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:

σ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

outside | 3 * 10^{-1} | 5 * 10^{-2} | 3 * 10^{-3} | 6 * 10^{-5} | 6 * 10^{-7} | 2 * 10^{-9} | 3 * 10^{-12} | 1 * 10^{-15} | 2 * 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, ...)*