The phrase law of large numbers is employed because, it concerns the way
results reflect underlying probabilities when we make a large number of
observations.
In real-life situations we often make the opposite error: we assume that a sample or a series of trials is representative of the underlying situation when it is actually far too small to be reliable.
The misconception—or the mistaken intuition—that a small sample accurately
reflects underlying probabilities is so widespread that Kahneman and Tversky
gave it a name: the law of small numbers.
The law of small numbers is not really a law. It is a sarcastic name describing
the misguided attempt to apply the law of large numbers when the numbers aren’t
large.
When people observe the handful of more successful or less successful years
achieved by the Sherry Lansings and Mark Cantons of the world, they assume that
their past performance accurately predicts their future performance.
Consider a situation in which two companies compete head-to-head or two
employees within a company compete. Think now of the CEOs of the Fortune 500
companies. Let’s assume that, based on their knowledge and abilities, each CEO
has a certain probability of success each year (however his or her company may
define that). And to make things simple, let’s assume that for these CEOs
successful years occur with the same frequency as the white pebbles or the
mayor’s supporters: 60 percent. (Whether the true number is a little higher or
a little lower doesn’t affect the thrust of this argument.) Does that mean we
should expect, in a given five-year period, that a CEO will have precisely
three good years?
No. Even if the CEOs all have a nice cut-and-dried 60 percent success rate,
the chances that in a given five-year period a particular CEO’s performance
will reflect that underlying rate are only 1 in 3! Translated to the Fortune
500, that means that over the past five years about 333 of the CEOs would have
exhibited performance that did not reflect their true ability. Moreover, we
should expect, by chance alone, about 1 in 10 of the CEOs to have five winning
or losing years in a row.
What does this tell us? It is more reliable to judge people by analyzing
their abilities than by glancing at the Scoreboard. Or as Bernoulli put it,
“One should not appraise human action on the basis of its results.
Going against the law of small numbers requires character. For while anyone
can sit back and point to the bottom line as justification, assessing instead a
person’s actual knowledge and actual ability takes confidence, thought, good
judgment, and, well, guts.
Executives’ winning years are attributed to their brilliance, explained
retroactively through incisive hindsight. And when people don’t succeed, we
often assume the failure accurately reflects their talents and abilities.
Full Disclaimer: I am not a financial planner. The views expressed in this post are all mine and they may or may not suit your needs. Please do you own due diligence. I do not make money on any of the products suggested in this post.
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