Here is the article in NY times called "Apple confronts the law of large numbers". It tries to explain Apple share price rise using law of large numbers. What statistical (or mathematical) errors does this article make?

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    $\begingroup$ I found this article via blog of @Epigrad: confounding.net/2012/03/12/…. $\endgroup$
    – mpiktas
    Commented Mar 13, 2012 at 7:17
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    $\begingroup$ (+1) Thanks for bringing attention to this article here. $\endgroup$
    – cardinal
    Commented Mar 13, 2012 at 12:47
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    $\begingroup$ My second most upvoted answer comes from question about article in NYTimes. Also I wanted to know how other people would answer this question. I have an answer with a bit different perspective than Epigrad, and wondered whether anyone else would post it. $\endgroup$
    – mpiktas
    Commented Mar 13, 2012 at 13:16

3 Answers 3


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.

This muddled jumble actually refers to three different phenomena!

  1. The (various) Laws of Large Numbers are fundamental in probability theory for characterizing situations where it is reasonable to expect large samples to give increasingly better information about a process or population being sampled. Indeed, Jacob Bernoulli was the first to recognize the need to state and prove such a theorem, which appeared in his posthumous Ars Conjectandi in 1713 (edited by nephew Nicholas Bernoulli).

    There is no apparent valid application of such a law to Apple's growth.

  2. Regression toward the mean was first recognized by Francis Galton in the 1880's. It has often been underappreciated among business analysts, however. For example, at the beginning of 1933 (during the depths of a Great Depression), Horace Secrist published his magnum opus, the Triumph of Mediocrity in Business. In it, he copiously examined business time series and found, in every case, evidence of regression toward the mean. But, failing to recognize this as an ineluctable mathematical phenomenon, he maintained that he had uncovered a basic truth of business development! This fallacy of mistaking a purely mathematical pattern for the result of some underlying force or tendency (now often called the "regression fallacy") is reminiscent of the quoted passage.

    (It is noteworthy that Secrist was a prominent statistician, author of one of the most popular statistics textbooks published at the time. On JSTOR, you can find a lacerating review of Triumph... by Harold Hotelling published in JASA in late 1933. In a subsequent exchange of letters with Secrist, Hotelling wrote

    My review ... was chiefly devoted to warning readers not to conclude that business firms have a tendency to become mediocre ... To "prove" such a mathematical result by a costly and prolonged numerical study ... is analogous to proving the multiplication table by arranging elephants in rows and columns, and then doing the same for numerous other kinds of animals. The performance, though perhaps entertaining, and having a certain pedagogical value, is not an important contribution either to zoology or to mathematics.

    [JASA Vol. 29, No. 186 (June 1934), pp 198 and 199].)

    The NY Times passage seems to make the same mistake with Apple's business data.

  3. If we read on in the article, however, we soon uncover the author's intended meaning:

    If Apple’s share price grew even 20 percent a year for the next decade, which is far below its current blistering pace, its \$500 billion market capitalization would be more than \$3 trillion by 2022.

    This, of course, is a statement about extrapolation of exponential growth. As such it contains echoes of Malthusian population predictions. The hazards of extrapolation are not confined to exponential growth, however. Mark Twain (Samuel Clements) pilloried wanton extrapolators in Life on the Mississippi (1883, chapter 17):

    Now, if I wanted to be one of those ponderous scientific people, and 'let on' to prove ... what will occur in the far future by what has occurred in late years, what an opportunity is here! ... Please observe:--

    In the space of one hundred and seventy-six years the Lower Mississippi has shortened itself two hundred and forty-two miles. That is an average of a trifle over one mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the “Old Oolitic Silurian Period,” just a million years ago next November, the Lower Mississippi River was upwards of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and threequarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.

    (Emphasis added.) Twain's satire compares favorably to the article's quotation of business analyst Robert Cihra:

    If you extrapolate far enough out into the future, to sustain that growth Apple would have to sell an iPhone to every man, woman, child, animal and rock on the planet.

    (Unfortunately, it appears Cihra does not heed his own advice: he rates this stock a "buy." He might be right, not on the merits, but by virtue of the greater fool theory.)

If we take the article to mean "beware of extrapolating previous growth into the future," we will get much out of it. Investors who think this company is a good buy because its PE ratio is low (which includes several of the notable money managers quoted in the article) are no better than the "ponderous scientific people" Twain skewered over a century ago.

A better acquaintance with Bernoulli, Hotelling, and Twain would have improved the accuracy and readability of this article, but in the end it seems to have gotten the message right.

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    $\begingroup$ That was my core takeaway. The author of the article isn't wrong. His "because Math" justification on the other hand, is way off base. $\endgroup$
    – Fomite
    Commented Mar 14, 2012 at 20:49
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    $\begingroup$ what an nice and well balanced answer! i wanna give this 100 marks $\endgroup$ Commented Nov 1, 2013 at 12:59
  • $\begingroup$ I disagree on P/E ratio bit. It's a very well known relative value investing trick and it works across many different assets. The idea being that low P/E suggests relative value over the peers. One reason why conventional statistics is tricky to apply to price series is the fact that the meaning of expectation is different from its mathematical equivalent. Consider this: my expectation of the temperature tomorrow cannot impact the temperature, but our expectation of the price tomorrow pretty much determines that price. It may sound absurd only when you treat the price like a temperature $\endgroup$
    – Aksakal
    Commented Oct 15, 2021 at 23:24
  • $\begingroup$ That claim: "500 billion market capitalization would be more than 3 trillion by 2022." is looking plausible as reported by companiesmarketcap.com. $\endgroup$
    – Galen
    Commented Dec 18, 2021 at 17:33
  • $\begingroup$ @2.7182818284590452353602874713 Yes, indeed--I have been tracking this thread and checked it when Apple's market cap crossed one trillion dollars. But another way to look at this situation would be to wonder just how over-priced Apple is going to get before investors figure it out :-). $\endgroup$
    – whuber
    Commented Dec 18, 2021 at 17:45

Humorously enough, I just wrote a blog post on this very subject: http://confounding.net/2012/03/12/thats-not-how-the-law-of-large-numbers-works/

Essentially, the Law of Large Numbers is that as the number of trials of a random process increases, the mean of those trials will approach the actual mean (or expectation, for more complex distributions). So while if you flip a coin once and get heads your probability of heads = 1.0, as you flip more and more coins, you'll head closer and closer to 0.50.

The author argues that Apple will have trouble in the future due to something that is not actually at all related to the Law of Large Numbers. Namely, that as Apple grows larger, the same % increase in share price, earnings, etc. get harder to reach in absolute dollar terms. Basically, to stay on course, Apple has to get larger and larger hits.

Linking that to the behavior of a random process converging to a mean requires some serious mental gymnastics. As far as I can tell, the assertion is that "The awesomeness of your products" is a random process, and while Apple has had a streak of "Above Average" awesome, they'll eventually have to converge toward a mean of "Middling". But that's being really charitable to the author.

Just because 500 billion is a large number doesn't mean the "Law of Large Numbers" is what's acting on it.

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    $\begingroup$ (+1) At first when I started reading the article, I thought the author was maybe conflating the law of large numbers with regression to the mean. Then, I got to the paragraph that starts "Also known as the golden theorem...". This reads like someone who skimmed L. Mlodinow's The Drunkard's Walk: How Randomness Rules our Lives (an otherwise interesting read) and then thought they knew something. $\endgroup$
    – cardinal
    Commented Mar 13, 2012 at 12:56
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    $\begingroup$ "The awesomeness of your products" as a random process, I can feel a new branch of statistics being created right now. $\endgroup$
    – asjohnson
    Commented Mar 13, 2012 at 14:18
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    $\begingroup$ Andrew Gelman's blog also has a discussion. andrewgelman.com/2012/02/… $\endgroup$
    – zbicyclist
    Commented Mar 14, 2012 at 1:21

There is no reason to think that stock price draws over time for a particular company represent independent, identically distributed random variables.

  • $\begingroup$ Well yes, but iid assumption can be relaxed considerably for lln to hold. $\endgroup$
    – mpiktas
    Commented Mar 13, 2012 at 12:05
  • $\begingroup$ But you still need independence, which makes no sense when talking about the DGP of a stock price, unless you view finance as a special case of roulette. But in that case, surely regression to the mean would be the more useful concept, not LLN. It's also not clear to me what the random process the LLN applies to. Is it the price itself, the change in the price, or Apple's market capitalization? Finally, I am not sure if the expected value to which the sample means supposedly converge over time is actually meaningful in any of the three cases above. $\endgroup$
    – dimitriy
    Commented Mar 13, 2012 at 15:04
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    $\begingroup$ Dimitriy, your remarks are well taken. Do note, though, that the article (as nonsensical as it is) refers to Bernoulli's work, which is the WLLN. So, for example, we can get away with uncorrelated rather than independent random variables, and, indeed even mild correlation as long as it doesn't grow too fast as a function of the number of variables. $\endgroup$
    – cardinal
    Commented Mar 13, 2012 at 15:25
  • $\begingroup$ @cardinal: I glanced at the definition at mathworld.wolfram.com/WeakLawofLargeNumbers.html (aka Bernoulli's Theorem) before posting that, which has $iid$ as an assumption. This agrees with Casella & Berger's definition of WLLN. But you are certainly correct. You can relax that to finite moments for $x_{i}$ and not too much dependence so that the random components cancel out. Independence is too strong. $\endgroup$
    – dimitriy
    Commented Mar 13, 2012 at 17:34
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    $\begingroup$ Yes, if someone wants to be somewhat ungrateful to Bernoulli, they can note that the WLLN is essentially a straightforward application of Chebyshev's inequality as long as all $X_i \in L_2$. Then, one sees that as long as $\mathrm{Var}(S_n) = o(n^2)$, everything works out. This doesn't even require the means or variances of the $X_i$ to be constant if we interpret the relevant statement of interest as $\bar X_n - \bar{\mu}_n \to 0$ in probability. Of course, more general forms of even the WLLN exist. (+1, by the way.) $\endgroup$
    – cardinal
    Commented Mar 13, 2012 at 18:02

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