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 Question from Anouska, a student: In category A, Annie gave 5 out of 6 responses. I make this 83% I the same category, Adam gave 4 out of 10 responses. I make this 40% Does this mean that Annie gave 43% more responses in this category than Adam?

That would not be a good way to say it, as I think you suspect (or you wouldn't have asked, right?)

Someone might say that. Adam might say Annie gave 25% more responses (5 vs 4). Annie might say she gave 107.5% more responses (43% is 107.5% of 40%). And without the data the listener wouldn't have a clue what was meant.

Percentages are used in so many different ways (and we aren't even getting into the percent-more vs percent-less minefield - in raw numbers Annie gave 25% more responses but Adam gave 20% fewer) that it is best to give raw numbers and/or explain "percentage of what". Percentages of percentages are generally a bad idea. When percentage comparisons are a problem, avoid them, especially percent differences.

"Annie responded to five (83%) of the 6 questions she had to answer, Adam to four (40%) of the 10 in his survey. Annie's raw number of responses was larger by 25% but her proportion of responses was more than twice as great as Adam's."

Finally, as they were apparently answering different questionnaires, the onus is on the reporter to justify any comparison at all.

Good Hunting!
RD

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.