Normal Distribution

A Gaussian, aka normal distribution or bell curve, is what most statistical averages converge toward. The "Square Root of N" rule-of-thumb says that if you toss 100 coins you'll get half heads, plus or minus 10 or so (since 10 = √100). Toss a million coins and you can expect half a million heads, plus or minus maybe 1,000 (= √1,000,000). Getting a little more precise: the standard deviation σ of N events, each of which is 50% likely, is half the square root of N. In general for probability p it's √(N * p * (1-p)).

How often does something happen more than a few standard deviations away from expectations? Worth remembering are the first few values: 68% of the time the result is within 1σ, 95% of the time it's within 2σ, and 99.7% of the time it's within 3σ. So for a thousand coins, or in a typical polling sample of a thousand evenly-split voters, about two-thirds of the time the result is within ~15 of the halfway point, 95% of the time it's within ~30 (= √N), and 99.7% of the time it's within ~45. The "Square Root of N" is thus a 95% guideline.

Beyond a few σ it's easier to look at how infrequently things happen — how often something occurs outside the window of that many sigma:

outside 3 * 10-15 * 10-23 * 10-36 * 10-56 * 10-72 * 10-93 * 10-121 * 10-152 * 10-19

Events therefore fall within 3σ more than "two nines" (0.99+ of the time), 4σ more than "four nines" (0.9999+), 5σ more than "six nines" (0.999999+), and 6σ more than "eight nines" (0.99999999+). (Then the pattern in the exponents breaks down: rare events are even rarer than that.) Note also that "six sigma", a popular quality-control mantra in some industrial circles, isn't the usually cited value of 3.4 failures per million; it's more like a few chances per billion. But "4.5 sigma" doesn't have the alliterative ring to it that sells management advice books!

(cf. Human Diffusion (2000-01-19), Square Root of Baseball (2005-05-13), ...) - ^z - 2008-04-26

(correlates: 2008-07-11 - MidSummer Night's Mile, SquareRootOfBaseball, ClubScience, ...)