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If I have two values A and B which are both expressed as a percentage of C, and I want to express the difference in magnitude between A and B as a percentage D, is it more correct to express D as a percentage of C, or as a percentage of B (or indeed A)?

50 unemployed people is obviously 50% bigger than 25 unemployed people, because it's clear that '%' here means '% of 25 unemployed people'. But how much bigger is 50% unemployment than 25% unemployment? It's an increase of 100% of 25% unemployment, but only an increase of 25% of total potential unemployment.

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    $\begingroup$ "50 unemployed people is obviously 50% bigger than 25 unemployed people, because it's clear that '%' here means '% of 25 unemployed people'". ???? 50 unemployed people is 25 bigger than 25 unemployed people which is 100 '% of 25 unemployed people' bigger than 25 unemployed people. $\endgroup$ – user20637 Feb 16 '17 at 16:35
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    $\begingroup$ Relevant XKCD $\endgroup$ – Gregor Feb 16 '17 at 18:46
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    $\begingroup$ The math here is all wrong, but that's irrelevant. Obviously, choose whichever figure makes the statistic seem most impressive and use that. In this case, the figure you want is 200%; 50% is 200% of 25%, obviously. Then you say "unemployment is up by 200%" and hope no one realizes that "up by 200%" is actually tripling, not doubling, and that it is impossible for 200% of a population to be unemployed. Remember, percentages exist in order to confuse readers and obscure a factual narrative. $\endgroup$ – Eric Lippert Feb 17 '17 at 6:05
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    $\begingroup$ Therefore, if you want to actually inform readers, avoid percentages whenever possible, and when not possible, back them up with solid numbers. "the unemployment rate fell from 7.0% to 5.8%" is clear, but it does not tell the whole story. Is that driven by an increase in retirement? Or by new jobs in construction? Those are very different stories. See blogs.msdn.microsoft.com/ericlippert/2005/09/01/… for some more thoughts on this. $\endgroup$ – Eric Lippert Feb 17 '17 at 6:12
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    $\begingroup$ This is why you shouldn't use percentages. Unless you also provide a lot of context, in which case why use percentages anyway? $\endgroup$ – matt_black Feb 17 '17 at 8:08
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There are percent (%) and there are percentage points (%p), which are two different things.

50% (of $X$) is 100% more than 25% (of $X$). At the same time,
50% (of $X$) is 25%p more than 25% (of $X$).

So if your bank promises to increase the interest rates on your deposit by 5%, that means nearly nothing; 5% of, say, 1% original rate is just 0.05%, resulting in 1.05% after the increase.
But if it promises to increase the interest rate by 5%p (or 500 basis points, as Chris Haug notes), then it is an attractive deal; 5%p on top of, say, 1% original rate gives 6%.

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    $\begingroup$ This is true, but I have personally never seen the symbol "%p" used (in North America). Most likely, on a personal savings account, the bank would say the first thing but mean the second (and explain in detail in the fine print). In a more technical setting, the bank would say "500 basis points", which are always interpreted additively. The answer's not wrong but it doesn't address the fact that there is really no universal standard in the language, and you always have to ask exactly what people mean. $\endgroup$ – Chris Haug Feb 16 '17 at 18:19
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    $\begingroup$ @ChrisHaug, interesting to learn this. When I was issuing reports for clients in an investment bank, we would always carefully distinguish between % and %p (yes, we used precisely that symbol), but that was in Europe and I do not know whether it is the same in U.S. (probably not, as you note). $\endgroup$ – Richard Hardy Feb 16 '17 at 18:22
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    $\begingroup$ @RichardHardy I have not seen it written that way, but I have seen the distinction between "percent" and "points" maintained — just without the shorthand "%p" for "points". $\endgroup$ – hobbs Feb 16 '17 at 19:02
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    $\begingroup$ Indeed, in my (US) experience, I've seen "percentage points" written out to convey what you're abbreviating to "%p", so it's not an unreasonable abbreviation, but the first time I used it in any given work, I'd explain it "percentage points (%p)" for those unfamiliar with it. $\endgroup$ – Monty Harder Feb 16 '17 at 20:32
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    $\begingroup$ I've seen it written as pp $\endgroup$ – James Feb 16 '17 at 21:27
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Both are correct, as long as the increase is described correctly. A common way of distinguishing the two cases is to say there is a 100% relative increase or a 25% absolute increase. However, this might not be clear to all audiences. Most laypeople probably expect the latter number, and quoting the multiplicative increase may be considered intentionally misleading.

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The expression "B is x % higher than A", implies that x is calculated as a percentage of A, because it is against A that B is being compared, not some unspecified third entity.

If A=25% of C and B=50% of C, then B is 100% higher than A.

It's also 2 times A. Confusingly, many people will say "B is 2 times more than A", which is completely illogical. For B to be 2 times more than A, it would be 2 * A + A, or 3 * A (in this case, 75% of C).

However, "B is 25 percentage points higher than A, when both are compared to C". If the context of the percentage being calculated relative to C is omitted (and not strongly implied), the statement is meaningless, because a percentage is always a percentage of something.

[If you doubt this, consider whether you'd rather have 50% of the money spent in a year on pork sausage in Jerusalem or 1% of the money spent in the same year on rice in China.]

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    $\begingroup$ " Confusingly, many people will say "B is 2 times more than A", which is completely illogical." Many people also say inflammable to mean flammable (bad example, but I don't know better ones in English). Natural languages are sometimes strange shouldn't be interpreted as certain mathematical syntax. $\endgroup$ – JiK Feb 16 '17 at 19:46
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    $\begingroup$ @JiK When expressing mathematical relationships, one ought to at least try to avoid confusing terminology. The "__ times {more|less|etc.} than" construct clearly admits of two different interpretations, and thus should be avoided. Arguably, the same can be said about percentages of items already expressed in percentages. $\endgroup$ – Monty Harder Feb 16 '17 at 20:29
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    $\begingroup$ @JiK inflammable does mean flammable. The one that really gets me though is A is 3 times less than B $\endgroup$ – A C Feb 16 '17 at 22:52
  • $\begingroup$ @AC and "A is 2 times more than B" does mean $A=2B$, if that is how people use English. $\endgroup$ – JiK Feb 17 '17 at 6:07
  • $\begingroup$ @JiK it might be interpreted as A = 2B but I do disagree with your claim that it means so. If I was to say "A is 2 times B" that would mean A = 2B literally, but none the less "A is two times more than B" literally means that A = B + 2B. It's not mathematics but the content of the saying, which people usually won't even realize. $\endgroup$ – Han Soalone Feb 17 '17 at 9:41
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The only valid approach here is to assume your reader does not know which version you are using, and make sure you build up enough context to make it clear.

One context may be to state what Richard Hardy suggested, differentiating between percentages and percentage points. That being said, I've never seen the %p notation before, so if you use that notation you might want to clarify it as well. Wikipedia suggests pp or p.p. as other possible notations.

Another context might be nearby numbers. If I say "Inflation went up .2% this year," people reliably understand that that number must be an addition of 0.2% to the inflation rate percentage, as opposed to a claim that inflation has been scaled by 100.2%. This holds true even if I was imprecise in my use of percent vs. percentage points. On the other hand, if I say "murder rates were at 3% last year, but went up 30% this year," you can be quite confident that that was a scaling factor, unless you knew there was some major insurrection in the area.

One of the easiest ways to maintain this context is to say the value in several different ways. If you say "unemployment went up 25%, to a record high of 18%," it's pretty darn clear that you intended to say unemployement rates were multiplied by 1.25. Incidentally, this is where the duplicated legal terms like "null and void" or "aiding and abetting" came from -- they were saying the same thing twice, once in common law speak and one in the official terminology which derived from French law.

We have a similar issue with the nuanced difference between an absolute temperature measured in Fahrenheit and a differential temperature measured in Fahrenheit. The conversions are different because one has to account for the fact that the Fahrenheit scale does not start at absolute zero. Dozens of notations have been suggested to resolve this, but nothing is quite as effective as maintaining a clear context so that the reader understands what you meant.

Communication is all about context, and different phrasing will mean different things to different people.

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