I am an accountant with a university mathematics degree. Although its been some years since I became heavily involved in Mathematics, I should be able to understand most things, however I can't get my head around this problem.

At work I have encountered a problem with fractions. I have to explain this to my Chief Financial Officer and to the board so I need to come up with a credible explanation that is say less than 30 seconds long and is in leyman's term's. I can't simply say the "The maths doesn't work" or go through on a whiteboard the proof. So any references to Math theory would be appreciated so that I can add credibility to my answer.

(e.g. the fraction rule - Fraction addition with different denominators - If one denominator divides evenly into the other, change to the higher denominator and add)

The problem - (I have used simple numbers below but the actual problem is in thousands of dollars and holds more sets of customers)

I have two sets of customers,

'Customer A' had expected revenue of 10 but has only renewed 8, therefore we have a renewal fraction of 4/5

'Cusotomer B' had expected revenue of 20 but has only renewed 10, therefore we have a renewal fraction of 1/2

The summarized renewal fraction is therefore 3/5 (e.g 18 / 30)

We then have this other Key performance indicator called ACV (Annual contract value) I won't explain further what this is but it is independently derived from the renewed dollars.

For this we have the following result

'Customer A' ACV = 5 and 'Customer B' ACV = 15, total ACV therefore equals 20.

So for my purposes we are going to apply the above fractions to these results.

'Customer A' renewal fraction X ACV = 5 x 4/5 = 4 'Customer B' renewal fraction x ACV = 15 x 1/2 = 7.50 4 + 7.50 = 11.50

But if we take summarized renewal fraction and multiply it by the total ACV we reach a different result of 12 ( 20 x 3/5 = 12)

Does anyone know of any theory of why we get different results? I've tried to explain to my CFO that the Maths doesn't work and you can't apply fractions to different sets of data, but he has asked for a more credible answer.

Thanks for your help in advance.

  • $\begingroup$ Without looking too closely, I would guess it's related to Simpson's Paradox (look it up on Wikipedia). $\endgroup$ – ConMan Dec 1 '16 at 2:42
  • $\begingroup$ Thanks ConMan, I can see some relation to Simson's Paradox, but I still don't feel confident with this especially when determing the right number 11.50 or 12, it be good if there was a similiar example in SP to my situation above. I will keep looking though $\endgroup$ – Timso14 Dec 1 '16 at 3:55
  • $\begingroup$ Why it has to be the same? The result show that - 5X(8/10)+15X(10/20) does not equal (5+10)X((8+10)/(10+20)). Is there any reason why the result should be equal? $\endgroup$ – Moti Dec 1 '16 at 7:34
  • $\begingroup$ Thanks Moti, obviously they are not the same, its just understanding why and being able to confidently explain to a board of directors or shareholders. As they will likely say "Hey of all your customers your total renewal rate is 60% (3/5) but when you apply that percentage to their annual contract values in total you get 12 (60% x 20), but in each set of customers individually you get 4 and 7.5 reaching a total of 11.5, and instinctively they will likely next say hey your maths does seem right, because where has the 0.5 gone? and which is the correct figure 11.5 or 12? $\endgroup$ – Timso14 Dec 1 '16 at 9:15
  • $\begingroup$ The area where I am finding it difficult to apply Simpson's Paradox (maybe I am not understanding it, and need to analyse it more) is that Simpson's Paradox is taking combining two of actual data, and discussing that the overall results can draw a different conclusion to the conclusion you would draw if you have a breakdown of those sets of data. For example overall drug A is better than Drug B, but actually if you look into the breakdown of Drug A and Drug B actually Drug B is better. Which is not really the explanation that I am looking for here. $\endgroup$ – Timso14 Dec 1 '16 at 9:33

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