I need to decide which secondary school I should send my child to. In the area where I live, two schools are available, St James School and St Giles School. Which school has the better track record of students obtaining at least five passes in the SEC exams? I look up last year’s statistics for both schools to check this. Both schools last year presented 100 fifth formers for these exams. St James had 60 students who obtained at least five SEC passes, that is, a pass rate of 60 per cent, whereas St Giles had 70 successful students, a pass rate of 70 per cent. It seems clear that, based on these figures, I should choose St Giles. Or is it?
Both schools are co-ed. So I analyse the above figures by gender. The table shows the results of my analysis.
This looks very odd. Overall, St Giles does have a better success rate, but if my child is a boy, I should choose St James, because 53 per cent of St James’ boys were successful in their SEC exams, as compared with 43 per cent for St Giles, while if my child is a girl, I should also choose St James since 90 per cent of their girls were successful as compared with 88 per cent for St Giles.
This is a simple example of what is called Simpson’s Paradox, which occurs when conclusions pointed to by data are reversed when the data are grouped by sub-categories. A famous case of Simpson’s Paradox occurred when the University of Berkeley was facing a gender discrimination case because it seemed that its student entry policy was gender-biased against females. When the student entry was broken down by departments it turned out that more departments were biased in favour of females than males. Why does this happen?
In the Berkeley case it was because females were applying to join the more selective courses which accepted fewer students. As a result, although a larger proportion of females than males were selected for entry into these courses, overall a larger proportion of males than females were accepted, because males were more likely to apply for courses with larger intakes.
The same phenomenon is happening in the St James and St Giles figures above. Girls got better SEC results than boys, but St James had more boys than girls, so that St James’ overall results were not as good as St Giles’, although the school did better when the data was grouped by gender.
So, which school should I choose for my child? And how should statistics be presented if they are to guide us in our decisions? The situation is made even more complicated because the same data can be grouped under different categories pointing to different conclusions. Are these the damned lies referred to by Disraeli?
(To be continued)
Did you know!
Mathematical curiosities about 2017:
• 2017 is a prime number (its only factors are 1 and itself). The next prime year is in 10 years’ time: 2027.
• 2017 is a sum of two squares: 9 x 9 + 44 x 44. A theorem of Fermat states that every prime number of the form 4n + 1 (for some whole number n) is a sum of two squares. We have 2017 = 4 x 504 + 1.
• 2017 is the hypotenuse of a rightangled triangle in which the lengths of the other two sides are also whole numbers: 792 and 1,855. This is because Pythagoras’s Theorem a x a + b x b = c x c is satisfied with a = 792, b = 1,855 and c = 2,017.
For more trivia see: www.um.edu.mt/think
Sound bites
• My favourite theorem of 2016: You are at the mercy of the Evil Captor. You are placed in the middle of a long corridor and you are to take steps forward or backwards in any order you desire with as many consecutive backward or forward steps as you want. At the front end of the corridor there is a pit full of poisonous vipers, while at the back end there is a pit full of poisonous spiders. You must avoid falling into one of the pits. If you take, for example, a step forward, then backwards, then forward, and so on, this sequence would be recorded as +1, -1, +1, -1, +1... But the deal is this. You write down your proposed sequence of steps and then the Evil Captor can require you to take every second step, or every third step, or every fourth step, for example. In this case, he can ask you to take every second step and you will end up with the poisonous spiders. If the sequence is -1, -1, +1, -1, -1, +1..., he can ask you to take every third step and the vipers will get you. Can you devise a sequence so that no matter what sub-sequence of regularly spaced steps the Evil Captor requires you to take, you will never fall into any of the pits? In 2015, Terence Tao, arguably the most brilliant mathematician in the world at the moment, proved that no matter how long the corridor is, if you give the Evil Captor a long enough sequence of steps, he will be able to force you to your doom. This solution eventually appeared in a journal in 2016. In spite of the frivolous popular description given above, this problem, posed by Paul Erdős over 80 years ago, involves some very deep ideas from various areas of mathematics. Tao also used some ideas from Polymath5, an online project (the fifth of its kind) started by Timothy Gowers for crowdsourcing a mathematical proof of this problem!
• For more science news, listen to Radio Mocha on Radju Malta 2 every Monday at 1pm and every Friday at 6pm.