Combining Percentages

The data on 4 examinations showed scores of 75 percent correct, 87 percent correct, 69 percent correct, and 93 percent correct. The overall grade was determined by taking the Arithmetic Mean (or 'average') of these scores, reporting a combined percentage of 81.

Consider that the 4 examinations had

24 correct out 32 questions for 75 percent

87 correct out of 100 questions for 87 percent

38 correct out of 55 questions for 69 percent

and 93 correct out of 100 questions for 93 percent.Thus if the questions are of equal merit across the examinations, there are

242questions answered correctly out of287for a percentage of84.3.

Discuss the discrepancy between 81 percent and 84.3 percent for these two computations.

What is the mathematical explanation?

Another example:

30 correct out 32 questions for 94 percent

75 correct out of 100 questions for 75 percent

50 correct out of 55 questions for 91 percent

and 69 correct out of 100 questions for 69 percent.Here, the AM = 82.3 percent whereas 224 correct out of 287 is 78 percent.