if we look at these five numbers? Well, if you have five numbers, if you have an odd number of numbers, you're going to have one middle number, and it's going to be the one that has 2 on either side. This has 2 to the left and it has 2 to the right. So the median of the first half, the middle of the first half, is 9 right over here.

In the middle of the second half, I have 1, 2, 3, 4, 5 numbers, and this 12 is right in the middle. You have 2 to the left and 2 to the right, so the median of the second half is 12. The interquartile range is just going to be the median of the second half, 12, minus the median of the first half, 9, which is going to be equal to 3. So if I was doing this on the actual exercise, I would fill out a 3 right over there.