Arrested for breaking the law of large numbers

So there’s this article in the New York Times today, by James B. Stewart, called “Apple Confronts the Law of Large Numbers,”1 and it’s been linked to a lot in the Apple-centric corner of the internet, including a Linked List mention at Daring Fireball.

Let’s cut to the chase:

The New York Times doesn’t know shit about the Law of Large numbers.…
  — Dr. Drang (@drdrang) Fri Feb 24 2012

I don’t really have a problem with the central premise of the article, which is that Apple can’t grow forever at the pace it’s been at recently. I’m not sure why such a self-evident truth needs to take up space in the New York Times—any company that’s growing faster than the economy as a whole can’t continue to do so forever because eventually it would have to become larger than the economy it’s part of—but then I don’t understand why the Times publishes a lot of things.2

After telling us Apple’s growth is mathematically guaranteed to slow, Stewart then drops this gem:

The law of large numbers may explain why, even at its recent lofty stock price, Apple looks like a bargain by most measures.

Emerson said a foolish consistency is the hobgoblin of little minds, but most people like to put at least a few paragraphs between their self-contradictory statements.

But let’s set aside the question of Apple’s future market value. The truly objectionable part of the article is these two paragraphs:

Here is the rub: Apple is so big, it’s running up against the law of large numbers.

Also known as the golden theorem, with a proof attributed to the 17th-century Swiss mathematician Jacob Bernoulli, the law states that a variable will revert to a mean over a large sample of results. In the case of the largest companies, it suggests that high earnings growth and a rapid rise in share price will slow as those companies grow ever larger.


Let’s start with what the Law of Large Numbers really states. Put simply, it says that the sample mean of a random variable will tend toward the underlying population mean as the number of samples grows larger. For example, Wolfram Alpha says the average height of an adult male is 5′ 9″. If you measured the height of a few randomly selected men, you might get an average for your sample that’s quite far from 5′ 9″. But if you increased the size of the sample, the tendency would be for your sample average to move closer to 5′ 9″.

The law does not state that “a variable will revert to a mean over a large sample of results.” The Law of Large Numbers says nothing about individual measurements; it’s all about averages. And it certainly doesn’t “suggest” anything about the future growth of large companies.

If the Law of Large Numbers worked the way Stewart says, you could repeatedly measure the height of Dirk Nowitzki and he’d eventually shrink down to 5′ 9″. I’m surprised the Mavericks’ opponents haven’t thought of this.

What really bothers me about the Times article is Stewart’s pretense of scholarship with the bit about Jacob Bernoulli. He obviously knows nothing about the subject and is basically just rewriting Wikipedia in an attempt to appear erudite. Unsuspecting readers may think Stewart knows what he’s talking about.

By the way, Jacob is not the Bernoulli of Bernoulli’s Principle. That was Daniel, who’s made two appearances in this blog, once in a post about the water level in my toilet and more recently in a post about a huge hydraulic forging press. Jacob, whose name is often Anglicized to James, is the Bernoulli of Euler-Bernoulli beam theory,3 a topic near and dear to my heart but unrelated to the Law of Large Numbers. More to the point, he’s also the Bernoulli of Bernoulli trials, which is pretty closely related to the Law of Large Numbers, and was discussed in this post about confidence intervals and the choice of sample size.

(And that, Mr. Stewart, is how you do erudite. Leave the Bernoullis alone. Stick with what you do best: transcribing your credulous interviews with analysts. You know, like the one near the end of the article with the analyst who thinks Microsoft is the company that best understands cloud computing.)

  1. That’s what it’s title was this afternoon when I first read it. It’s since been renamed to “Confronting a Law of Limits.” 

  2. Maureen Dowd’s column, in particular. 

  3. Beam theory is discussed on pages 5 and 6 of the linked PDF, which is, essentially, the Encyclopedia Britannica entry on solid mechanics. The author, James R. Rice, is one of the giants in the field of solid mechanics, having invented the J-integral method for elastic-plastic fracture mechanics analysis.