Prices, location and spread
Prices, location and spread

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Prices, location and spread

2.1 The mean of a combined batch

This first subsection looks at how a mean can be calculated when two unequally weighted batches are combined.

Example 7 Alan’s and Beena’s biscuits

Suppose we are conducting a survey to investigate the general level of prices in some locality. Two colleagues, Alan and Beena, have each visited several shops and collected information on the price of a standard packet of a particular brand of biscuits. They report as follows (Figure 9).

  • Alan visited five shops, and calculated that the mean price of the standard packet at these shops was 81.6p.

  • Beena visited eight shops, and calculated that the mean price of the standard packet at these shops was 74.0p.

Described image
Figure 9 Means of biscuit prices

If we had all the individual prices, five from Alan and eight from Beena, then they could be amalgamated into a single batch of 13 prices, and from this combined batch we could calculate the mean price of the standard packet at all 13 shops. However, our two investigators have unfortunately not written down, nor can they fully remember, the prices from individual shops. Is there anything we can do to calculate the mean of the combined batch?

Fortunately there is, as long as we are interested in arithmetic means. (If they had recorded the medians instead, then there would have been very little we could do.)

The mean of the combined batch of all 13 prices will be calculated as

fraction sum open bracket of the combined batch prices close bracket over size open bracket of the combined batch close bracket end .

We already know that the size of the combined batch is the sum of the sizes of the two original batches; that is, 5+8=13. The problem here is how to find the sum of the combined batch of Alan’s and Beena’s prices. The solution is to rearrange the familiar formula

mean = fraction sum over size end

so that it reads

sum = mean times size .

This will allow us to find the sums of Alan’s five prices and Beena’s eight prices separately. Adding the results will produce the sum of the combined batch prices. Finally, dividing by 13 completes the calculation of finding the combined batch mean.

Let us call the sum of Alan’s prices ‘sum(A)’ and the sum of Beena’s prices ‘sum(B)’.

For Alan: mean = 81.6 and size = 5, so sum open bracket A close bracket = 81.6 times 5=408.

For Beena: mean =74.0 and size =8, so sum open bracket B close bracket =74.0 times 8 =592.

For the combined batch:

mean = fraction combined sum over combined size end = fraction 408 +592 over 13 end = fraction 1000 over 13 end simeq 76.9

Here, the result has been rounded to give the same number of digits as in the two original means.

The process that we have used above is an important one. It will be used several times in the rest of this course. The box below summarises the method, using symbols.

Mean of a combined batch

The formula for the mean overline x subscript uppercase C end of a combined batch uppercase C is

overline x subscript uppercase C end = fraction overline x subscript uppercase A end n subscript uppercase A end + overline x subscript uppercase B end n subscript uppercase B end over n subscript uppercase A end + n subscript uppercase B end end comma

where batch uppercase C consists of batch uppercase A combined with batch uppercase B, and

overline x subscript uppercase A end = mean of batch uppercase A comma n subscript uppercase A end = size of batch uppercase A comma overline x subscript uppercase B end = mean of batch uppercase B comma n subscript uppercase B end = size of batch uppercase B.

For our survey in Example 7,

overline x subscript uppercase A end = 81.6 comma n subscript uppercase A end = 5 comma overline x subscript uppercase B end = 74.0 comma n subscript uppercase B end = 8.

The formula summarises the calculations we did as

overline x subscript uppercase C end = fraction open bracket 81 .6 times 5 close bracket + open bracket 74 .0 times 8 close bracket over 5 +8 end .

This expression is an example of a weighted mean. The numbers 5 and 8 are the weights. We call this expression the weighted mean of 81.6 and 74.0 with weights 5 and 8, respectively.

To see why the term weighted mean is used for such an expression, imagine that Figure 10 shows a horizontal bar with two weights, of sizes 5 and 8, hanging on it at the points 81.6 and 74.0, and that you need to find the point at which the bar will balance. This point is at the weighted mean: approximately 76.9.

Described image
Figure 10 Point of balance at the weighted mean

This physical analogy illustrates several important facts about weighted means.

  • It does not matter whether the weights are 5 kg and 8 kg or 5 tonnes and 8 tonnes; the point of balance will be in the same place. It will also remain in the same place if we use weights of 10 kg and 16 kg or 40 kg and 64 kg – it is only the relative sizes (i.e. the ratio) of the weights that matter.

  • The point of balance must be between the points where we hang the weights, and it is nearer to the point with the larger weight.

  • If the weights are equal, then the point of balance is halfway between the points.

This gives the following rules.

Rules for weighted means

Rule 1 The weighted mean depends on the relative sizes (i.e. the ratio) of the weights.

Rule 2 The weighted mean of two numbers always lies between the numbers and it is nearer the number that has the larger weight.

Rule 3 If the weights are equal, then the weighted mean of two numbers is the number halfway between them.

Example 8 Two batches of small televisions

Suppose that we have two batches of prices (in pounds) for small televisions:

Batch uppercase A has mean 119 and size 7. Batch uppercase B has mean 185 and size 13 .

To find the mean of the combined batch we use the formula above, with

overline x subscript uppercase A end = 119 comma n subscript uppercase A end = 7 comma overline x subscript uppercase B end = 185 comma n subscript uppercase B end = 13.

This gives

overline x subscript uppercase C end = fraction open bracket 119 times 7 close bracket + open bracket 185 times 13 close bracket over 7 +13 end = fraction 833 +2405 over 20 end = fraction 3238 over 20 end = 161.9 simeq 162.

Note that this is the weighted mean of 119 and 185 with weights 7 and 13 respectively. It lies between 119 and 185 but it is nearer to 185 because this has the greater weight: 13 compared with 7.

Example 8 is the subject of the following screencast. [Note that references to ‘the unit’ and ‘the units’ should be interpreted as ‘this course’. The original wording refers to the Open University course from which this material is adapted.]

Download this video clip.Video player: Calculating a weighted mean
Skip transcript: Screencast 2 Calculating a weighted mean

Transcript: Screencast 2 Calculating a weighted mean

INSTRUCTOR: 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.

End transcript: Screencast 2 Calculating a weighted mean
Screencast 2 Calculating a weighted mean
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