David R. Henderson  

The Wall Street Journal's Math

Sovereign Debt and Confidence... Capturing the Dynamic...

In today's Wall Street Journal, one of the editorials makes the following statement:

Arizona got into this crisis because during the boom years--2003 to 2007--then-Governor Janet Napolitano, a Democrat, and Republicans in the legislature let spending climb by more than 100% to $10 billion from $6.6 billion.
Of course, the increase was 52%, which is distinctly less than 100%.

But how did the Journal's editors make this mistake? I think it's because they've succumbed to the incorrect modern usage in discussing increases. Here's an example:

GDP in a poor country rises from $10 billion to $40 billion. The modern usage is that it "rose fourfold." But it didn't. It rose threefold. When I've made this point to my students, I've pointed out that when a number rises from 50 to 90, by the modern usage one would say that it rose 1.8 fold. But it didn't. It rose 0.8 fold.

I was wondering when I would first see someone make the error that arises from using the modern usage consistently. I just did.

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CATEGORIES: Fiscal Policy

COMMENTS (24 to date)
dWj writes:

And when it goes from 50 to 40, one would say it rose 0.8 fold.

Actually, that's how I understand "fold" to be correctly applied; it's a straight multiple. ("Rose" is incorrect in my example, though.) A four-fold increase is a 4-times increase, or, additively rather than multiplicatively, an increase of $30 billion, or 300% of the initial value. Perhaps they made the mistake of translating a 1.5 fold increase into a 150% increase, when it's really a 50% increase. I think if they'd said the former, though, they would be correct.

(If I said "increased by a factor of 4", would you agree with that?)

Ted writes:

Alternatively, the editors are just stupid - which seems equally plausible.

David R. Henderson writes:

dWj asks:
If I said "increased by a factor of 4", would you agree with that?

Me: No, I wouldn't. If it rose from 10 to 40, it increased by a factor of 3.

michael writes:

I'm a college student, and for the past few years I've been an SAT tutor to pay the bills. 95% of my students do not know how to calculate percentage change. I spend hours teaching kids this.

They usually start to get it after working with the formula several times, and then I show them this:

for any number "n"

100% increase = n*2 = double

200% increase = n*3 = triple

300% increase = n*4 = quadruple


x*100% increase = n*(x+1)

Maybe I should forward this to the _Journal_.

David R. Henderson writes:

Thank you for the work you do. You are performing a valuable service.

Radford Neal writes:

Sorry, I can't agree with this post.

First, I don't think your explanation for what the Wall Street Journal said works. Would they have said "increased by more than 100%" if it had gone from $6.6 billion to $6.7 billion? I don't think so. They were just totally stupid/confused.

More generally, percentage increase/decrease is a good way to present changes only when the percentage is small. I have to always mentally translate "300% increase" to "increased by a factor of 4", which is more intuitive. And it is indeed correct, as judged by common usage. Certainly using "increased by a factor of 2" to describe a change from 100 to 300 is NOT correct, in that you would not be successfully communicating to the vast majority of readers.

And if you don't like any of the ways suggested for expressing multiplication by 4, how would you say it? Note that this is the best way of expressing and thinking about such large changes, and so ought to be communicated easily. The reason it is the best way is related to why many quantities are best looked at in terms of their logarithms, in which multiplicative factors become additive.

Richard A. writes:

Over the years I have come to the conclusion that those who write the editorials for the WSJ are not very bright. I suspect that few WSJ editorial writers have degrees in economics or relevant fields.

Sean A writes:

I'm just curious what you mean by the "modern use." Is there a formal school of though (or gov figure) that uses statistics this way? Or are you just saying that people are becoming worse at basic calculations?

caveat bettor writes:

The WSJ should remove all numbers from its op eds, thereby avoiding the risk of errors, and more importantly, being caught in them.

It's been working great for the NYT for a long time now.

Norman Pfyster writes:

Another question would be where they got those numbers. According to usgovernmentspending.com, Arizona's spending rose from $13B to $18.5B over that time period. Of course, that's not more than 100% either.

I disagree that the WSJ's mistake is due to an incorrect usage of percentage increase, unless you are suggesting that they thought anything over $6.6B represented over 100% increase, and thus the increase would be 150% (as in 10 represents 151.5% of 6.6), which represents a confusion of what "percentage" means.

Todd Kuipers writes:

Michael's list is clear - and appreciated. I also agree that whoever wrote the piece is plainly bad at math - a real shame for the WSJ.

Some of the issues described are due to the simple (but obviously subtle) difference between:
- x times larger (e.g. 100 to 200 is 2 times larger)
- an x-1 times increase (e.g. 100 to 200 is a 1 times increase) or a 100% increase (e.g. 100 to 200 is a 100% increase)

The difference being the use/exclusion of the word increase. Increase implies the measure of the change between the two numbers, as opposed to indicating the relative size of the two numbers. I've been caught arguing over correct numbers when I didn't read clearly enough.

These are annoying but the winner for these types of gaffs are those that can take 2 positive numbers describing a decrease and get greater than 100%. (e.g. 200 to 50 yields a 150% decrease...)

Matt Flipago writes:

This is the very simple and extremely common example of somebody who likes the idea of math, more then the actual math itself.
It rose 52% , and is 152% of its original size. No where can you get the number 100%. Clearly the editors were asleep at the wheel, or if they are like the SEC, were looking at porn.

Andy Hallman writes:

I'm a small town newspaper reporter, and I made this same mistake about confusing "perfect of x" and "percent x increases." I caught the mistake on my own when I went back to re-read the story and we published a correction on the front page the next day.

David R. Henderson writes:

@Radford Neal,
And if you don't like any of the ways suggested for expressing multiplication by 4, how would you say it?
Me: I didn't say I don't like any. I would say that it quadrupled.

@Andy Hallman,
Good for you. Really. And I particularly appreciated, in your link above, your trying to shed light on the point Steven Landsburg really should have made, which is that citizenship is not morally relevant.

mulp writes:

The WSJ didn't make a mistake, it wrote the lie the WSJ editors intended: a Democrat increased the spending by much more than the Democrat actually did.

Yep, the Democratic governor forced the 60% Republican legislators to vote like liberals....

Actually, they did pretty much what Washington Republican majorities did, cut tax and increase spending, but with incomes rising rapidly in AZ, the budget went from deficit to surplus in Napolitano's first term.

botogol writes:

the problem is not our understanding of math. The concepts of doubling or quadrupling are easy enough,. It's the english language - our words are ambiguous and lack a universally agreed meanings.

pace David here's a dictionary that holds
-by factor of 4
-four times as many
to all mean the same thing.

Here's another

So it's not all all clear to me that David is right when he says that increase from 10 to 40 is a 'threefold' increase. (although I do agree that it is a 300% increase)

Ryan writes:

The words aren't ambiguous. Pay attention to what they are modifying. A threefold increase means that that the amount increased is three times the base. The total would be the base plus the increase, the sum of which is four times the base.

Fourfold does mean the same thing as quadruple. 40 is fourfold 10. However, a fourfold increase of 10 is 50--exactly like a 400% increase of 10 is 50.

David R. Henderson writes:


botogol writes:

OK then let's pay precise, pedantic, attention to detail.

David said:

GDP in a poor country rises from $10 billion to $40 billion. The modern usage is that it "rose fourfold." But it didn't. It rose threefold

"It" means GDP
GDP rose from 10 to 40
GDP (indisputably in my mind) rose fourfold

Ryan - I understand your logic that 10 to 40 is a threefold increase. But, paying attention, that's not what David said.

Kevin writes:

Does OED or Fowler's weigh in on this subject?

Mark Brady writes:

Has the Wall Street Journal issued a correction in today's issue? The Financial Times would have done so.

Radford Neal writes:

I think it's quite clear that many people think going from 10 to 40 is a "fourfold increase". So calling it a "threefold increase" is bound to cause them to misinterpret the remark. So you shouldn't say that.

The only remaining question is whether "n-fold increase" is a useless term that should never be used, because there's no agreement on its meaning, or whether it can reasonably be used for a change from 10 to 40. If lots of people will interpret it incorrectly, then it shouldn't be used.

But personally, David is first person I've encountered who interpreted it differently from me. I'll keep saying a change from 10 to 40 is a "fourfold increase" or "an increase by a factor of 4" until I see evidence that more than a tiny minority interpret it differently than I mean. (Practically anything you say will be interpreted wrongly by somebody or other...)

Snorri Godhi writes:

I agree with Radford Neal, in the sense that for me, a fourfold increase means an increase by a factor of 4, which means that the original value should be multiplied by 4 to get the final value. For the sake of clarity it's best to include the actual figures, as the WSJ does.

Otherwise, the WSJ math does seem bizarre. The only way in which spending can be said to "climb more than 100%" in this case, is if climbing more than 100% means increasing "more than one-fold", i.e. increasing more than 0% by most people's math.

botogol writes:

Usage -

OK I googled [fourfold] and took the first five articles where
- the word four-fold is used in a sentence to describe a rise in a quantity
- the start and end figures are specified

In every case the quantity referred to had quadrupled, ie a 300% rise.

So I would maintain that in ordinary common usage no matter how the sentence is constructed, or what the word qualifies the word fourfold consistently denotes a quadrupling in the base quantity.

Now, David may argue that everyone is using the word wrongly. But if everyone uses a word wrongly then eventually the meaning of the word changes

Examples (sorry the links are not clickable, but the blog software has no shortcut other than manually messing with a -tags.

Android market share goes from 3% to 12.3%

Stillbirths from 4.3 to 16.2

Pirate bay users go from 1m to 4m

Hyundai profits go from 243 to 945

Addicts from 4,800 to 19,300

[links made clickable. Note that the first two links are the same.--Econlib Ed.]

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