September 25, 2007

Exponentially Larger


Yesterday, I saw that Barry of Staring at Empty Pages referenced a story in the Washington Post. The story is entitled "Protest Worth Covering." It's a response to criticism that WaPo's coverage of the recent national anti-war protest was unsatisfactory. It's a good response. However, I immediately found something in it that bugged me. Early on in the piece, Deborah Howell writes the following:
The story did not say that the antiwar protest was exponentially larger than the pro-war demonstration. The headline and photo display exacerbated the problem.
As the title of my post indicates, my problem with this paragraph is the use of the phrase "exponentially larger." I'm trying to understand what she means by using this phrase.

The only safe thing to conclude, by conservatively and charitably interpreting the phrase, is that it means the same as the vague phrase "a lot larger."

I don't like the phrase "exponentially larger" here; its misused mathematical terminology makes it sound like it is saying something precise, when it is the opposite of precise. I'll explain why.

Taken literally, "exponential" means "of or relating to an exponent." So, we might conclude that the author means that group B is equal to group A raised to some power. But what is that exponent? It could be anything. If y = xn, let's plug in some values. Assuming group A comprises a modest 100 members, what happens when we use a few different exponents?


y = xn
Exponent (n) Group A Size (x) Group B Size (y)
1 100 100
10 100 100,000,000,000,000,000,000
.5 100 10
0 100 1


In fact, you can get any size you want for group B by choosing the right exponent. But one thing you do notice is that even using a modest-sized exponent (10), you get an enormous size for group B. Just "having an exponent" has nothing to do with the relationship between the sizes of these two groups.

Exponentiality is a phenomenon of change. It's a way to describe a dynamic relationship. For instance, when x represents time, you can see exponential growth over the passing of time. Unfortunately, this story is about two distinct groups of people at some fixed point in time. One group is not changing into the other group. The aspect of "change" is really missing1.

The formula I used above isn't even an exponential function, really. It's a polynomial function of the nth degree where the other coefficients are zero. But we were playing with "exponential" in the sense of "an exponent." "Exponential function" generally refers to ex where the variable x is the exponent. You can read about the base, e, here. But for our purposes, assume the base doesn't make too much difference.

It's true that with y = ex, the y values can get big really quickly when you increase x by modest amounts.

This time, let's assume that the two group sizes are 500 for group A and 3000 for group B. I think we would all agree that group B is much larger. I've plotted these as y values on a graph (with arbitrary x values -- assume they have some meaning that has to do with a different attribute of the groups) to see if there can be an exponential relationshiop between these two values. The pinkish dot is group A, the blue dot is group B.

It's a graph of a linear function and an exponential function

The red plot shows an exponential function which passes through the two values. So, the two values can lie on an exponential function. But, wait. I've also drawn a green line through the two dots, representing a linear function in the form of f(x) = mx+b.

You can make an exponential function that increases from the smaller value to the larger one. And you can also make a linear function that does the same thing (in a different way).

Would it make any sense to say "group B is linearly larger than group A?" In this situation, it would make just as much sense as saying "exponentially larger" which is to say, not much sense at all.

None of this makes much sense because the relationship between the sizes of these groups has nothing to do with exponentiality. On the information we're given, we can't see exponentiality in action.

We can't really make the assumption, but it is possible that what she meant was that the size of group B was an order of magnitude larger than the size of group A. That would mean that group B was very roughly 10 times larger. She didn't say that, so we can't assume it. She could have said "an order of magnitude." She could have specified two, or even three orders of magnitude. That would be roughly 100 times the size and roughly 1000 times the size, respectively. But she didn't say any of those things.

In the end, we're really left assuming she meant "a lot larger." In Massachusetts, we ahve a phrase specifically for this. "A wicked lot bigger." What she really meant to write was that they failed to report that group B was a wicked lot bigger than group A.

"Ack!"
you say. "That's not English!" True, but at least it doesn't pretend to be precise when it isn't. It doesn't imply a dynamic mathematical relationship that doesn't exist. "A wicked lot bigger" is more honest.

Also, "a wicked lot bigger" doesn't confound people's possible understanding of "exponential" and what exponentiality is. Of course, that's a topic for another blog post, or you can read this .

I don't recommend you say or write "a wicked lot bigger." But if you really only mean "a lot bigger" why not just say that? Or, if you have more specific information, give a more precise description.


1. If there were some reason for us to believe that the size of one group had something to do with the size of the other group, that would be interesting. That could involve exponentiality. But this is neither supported nor implied in any part of the story. Posted by James at September 25, 2007 12:49 PM
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Comments

Totally off-topic waring: I didn't realize Barry's blog title was a Traffic song. What a great band that was.

Sorry for the tangent!

Posted by: Patti M. at September 25, 2007 1:09 PM

This post makes my brain hurt.

Posted by: Keri at September 25, 2007 1:10 PM

That graph does not have any fish.

Using fish to explain exponentiality (or exponentialism) is required, I believe.

Posted by: Derek at September 25, 2007 1:21 PM

Nor does it reference pirates so it cannot be true.

I think the misuse of exponential stems from the use(misuse?) of the term exponential growth. Generally that term is probably used correctly more often than not Then people who don't kow better drop the growth and use exponential for something else.

Posted by: B.O.B. (bob) at September 25, 2007 1:35 PM

The misuse of "exponential" made my brain hurt. A wicked lot.

Posted by: James at September 25, 2007 1:45 PM

I think they probably were looking for "order of magnitude" which is much more precise.

Posted by: briwei at September 25, 2007 1:53 PM

If you are going to mention pirates, you must also mention ham.

Thanks to this post, I figured out how to get the Microsquish calculator to show in scientific rather than standard format. Yay!

Start
Run
Calc
View
Scientific

Posted by: Patti M. at September 25, 2007 2:48 PM

Based on the crowd counts there was an order of magnitude difference between the groups.

However, we can still only guess at what she meant.

Order of magnitude comparisons give you a good sense of scale. I hope "exponentially larger" doesn't catch on and supplant "order(s) of magnitude."

Posted by: James at September 25, 2007 3:11 PM

Yeah, I noticed it too but it didn't bother me. I guess this is why:
I figured that {1, 10, 100, 1000, ...} is a geometric series. Calling that an "exponential series" is fine with me. Then saying that an entry in the series is "exponentially larger" than one before it isn't too much of a stretch.

In the end, I suppose that I didn't think it was "right", but I thought it was better than what we usually get from media people who try these things. "An order of magnitude larger" would have been better. But I knew what she meant, and I just didn't think it was that bad.

[P.S. Thanks for the love. And to Patti, yes... ahhhh, Traffic. Did you read my first post, then, or did you find that it's a song title somewhere else?]

Posted by: Barry Leiba at September 25, 2007 5:28 PM

Doc, Off on something completely different. I could use you and your readers help. Papamoka Straight Talk has been nominated for an award for besst political blog and I need votes.

It's called Bloggers Choice Awards and I have the link in the side bar on the site. I appreciate you and your readers help!

Blog on my friend...

Posted by: Papamoka at September 25, 2007 6:21 PM

I can easily believe your explanation, Barry (BTW, I think it's a sequence or a progression, not a series, unless you are going to sum the terms) where it applies to you. However, I still have a few rather large problems with it. One small problem is that I'm not sure all readers are applying the same knowledge of mathematics you are when they read the articles in the WaPo. I could be wrong, but I'd bet not. More worrying to me:

It creates a new meaning for "exponential" when "exponential" already means something else *and* there already are terms for what she's talking about. Not only "order of magnitude" but also, as you mention, the idea of a geometric progression with a ratio of 10. Even though such a progression does grow exponentially.

Nany aspects of our world are best understood as exponential phenomena. Mislead people about that and it only makes it harder to understand exponentiality. It's already something that people have some trouble with.

Hijacking "exponential" to mean "big" reinforces a specific misunderstanding that confounds half of the uses of exponentiality: exponential decay. A cooling curve is exponential, but nothing is getting bigger. Exponential is much more about proportionality -- rate of change being proportional to the value at any given point -- than it is about "getting bigger" or even "getting bigger or smaller by a lot."

And while, as I said before, I certainly believe you made the leap of understanding, I think it is in spite of the language rather than because of. You can't even make a geometric sequence with two data points. By definition such a sequence requires three values, since it is defined by the ratios of successive terms. You can't compare ratios if you only have one.

I can argue this all day. But the short version is that the phrase implies a nonexistent relationship, and I'm against turning it into American English shorthand because it is both superfluous and ultimately misleading.

I do applaud the piece in general, though. I'm glad they saw fit to correct their earlier reporting, which was more misleading than my mathematical nit. However, you'd already done a fine job covering that aspect.

Posted by: James at September 25, 2007 7:56 PM

I've traded mathematical nits for a handful of grammar disasters. Talk about being unclear! Please excuse.

Posted by: James at September 25, 2007 7:58 PM

Barry, when I saw James's post in which he wrote "Yesterday, I saw that Barry of Staring at Empty Pages referenced a story..." I said to myself, "Hmm. That's a Traffic song. I wonder if that's what he means by calling his blog that."

It's funny. Just the other day, an XM channel was playing "The Low Spark of High-heeled Boys" and I remarked to Bob that I never liked the light, poppy stuff Steve Windwood did after Traffic.

Posted by: Patti M. at September 26, 2007 8:17 AM

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