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The law of small numbers

People often make big decisions without much information to go on.

By Karen Datko Jan 17, 2011 4:41PM

This guest post comes from Pop at Pop Economics.


Quick game: I have just invented a device that will show me a face or a tree at random every time I use it.

I'm going to use it 16 times, and you're going to guess the probability that the device will show a face the next time I use it.


OK, here are my results:

  1. Tree
  2. Face
  3. Face
  4. Face
  5. Face
  6. Face
  7. Face
  8. Face
  9. Face
  10. Tree
  11. Face
  12. Face
  13. Face
  14. Face
  15. Face
  16. Face

So, in total, the device showed 14 faces and two trees. OK, time to decide. Do you think it's going to show a face or a tree next time? What are the chances?


If you're like most people, you might have said "a face!" And maybe you would've put the probability at 88% or so.


That is, until I told you I was just flipping a quarter from Connecticut.


But now, you might make the opposite guess. The probability of a quarter landing heads or tails is 50%, and Pop just flipped a bunch of heads. That means a string of tails must be coming up, right?


Except it doesn't mean that. Sure, after thousands of flips, you'll start to see the balance of heads and tails get closer to 50%, but that next flip has the same 50% of landing on heads as the last flip did.


Thinking it through, we know this. But our gut tries to take a small sample size and turn it into a trend. With the quarter, it's just fun and games, but where does this tendency threaten our money?


Starting a business

Around half of small businesses fail. Well, so says the governor of Texas, quoting the Small Business Administration. I can't find that actual stat on the SBA website -- it might be apocryphal. But anyway ....


Start-ups are risky. And yet, entrepreneurs take the risk anyway. Is it because they tolerate risk better or because they're not aware of the risk?


A few researchers from Georgia State University and Oakland University tried to figure that out by having 191 MBA students read a Harvard Business School case study and decide whether or not the subject of the study should quit his job and pursue a theoretical venture.


Granted, these students were faking it. They didn't have to actually feel the pit in their stomach before making the decision to jump or not to jump. Somewhat hilariously, the proposed venture was for contact lenses for chickens. Better eyesight made the chickens fight less, which had financial implications for farmers.


Anyway, the students were asked to pick three important facets of the case that led them to suggest starting or abandoning the venture and explain why. If a student gave a reason that suggested he or she believed in the law of small numbers -- that is, that he or she relied on feedback from a couple customers in the case in making a decision -- they marked that down.


It turned out that the students who let small sample sizes guide their decisions were much more prone not to perceive the high risks associated with the venture.


Meta question of the post: Are 191 MBA students with a median age of 28 representative of the risk attitudes of the entrepreneurial population? Eh.


Picking a mutual fund manager

Pick index funds. Low fees. Blah blah blah -- but how do investors actually go about picking a fund? As you might have guessed, they don't go through the empirical methods they think they do, and it's true even among "sophisticated" investors.


Take hedge funds, for example. To invest in a hedge fund, you need a minimum net worth of $1 milllion or an income of $200,000. So, these guys are supposed to know their way around a bank account.


But a couple researchers found that they chase returns like everybody else. Even though funds that perform well do tend to outperform funds that perform poorly (in their sample), investors piled even more money into the funds after a "winning streak" than predicted.

That less-sophisticated investors also chase returns is well-documented.


But it's not all strawberries and cream for index investors. Even our basic assumptions about stock returns are based on a tiny amount of market data.

We have 200 or so years of somewhat reliable market data upon which to estimate how well stocks perform over long periods of time. And let's say you wanted to show how they tend to perform over 20-year periods, since you're planning to hold on to your index fund until you retire.


That gives you only 10 non-overlapping sets of data upon which to base your retirement decisions. It's like wagering your life savings on info from the first 16 flips of a face-tree device. Maybe they were representative of the actual probability of stocks performing well in the future, but we don't really know yet.


It's going to be impossible to root out "small numbers" from every decision you make in life. I'm still going to go to restaurants and movies that a small sample of friends suggest.

But when it comes to making big decisions, take a little extra time to find as much data as you can, and if the data doesn't exist, at least be aware that you don't really know how things might turn out.


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