Twice as good?

Headline from the Independent newspaper, dated 1st August 2013: "Schools ask pupils to sit maths exams twice to boost their league table scores".

The story which follows explains that several schools are making their pupils sit the GCSE maths examination twice in the same term, and then recording the better result for the school performance league tables.  It also applied to english exams.  School league tables record the percentage of students who achieve a grade "C" or better in maths and english, and such results are critical for the progress of those students.  (GCSE exams are at age 16, year 10)

For years, schools realised that students needed such grades for their careers, and so students would sit the examinations until they passed, in June, in December, in June, ad infinitum.  Those who wanted to teach, for instance, needed these grades.  As electricians might say, they sat the exams in series.  The headline was highlighting the practice of sitting exams in parallel, albeit with different examination boards, and therefore different examiners.  And this, not simply for the benefit of the pupils, but also for the school's reputation. 

It doesn't take a genius to work out that if a pupil has probability p of passing at grade "C" or better, and q=(1-p) of failing, then - assuming independence, justified by the different examiners - taking the exam twice in parallel means probability 1-q^2 of passing at least once, and inevitably, 1-q^2 > p.

Why stop at two exams in parallel?  Take the exam with n examiners, and the probability rises to 1-q^n.  It is a constrained optimisation problem - constrained by time and psychological pressure on the pupils.  Presumably, most schools are split between n=1 and n=2, but I wonder if somewhere, there is a school where n=3?

If you are reading this, you are probably numerate, and for you, p is very close to 1, and your school wouldn't have needed to enter you more than once. 


  1. And thus is born the Large Law of Weak Numbers (a large number of attempts will almost surely improve a weak score). We're more efficient here: we just record fictitious scores, without the effort of retests.


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