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Print Weighted Averages – Redressing the Balance

Here’s a little problem that I’ve seen in Numerical Reasoning questions which shouldn’t cause a problem but somehow does.

Maybe it’s the wording that throws some of you; I’m not sure. I’m referring to weighted averages.

It’s easiest to look at an example to explain what I mean:

Weighted Averages

Question: What is the average income of a worker from Belgium and France combined in 2005?

A. - 26930; B. – 27831; C. – 27992; D. – 28067; E. – 28140.

So, what does the question mean by ‘average income’. My dictionary defines ‘average’ as ‘a quantity intermediate to a set of quantities’ which is not particularly useful.

In general, to calculate the average of a set of numbers you would add them together and divide by the number of numbers which make up the total to get the average. So, if I asked what is the average of 12, 24 and 30 then you would add them together to get 66, divide this by 3 and get the answer of 22.

Correct.

In the question above it asks for the average income over two quantities. If you added these together you would get 54661 which, when divided by 2, gives an average of 27831 (when rounded to the nearest euro).

So what’s the problem?

The problem is that the question is asking about the average income over the whole populations of France and Belgium combined and each of these data values specified is, itself, and average over a number of people. If the question had said, ‘Suppose there were a group of people consisting of 38 Belgian and 219 French workers, what would their average income be?’ This is, in effect, what the question is asking.

We cannot simply add together the two values and divide by two because there are more French workers contributing to the overall average than Belgian workers so the overall average will be skewed, or weighted, towards the French figure. What we need to do is to add up all the French contributions to the average and all the Belgian contributions and divide by the overall total number of contributions. How do we do this?

We must multiply each of the contributing numbers by the number of people contributing to it. For Belgium this would be:

26181 * 3.8 = 99488

And for France it would be:

28480 * 21.9 = 623712

The total would now be:

99488 + 623712 = 723200

And the total number of contributors would be:

21.9 + 3.8 = 25.7

Finally we can calculate the average income which will be:

723200 / 25.7 = 28140

This means that answer E is the correct answer and not answer B.

Note that the average lies much closer to that of the French contribution than the Belgian contribution because it has been weighted towards the French.

Lost in numerical reasoning? Check out some practice tests and boost your performance!

PA 08 May 2014

Am I missing something or do I have a problem with my calculator?

Your statement says:

54661 which, when divided by 2, gives an average of 27831 (when rounded to the nearest euro).

My calculator will only provide the answer 27330.5

What am I doing wrong?

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