David R. Henderson  

Can a Positive Number Fall by over 100 Percent and Still be Positive?

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Answer: No.

But it has become increasingly common for people, even otherwise numerate analysts, to write as if it can.

Consider a recent instance. In the Spring 2014 issue of Regulation, Sam Batkins and Mitch Boynton discuss a case in which an estimate of a regulatory cost fell from $672 million to $89 million. That's a drop of 87 percent.

But that's not what they wrote. They call this "a 750 percent drop in costs." How did they get that? It looks as if they take the drop as a percent, not of the number from which it dropped, but of the number to which it dropped. The drop was $583 million. That's 655 percent of the number it dropped to. Then, for some reason, they add another approximately 100 percentage points.


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COMMENTS (19 to date)
Matt writes:

What they mean is "the new budget number would need to be increased ~7.5x to reach the old budget number" (ie, ~89 million increased by 750% is ~667).

However, that is indeed a bizarre way to make the point.

David R. Henderson writes:

@Matt,
Right. I know that’s what they meant. But notice that they got even that wrong--by about 100 percentage points.

Greg Heslop writes:

Annoying, that. Maybe they had in mind $672 million's being approximately 750 per cent of $89 million? 89*7.5 is not much less than 672.

At least they did not call it a drop of 750 percentage points.

Martin Ringo writes:

"Drops," "increases" and the like have an arrow of time implied. And if we are semantically honest -- probably an all to rare occurrence in policy debates -- the arrow of time implies a base for percentage calculations. Maybe we -- the semantically honest -- could flaunt our virtue by always using constructions like "and 87 percent drop from $672 million."

David R. Henderson writes:

@Martin Ringo,
Maybe we -- the semantically honest -- could flaunt our virtue by always using constructions like "and 87 percent drop from $672 million."
I think you meant “an,” not “and.”
I don’t have reason to think they were dishonest. I doubt they were. They were just unclear and incorrect. For me, it’s not primarily about virtue: it’s about correctness and clarity.

Big Dubya writes:

In a similar vein, a radio ad for a residential alarm company asserts: "Homes with an alarm system are three times less likely to be robbed."

I take them to be representing that my odds of being robbed will drop to less than zero. Very impressive protection!

David Cushman writes:

You're all wrong! It's a 202% decline!

ln(89) - ln(672) = -2.02.

Furthermore, I like my answer because it's sorta Solomonic, being (very, very) roughly half way between 750% and 87%!

Or we could use use the midpoint formula like they do with elasticities in Principles books, but that only gets us a 153% decline, so I'm stickin' with my log approach.

John Jenkins writes:

David:

I believe you are wrong about how they calculated their erroneous number. It's much simpler than you are giving them credit for:

89/672=0.1324

1-0.1324=0.8676 or ~87%

John Jenkins writes:

Sorry, I completely misread this. please disregard my prior post.

Rob Rawlings writes:

I think they just mean that 672 is (roughly) 750% of 89.

It incorrect to say this is a "drop" of 750%" but its not (as far as I can see) out by 100.

NZ writes:

Just for clarity, so it's all in one spot:

655% of 89 is 582.95

750% of 89 is 667.5

755% of 89 is 671.95

Sam Batkins writes:

As the author, I'll note that we never claimed the drop between those two numbers was 750%. The two numbers David references were from the previous paragraph. Anyway, our language there was not precise.

David R. Henderson writes:

@Sam Batkins,
Thanks for replying. The bigger question is: how can any number fall by 750% and still be positive?

emerich writes:

I thought it was a trick question and answered, well, a number could fall by 1% 87 times and still be positive (or by 8.7% 10 times...)...

emerich writes:

Sorry, should have said a number could fall by 1% 750 times, etc.

Eli writes:

Wouldn't that make any fall by any amount a fall of more than 100%?

My weight dropped 108% in the last 8 weeks. Frames can make you feel so good.

Jay writes:

@Sam Batkins

Thanks for responding, but then what two numbers were you referring to when stating the 750% number if it wasn't the values David posted?

Jay writes:

David,

I believe you have the 2 values Sam was referring to in the article wrong. Within the same paragraph he mentioned the cost being $70 million versus $529 million.

529 = 755% of 70

These are the numbers I think the author was referring to.

This won't change your conclusion any, it is still bad language.

Sam Batkins writes:

The article was about changes in costs and benefits during the life of rulemakings. The word "drop" was inaccurate and I thank everyone for their feedback. This is an ongoing project and we will be more precise in our next iteration.

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