Screencast 3 Calculating quartiles

Transcript

INSTRUCTOR
Here’s another batch of data from the unit. This time, it’s the prices of a particular model of digital camera. And there are the prices along the top in pounds. And what we’re asked to do is to find the lower and upper quartiles of this batch of data.
So usually the first thing you have to do when you’re finding the median or the lower and upper quartiles is to put the prices in order. But if you have a look up at the top there, luckily, somebody’s already done that for us. They’re in order.
So we can bash straight on with the next step, which is actually finding the quartiles. Now, what I’m going to do to begin with is to use the notation that we use in defining what the quartiles are, just so you can relate what I’m doing to what’s in the unit on that. Remember how that works. We take the first number, the smallest number in the batch, which in this case is 53, right over the left there.
And we write that as x, within brackets, 1. The 1 in brackets means it’s the smallest value in the batch. And the next one, which is 60, that’s x, brackets, 2. And then you just carry on like that.
And the last one is x10. That means there’s 10 values in this bunch. We need to know that.
And in fact, that’s the next thing we need to worry about. There are 10 observations, 10 camera prices. So in the notation, we call that n. n’s 10.
Now, remember what you’ve got to do to work out this. We want for the quartiles a number that’s 1/4 of the way along the batch when we’ve arranged it in order and another number that’s 3/4 of the way along the batch. But of course, it’s a bit more complicated than that. You don’t look at 1/4 of n. You look at n plus 1 divided by 4.
And that gives you the position of the lower quartiles. n plus 1 divided by 4 is 11/4, which is 2 and 3/4. And that defines where the position of the lower quartile is in a way that I’ll come to.
And to get the upper quartile – well, this time, you’ve got to take 3/4 of n plus 1. That is 3 times n plus 1, over 4. And that comes to n plus 1 is 11.
So it’s 33 – 3 times 11 – over 4. And if you work that out, that’s 8 and 1/4. And that defines the position of the upper quartile.
OK, so moving on, then, Q1 is the lower quartile. And it’s defined by this number here, this 2 and 3/4 number. And what that tells us is it’s between the second number in order and the third number. And it’s 3/4 of the way between them.
So I’ll just write down what that is. Q1 is 3/4 of the way from x2 to x3. And you can work that out just in arithmetic – but it’s probably easier if we have a look and see what it looks like graphically.
Here is a bit of a number line. x2 is going to be there. And x2 is 60.
And x3’s going to be up here somewhere. And x3, if we look up here, is 65. So that one’s 65.
And what we want is a number that’s 3/4 of the way from 60 to 65. Well, that’s halfway there. That’s going to be 62 and 1/2. And 3/4 of the way is about here somewhere. And that’s going to be 63.75.
OK, but how do we write that down in algebra? The way to do it is like this. We say you start at x2, which is 60. And then you go 3/4 of the way from x2 to x3.
Well, how far is it from x2 to x3? Well, again, if you look on this, it’s 65 to 60. That is 5.
And you can calculate that as 65 minus 60. So it’s 3/4 of 65 minus 60. So that’s 60 plus – we’ve got 3 times 5 over 4. That is 15/4. And if you work that out on your calculator or something, it comes to 63.75.
So we’ve worked out the lower quartile. It’s 63.75. And luckily, that’s the same as I showed you over here.
So now we’ve got to go and find the upper quartile. Remember, we denote that as Q3. And if you look at the thing over here, that says it’s at position 8 and 1/4, which means it’s 1/4 of the way from the eighth one to the ninth one. That is, it is 1/4 of the way from x8 to x9.
And again, it’s helpful to draw what’s going on here. So here is a number line. And here we’ve got x8, which is 81. And here we’ve got x9, which is 85.
And again, we want the number that’s 1/4 of the way along, bit different. So here’s halfway. That’s going to be 83. And here is 1/4 of the way, halfway between 81 and 83. That’s 82.
So it’s going to be 82. Let’s just see how we can write that down as a bit of algebra. So again, you start from x8, which is 81. And then you go 1/4 of the way up to x9.
Now this time, the distance between x8 and x9 is 85 – that’s x9 – minus 81, which is x8. So the arithmetic’s actually a lot easier in this case. That’s 81 plus 1/4 of 4. And that just comes to 81 plus 1, which is 82. Again, it’s the same.
So just one final step to do here. We’ve got to write down the answers in an appropriate way. Now, what we have to do is round them to an appropriate accuracy. That’s all we haven’t done.
And what we’ve got is that Q1 – what accuracy do we use? Well, the usual thing we do is to round quartiles to the accuracy of the original data that we had. And the original data we had up at the top there is in whole pounds. It’s rounded to whole pounds. So we need the quartiles in whole pounds, as well.
And if you look at this one, it’s not in whole pounds. It’s 63.75. So we’ve got to round it to the nearest pound. The bit in pence, essentially, here that we’re going to get rid of in the rounding is bigger than 1/2.
So we’ve got to round up. That is, this one is– let’s write in the pound sign. It’s £64 to the nearest pound.
And what about Q3? Well, you’ve got to think about rounding that, as well. But actually, you needn’t bother. It’s already a whole number of pounds, so no rounding involved. We can just write down it’s 82. And that’s that.

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