The numbers behind the standard deviation.
The table below is provided courtesy of Wikipedia although the information will be available in any good statistics text book. I've included here to show the probability of results returned that fall outside of +/-3σ either side of the EV. As you can see, when it does happen it's a pretty rare event, although it can happen, and therefore will happen at some point - someone wins the national lottery most weeks? How many tickets are sold, how many does the winner buy and what are the odds of their win?
The link to the specific Wikipedia page is given at the bottom - there's some seriously heavy duty maths on the end of it !
| zσ | Percentage within CL |
Percentage outside CL | Fraction outside CL |
|---|---|---|---|
| 0.674490σ | 50.000000% | 50.000000% | 1 / 2 |
| 0.994458σ | 68.000000% | 32.000000% | 1 / 3.125 |
| 1σ | 68.268949% | 31.731051% | 1 / 3.1514872 |
| 1.281552σ | 80.000000% | 20.000000% | 1 / 5 |
| 1.644854σ | 90.000000% | 10.000000% | 1 / 10 |
| 1.959964σ | 95.000000% | 5.000000% | 1 / 20 |
| 2σ | 95.449974% | 4.550026% | 1 / 21.977895 |
| 2.575829σ | 99.000000% | 1.000000% | 1 / 100 |
| 3σ | 99.730020% | 0.269980% | 1 / 370.398 |
| 3.290527σ | 99.900000% | 0.100000% | 1 / 1,000 |
| 3.890592σ | 99.990000% | 0.010000% | 1 / 10,000 |
| 4σ | 99.993666% | 0.006334% | 1 / 15,787 |
| 4.417173σ | 99.999000% | 0.001000% | 1 / 100,000 |
| 4.891638σ | 99.999900% | 0.000100% | 1 / 1,000,000 |
| 5σ | 99.999943% | 0.000057% | 1 / 1,744,278 |
| 5.326724σ | 99.999990% | 0.000010% | 1 / 10,000,000 |
| 5.730729σ | 99.999999% | 0.000001% | 1 / 100,000,000 |
| 6σ | 100.000000% | 0.000000% | 1 / 506,797,346 |
| 6.109410σ | 100.000000% | 0.000000% | 1 / 1,000,000,000 |
| 6.466951σ | 100.000000% | 0.000000% | 1 / 10,000,000,000 |
| 6.806502σ | 100.000000% | 0.000000% | 1 / 100,000,000,000 |
| 7σ | 100.000000% | 0.000000% | 1 / 390,682,215,445 |
Here's the link to Wikipedia's page on standard deviation:
http://en.wikipedia.org/wiki/Standard_deviation
(the above table is about half way down the page)
There's also this one covering the "empirical rule".
http://en.wikipedia.org/wiki/68-95-99.7_rule