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Edited to 8 testers (not 9)

I feel like this is probably an extremely easy question but I am unable to articulate my question to google to find an answer.

Basically, referring to the image below, there are 5 tests and 8 testers, each test has a percentage success rate. What I want to know is why the overall percentage success rate, 83% (417%/500%) differs from the individual counted success rate, 75%. The individual counted success rate is the sum the individual successes (6) over overall testers (8).

Your help is much appreciated.

Chart image

Philip Craig
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  • Aren't there 8 testers? Anyway, the reason is because there are different numbers of testers in each test. If each test had the same number of testers, it would match. In general, you can combine the test success percentages in a way to match the overall sucess rate: by taking a weighted average of them, where you weight each test's success by the proportion of the total group who was in that test. I.e. weight test 1's success rate by $2/8$ (since there were $2$ testers and $8$ testers total), weight 2's success rate by $3/8$, and the others by $1/8$. – Minus One-Twelfth Mar 25 '19 at 08:57
  • Because the % of the sum is not equal to the sum of %. You have to take into account the "population: simple case with $2$ and $100$ test respectively. If the % success is 100% and 50% we have $52$ positive results vs $102$ cases, while if we swap the percentage we have $101$ positive vs $102$ cases. The total % is very very difefrent. – Mauro ALLEGRANZA Mar 25 '19 at 08:57
  • From an "abstract" point of view, let $r_1$ and $r_2$ the two ratios (the percentage of success expressed as number between $0$ and $1$. We have that $(r_1 \times a + r_2 \times b) \ne \dfrac {r_1 + r_2}{2}$. – Mauro ALLEGRANZA Mar 25 '19 at 08:59
  • By the way, when I said "overall success rate" before, I meant what you called the "individual counted success rate". – Minus One-Twelfth Mar 25 '19 at 09:04
  • Edited to 8 testers! Thanks everyone, starting to get my head around it. So 75% is the correct answer which is what I suspected. – Philip Craig Mar 25 '19 at 09:34
  • For more on this, read about "Simpson's Parados" ... https://en.wikipedia.org/wiki/Simpson%27s_paradox – GEdgar Mar 25 '19 at 10:18

1 Answers1

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Because the $67$% (well, actually $\frac{200}{3}$) and $50$% data is from $3$ and $2$ testers as opposed to the $100$%s which are only from $1$. In other words, the lower data counts for more of the data.

You should multiply the % by the corresponding number of testers when doing your calculation, so:

$$\frac{(50\% \cdot 2)+(\frac{200}{3}\%\cdot3)+3(100\%\cdot 1)}{2+3+3}=\frac{600}{8}=75\%$$

Rhys Hughes
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