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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 e^x 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 = e^x, 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.


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.

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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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