The Myth of Luck: Part One

If you have any level of familiarity with statistics, you can skip this post.

The first part of the Myth of Luck is the Myth of the Bell Curve.  We hear about the bell curve all the time.  Here’s a picture:  http://classes.kumc.edu/sah/resources/sensory_processing/images/bell_curve.gif

The fallacy that surrounds the Bell Curve is that it’s naturally occuring.  (It is, in certain ways that I’ll get to later.)  That huge 68% clump in the middle?  People claim that all sorts of things naturally fall in there:  fish weights, IQ scores, incomes, whatever.  The point is that the Bell Curve is NOT the “default” distribution of stuff.  It has a very specific role in statistics, one that people don’t understand and misinterpret.

Let’s suppose you have the numbers one to a hundred, one of each.  (Or to make it less abstract, you have one hundred index cards, numbered from one to a hundred.)  What does the distribution of that group of numbers look like?  It’s a straight line- you have an equal quantity of each number.

What does the distribution of the average of your numbers look like?  It’s a single point- there’s only one average, and it happens to be 50.5.  Of course, none of the cards actually say 50.5, so if you drew a card at random, you’d never achieve an “average” result.

Now we’re getting into bell curve territory.  Suppose you take 10 random numbers from 1 to a hundred, and then take the average of those numbers.  (This equals 10d100 divided by 10.)  What conclusions can we draw about what these averages are likely to look like?

First off, the average of ten random numbers (the sample) is NOT always going to be the average of the group they came from (the population).  Much more likely than not, it’s going to be some different number.

However, unlike taking one number, which gave results all over the spectrum, a sample of ten numbers is going to have an average that clumps.  Without doing any actual math, I’d say that a noticeable majority of those averages will fall between 40 and 60.  (If I wanted to calculate the standard deviation for this group, I could calculate the exact percentages, but that would be really boring.)

Suppose we do that a hundred times or so.  What we get NOW is a bell curve.  50.5 is the most likely single result, and most results will clump towards the middle.  However, we’ll still have the occasional “outlier”- this is when you roll lots of low or high numbers, and get an average that’s way out there.  The farther from 50.5, the less common a result is going to be.  Results at the far end, like 1 or 100, which require 10 1’s or 10 100’s to be rolled, will very rarely happen- 1 in a Quntillion times, in fact.  (That’s 1,000,000,000,000,000,000)  But they can happen.

And that’s what the Bell Curve is all about.

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