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Say I have these values :

Value 1 Value 2 Absolute difference (Value 1 - Value 2) Relative difference (Value 1 / Value 2 - 1)
0.80 0.30 0.50 167%
0.75 0.20 0.55 275%

I'm interested in calculating the overall relative difference. I see two options :

Option 1: Average of relative differences

$$(167\% + 275\%)/2 = 221\%$$

Option 2: Relative difference of averages

$$Average\ Value\ 1 = (0.80+0.75)/2 = 0.78$$ $$Average\ Value\ 2 = (0.30+0.20)/2 = 0.25$$ $$Average\ relative\ difference = (0.78/0.25-1) = 210\%$$

I've done the algebra and know that the two options will give different numbers, but I can't see why they are different. Is one better that the other? Is there some weighing happening that I don't see? I wish to have equal weighing across the observations.

Thank you.

pollytatouin
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1 Answers1

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Upon further research, I found the following resources which essentially conclude that no method is better than the other, they simply compute different things.

Of interest, the inequality comes from Jensen's inequality, which states that the expectation of a non-linear transformation is not the same as a non-linear transformation of the expectation.

Links: https://stats.stackexchange.com/questions/555182/taking-percentage-difference-b-w-average-of-values-vs-taking-average-of-individu

https://web.archive.org/web/20230530200251/https://hbiostat.org/doc/BBR.pdf (see section 14.5)

https://stats.stackexchange.com/questions/21587/how-to-find-mean-relative-differences

In my case, I'm interested in the average difference between pairs, not the average difference between groups. So Option 1 is preferred.

pollytatouin
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