OK, here’s an example that you’ve seen in the units about the price of two batches of small TVs. The first batch, that’s Batch
A, that’s got a mean of 119, that’s £119 actually, and size 7, there are 7 TVs in that batch. Batch B, the second batch, has
got a mean of £185, good bit more expensive, and the size of that batch is 13. And what we’re asked to do here is to find
the mean of the combined batch. OK, so how are we going to do that?
Well, there’s a formula in the unit which I’ll write down in a minute. But I want to build it up a bit at a time to show you
where the formula comes from, and the formula looks like this. We’ve got x bar, that means mean. And c, that’s because it’s
the mean of the combined batch.
Now, we usually calculate a mean by calculating the total of the values in the batch, and dividing it by the size of the batch.
And essentially, that’s all that’s happening here – we just have to do it in a bit of a complicated way. So how does that
work?
Well, first of all we need the sum of the values in Batch A. Now, we’ve got the mean, we haven’t got the sum. But that mean,
£119, is the sum divided by the size, which is 7. So you can get back to the sum by multiplying the mean by the size. So that
is it’s the mean of the values in Batch A, x bar A, times the size of Batch A.
Anyway, that deals with the TV prices in Batch A, and now we’ve got to add on the TV prices in Batch B. So you do the same
trick. You add on x bar B, the mean of the values in Batch B, times the size, nB, of Batch B, and that gives you the sum part
of this mean calculation.
Then you’ve got to divide that by the size, so there’s the division sign. And the size of the combined batches is the total
number of TVs in this combined batch. So it’s the number you’ve got in Batch A, plus the number you’ve got in Batch B – nA
plus nB. So that’s the formula, and that’s the same one as in the unit.
The next step is to convert the numbers we have to make clear how the notation we got in this formula fits with the numbers
we’ve got. So let’s just translate the mean and the size of the batches into the notation we need for the formula. So for
Batch A, the mean’s 119 – that is, in this notation we got x bar A is 119, and nA the size of Batch A is 7 because there’s
7 TVs. And then we do the same with Batch B, x bar B is 185 to match what it says over the left there, and nB is 13.
So we’ve got the formula here, and we’ve got the values we need to put in the formula. So now we’ve just got to go ahead and
put the values in and do the arithmetic. So the first step, let’s work out what we’ve got to put on the top. Well, on the
top of this expression we’ve got x bar A times nA, and that is 119, that’s x bar A, times nA, which is 7, and I’ve put the
multiplication sign in there to make it clear what’s going on.
And then the other aspect of this is we’re going to add something on in a minute, but we need to do the multiplication first.
That is, we need to do this multiplication before we get to this plus. So to make that absolutely clear, we put this in some
brackets to indicate that’s what we’re going to do first.
So we do that, and then we add on what we get from the B batch, that is x bar B times nB. And again, we’re going to put that
in brackets, so that’s 185 times 13. Close the brackets, and then divide the whole thing by nA plus nB, which is 7 plus 13.
So that’s put in the values we got into the formula we had. And now it really is just a case of bashing through the arithmetic.
So I’ve got my trusty calculator here – 119 times 7, that comes to 833. And then we’ve got to add on what we get from this
bit here – 185 times 13, and that comes to 2,405. So that’s the top of the thing we’ve got to calculate. We’ve got to divide
by the bottom, 7 plus 13, and doing that all in my head it’s 20 and we’re nearly there.
What we’ve got now is we added 833 to 2405, that comes to 3238, and we’ve got to divide that by 20. And calculating that,
that comes to 161.9. Well, we’re nearly there now. It looks as if it’s calculated the mean of the combined batch but there’s
a final step we got to do, and that’s to relate it back to what’s really going on here.
If you look at these means up here, or they were here originally, they’re given in whole pounds. That is, these means are
rounded to the whole pound. Well, it might have been coincidence that they actually come to whole numbers, but what’s really
happened is that it’s not appropriate to express the mean to the full accuracy you get from your calculator, because that’s
just too much accuracy. The data don’t support it.
So what we got to do is we got to round the means to an appropriate value. And in this case, the original means were rounded
to the nearest pound, so our combined mean also needs to be rounded to the nearest pound. So let’s have look at what it is
– it’s 161.9. And to round that to the nearest pound, is it going to go up or down?
Well, look at the last figure here, the one we’re going to take away in the rounding – it’s 9, that’s bigger than 5. So to
round it to the nearest pound, we have to round it up to 162. That is, we finish off by saying this is £162 rounded to the
nearest pound, and that really is the finish of the calculation.
But before we leave this, I just want to point out one thing to you. Look at this value here of 162 and compare it with the
means of the two batches we had – Batch A: 119, Batch B: 185. The 162 we’ve got here is a good bit nearer the mean of Batch
B than it is to the mean of Batch A. And that’s because Batch B is bigger. It’s got more TVs in it – it’s got 13, and this
has only got 7.
So what’s happened is that the Batch B has dominated in the calculation because there’s just more TVs in Batch B. There are
13 of them compared to 7. So Batch B has had more influence on this number than Batch A has.
Now in the unit, it explains that this formula here that we had for a mean of a combined batch is a kind of weighted mean.
And the unit also describes that there are various rules that weighted means follow, various properties they have. And one
of the properties they have is that the value of the weighted mean of two numbers is closest to the value that had the largest
weight.
Now what’s playing the role of weights here is the batch sizes, nA and nB. nB is bigger than nA, so this rule tells us that
the combined value, this weighted mean, is going to be nearer to this, the mean of Batch B, than it is to this. And indeed
that’s what we find – 162 is nearer to 185 than it is to 119.
So I’ll just record the fact that we found that. This is closer to the mean x bar B, the mean of Batch B, as Batch B is bigger.
And that’s that.