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# How can your scores be worse than mine every year but still come out higher on average than mine?

Fred and Jane study on the same course spread over two years. To complete the course they have to complete 10 modules. At the end, their average annual results are:

 Year 1 average Year 2 average Overall Average ? Fred 50 70 60 Jane 40 62 51

So how is it possible that Jane got the prize for the student with the best grade?

It’s because the overall average figure given in the column is an average of the year averages rather than an average over all 10 modules. We cannot work out the average for the 10 modules unless we know how many modules each student takes in each year.

In fact:

Fred took 7 modules in Year 1 and 3 in Year 2

Jane took 2 modules in Year 1 and 8 modules in Year 2.

Assuming each module is marked out of 100, we can use this information to compute the total scores as follows:

 Year 1 total Year 2 total Overall total Real Overall average Fred 350 (7 x 50) 210 (3 x 70) 560 56.0 Jane 80 (2 x 40) 496 (8 x 62) 576 57.6

So clearly Jane did better overall than Fred.

This is an example of what is commonly known as Simpson’s paradox. It seems like a paradox – Fred’s average marks are consistently higher than Jane’s average marks but Jane’s overall average is higher. But it is not really a paradox. It is simply a mistake to assume that you can take an average of averages without (in this case) taking account of the number of modules that make up each average. Look at it the following way and it all becomes clear:

In the year when Fred did the bulk of his modules he averaged 50; in the year when Jane did the bulk of her modules she averaged 62. When you look at it that way it is no surprise that Jane did better overall.

There are many example of Simpson’s paradox arising in real-life situations. Worryingly, there are several well known medical studies that have drawn exactly the wrong conclusion as a result of this ‘paradox’.

For example, a medical study compared the success rates of two treatments for kidney stones. Each treatments was applied to two groups of people – one group in which each subject had a small stone and one group in which each subject had a large stone. The ‘average’ success rates were:

 Small stones Large stones Overall Average ? Treatment A 93% 73% 83% Treatment B 87% 69% 76%

It was concluded therefore that treatment A was more effective.

But on inspecting the number of patients in each of the four groups it becomes clear that exactly the opposite was true:

 Small stones Large stones Overall successes Overall success rate Treatment A 81/87 192/263 273/350 78% Treatment B 243/270 55/80 289/350 83%

For more on Simpson’s paradox see here.

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Norman Fenton

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