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I came across the following while doing some questions from my textbook.
For eg. lets say we need to calculate the total percentage of marks for the following subjects:

  1. Maths 64/100
  2. English 78/100
  3. Science 30/50

Total percentage being calculated as (64+78+30)/250 = 68.8 %

The other method I followed was to calculate individual percentages and then take their average

  1. Maths 64/100 = 64%
  2. English 78/100 = 78%
  3. Science 30/50 = 60%

Their average being (64+78+60)/3 = 67.33 %
I know by forming equations I can prove the second method is different from the first, but is there any logical statements that proves the second method is wrong and the first correct.

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    Given that, as you've stated, the "total percentage" is calculated by the first method, then it must be "correct", right? You have an example for which the second method gives a different answer; that fact alone *proves* that the second formula is "wrong", right?2012-06-30
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    @DayLateDon I know the second formula is wrong, but is there any logical statement that says percentages shouldn't be calculated like this, that's what i want to know. I was trying for a shortcut in my exam by following the second method which proved wrong in the end.2012-06-30
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    One reason to *suspect* that the answers are going to be different (in general) is that the second formula depends on the number of subjects involved, while the first does not.2012-06-30
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    You might be interested in Simpson's paradox http://en.wikipedia.org/wiki/Simpson's_paradox2012-06-30
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    If you're just looking for an intuitive way to see the difference: suppose there are just two subjects. In the first, a student got a *million* points out of a *million*; in the second, she got *zero* points out of *one*. The first method takes into account that the first subject has a lot more weight, giving a "total percentage" of 99.9999%; the second method treats each subject equally, and since the student aced the first and tanked on the second, the method reports an average of 50%.2012-06-30
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    @MarkBennet That is seriously an interesting read. But then what method should we follow? Is there some "formal" way which says which method to follow, or is it up to us.2012-06-30
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    You have to decide whether it is "right" to treat the three subjects on an equal footing, or treat points earned in the three subjects on an equal footing. If Science is as important as English, then the second method is correct. If 1 mark in Science is equivalent to 1 mark in English (so Science overall is only half as important as English) then the first method is correct. And of course there are other possibilities. You can't say which method to follow until you confront the "relative importance" question.2012-06-30
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    @GerryMyerson The problem is this question is given in my textbook, where nothing is mentioned about how to "treat" subjects and points. Because the "conventional" way is to follow the first method, that's why second method is wrong.2012-06-30
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    If the textbook has *defined* "total percentage" as the value computed by the first method, and if you are asked to compute "total percentage", then clearly you need to use the first method (because you've determined that the second gives a different answer). If you're asking which method is *more appropriate* to compute a value worthy of the term "total percentage", then --as @GerryMyerson says-- you must "confront the 'relative importance' question". Each formula says something useful about the data; you have to decide which is *more*-useful for your purposes.2012-06-30
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    @DayLateDon The text book hasn't defined anything, it just contains questions with respective answers, moreover it said to calculate overall percentage which is guess is the same as total percentage :)2012-06-30
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    If anyone is having you compute "overall percentage" without saying *exactly* what that phrase means, then you are well within your rights to be confused. To me, either of the methods you've given --not to mention, *lots* of others-- could reasonably be described as an "overall percentage"; neither is inherently "correct" or "wrong" in this role.2012-06-30
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    @DayLateDon that's the main problem, that is why my first thought was there "may" exist some formal text which says which method to follow.2012-06-30

2 Answers 2

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In my opinion, the second method is a trick.

For the first method, we can simply get the answer is $$\frac{64+78+60}{250}=68.8\%.$$

For the second method, we get the average point of each subject like

  1. Maths $\frac{64}{100}=64\%$
  2. English $\frac{78}{100}=78\%$
  3. Science $\frac{60}{50}=30\%$

Now, we should calculate the average in the following way

$$64\% \times\frac{100}{250}+78\% \times\frac{100}{250}+60\% \times\frac{50}{250}=68.6\%.$$

It called weighted average, and $\frac{100}{250},\frac{100}{250},\frac{50}{250}$ is called weights.

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No, there exists no (meaningful) way to prove the second method wrong. Science could get weighted twice as much as English and Maths, as in Science could come as twice as important as English and twice as important as Maths, in which case both methods yield the same result.

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    But i am getting different answers for the two2012-06-30
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    If science is weighted twice as much as the other two, then 50 points in the science category equals 100 in the other categories. So, then, science percentage marks equal 60, and (((60+64)+78)/300)=.67333... the same result you got. It will also yield the same result in general.2012-06-30