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. 

16 Responses to “Arrested for breaking the law of large numbers”

  1. Eddie says:

    Honestly he could have written the entire article in one (logically flawed) sentence:

    “Apple can’t continue on its current pace for much longer because that’s never happened before.”

    There are many psychological pitfalls at play, including conservatism bias and reference point bias. The human mind is hard wired to make poor investing decisions, and the NYT article is a great example of it. 

    I’m not saying anything about Apple’s prospects for the next decade. I’m just saying the reasoning in that piece was flawed. It ignores all fundamentals specific to Apple as a firm and implies that a stock is simply a statistical commodity—just another trial in a game of dice. 

    Many an amateur technician has been burned in that game. 

  2. Jimm says:

    Did you first see the link on Daring Fireball?

    When I saw it, I thought, I don’t see how the Law of Large numbers would be relevant so I skipped it.

    I don’t really respect the New York Times. The quality of their articles is not that good and I disagree with their editorial policies.

  3. Dr. Drang says:

    I read it via a Twitter link before seeing the DF post. Was a little surprised Gruber posted an excerpt with no comment.

  4. Eric says:

    The NY Times seems intent on publishing an article designed to use Apple to get reader “engagement”. You can understand this from hacks at Business Insider but the grey lady has really become a bit of a slut recently. Today they’re running a new article “Apple riding high but for how long” in which they interview a few people who switched from iPhones. When I was a kid the New York Times was journalism. Now the NY Times is just another linkbait hag. Expect more of the same sadly.

  5. Harold K says:

    The law of large numbers as relating to the growth of a company does not apply to Apple at this time because Apple is creating new markets for itself. For example, if Apple had brought out the iPhone but not the iPad eventually everyone who wanted an iPhone would buy one and the the sales would stabilize at a certain number. Therefore the growth of the company would slow. However, this is not what is happening to Apple because several years after the iPhone it introduced the iPad. This is another new market which will go through its growth phase. In fact, most of Apple’s sales and profit are with products that did not exist prior to 2007. As long as a company can create new markets it can remain in a rapid growth phase almost indefinitely!

  6. Tim L. says:

    I’m glad to see someone else who was bothered by the incorrect statements about Bernoulli in that article.

    I’d like to thank you for the pointer to that Solid Mechanics PDF. As someone who spends his days with Euler-Bernoulli and Timoshenko beams (and is a teaching assistant for Solid Mechanics courses) it’s nice to have something to point people to.

  7. George K. says:

    “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″.”

    Reminds me of an annoying manager I had many moons ago when I made pizza. After seeing me, in the midst of a rush, peeking in at pizzas several times over the course of a few minutes, she said, “You know, every time you open that door, the temperature drops 50 degrees.” I replied, “Does that mean I can open and close it 13 times and turn it into a freezer?”

  8. Tem Kuechle says:

    U.S.A. = The United States of Apple? I hope Apple doesn’t become larger than the economy. As much as I like many of their products, and own many, I would like to see the competition make some inroads too.

  9. Jeff B. says:

    Thanks for this. The article title and substance — if one can call it that — were down right irritating to me, but I just couldn’t summon the will to straighten out the twisted thinking behind it. You did a creditable job.

    I think the NYT should have just called the thing “Malthusian Limits to Apple’s Growth”. Then they could have skipped writing the whole article.

  10. James Katt says:

    So what the Law of Large Numbers actually implies is that: 1. Apple can keep growing and growing and growing. 2. Other companies will shrink or grow or stay the same as Apple grows. 3. In order to maintain the average corporate profit, unless the economy grows, then the other companies (such as Dell) will shrink. If the economy grows, then they can also grow along with Apple.

  11. airmanchairman says:

    This is why Apple avoids going for market share like the plague.

    It actually works in their favour that their market share in various markets where they dominate the higher end and the profit share is measured in single-digit percentages. It translates to vast (though not unlimited) “headroom” to grow into.

    Their tried-and-tested modus operandi gives them the perfect springboard to execute in this scenario with a predictable result - massive sales, profit and growth, provided their logistics and distribution can keep up the pace of demand. Which is why their high-performing COO has become the current CEO

  12. ajay says:

    thanks. that total poo-pooing of the law of large numbers was really hacking me off. cheers to an eloquent and fun explanation.

  13. seven2521 says:

    Why does the NYT publishes stuff like this? Hate for those who show that it’s possible to succeed without government favors. Hate of the good for being the good.

    Apple’s of the world bad, Solyndra’s of the world good.

  14. Dr. Drang says:

    You gotta love people who use the internet to complain about “government favors.”

  15. Costanza says:

    Hear hear seven2521!

    Also, let’s get some more coverage of all the oil companies who are doing very well with no government assistance while we’re at it.


  16. Chris says:

    James Stewart is correct that the “Law of Large Numbers” will cause Apple’s growth to slow. But I think he’s wrong that it’ll happen soon.

    And, even if Apple were to slow to 10% growth in sales & earnings, it’s still too-cheap.

    Assume Apple earns $50/share this year (about 50% more than in 2011). At a market P/E of 13, Apple should sell for $650/share (about 20% more than today’s price).

    So…if you assume Apple’s prospects are similar to the average of all companies in the S&P 500, then Apple is 20% undervalued.

    Here’s where APPL should be treated differently because of its size: Fair valuation is often considered to be a PEG ratio (P/E divided by growth rate in earnings) of 1.0

    APPL is growing earnings over 50%/yr. That implies it’s worth a P/E of 50, which, multiplied by 2012 earnings of $50 would be a $2500 stock price.

    But, at APPL’s size, it just can’t grow earnings at 50% compounded each year indefinitely. So we can’t use PEG to value APPL.

    Using conservative valuation metrics (like the market P/E), APPL is fairly valued or even undervalued…Chris.