Higher Deviations
Because of the exponential tails of the normal distribution, odds of higher deviations decrease very quickly. From the Rules for normally distributed data:
| Range | Population in range | Expected frequency outside range | Approx. frequency for daily event |
|---|---|---|---|
| μ ± 1σ | 0.682689492137086 | 1 in 3 | Twice a week |
| μ ± 1.5σ | 0.866385597462284 | 1 in 7 | Weekly |
| μ ± 2σ | 0.954499736103642 | 1 in 22 | Every three weeks |
| μ ± 2.5σ | 0.987580669348448 | 1 in 81 | Quarterly |
| μ ± 3σ | 0.997300203936740 | 1 in 370 | Yearly |
| μ ± 3.5σ | 0.999534741841929 | 1 in 2149 | Every six years |
| μ ± 4σ | 0.999936657516334 | 1 in 15,787 | Every 43 years (twice in a lifetime) |
| μ ± 4.5σ | 0.999993204653751 | 1 in 147,160 | Every 403 years |
| μ ± 5σ | 0.999999426696856 | 1 in 1,744,278 | Every 4,776 years (once in recorded history) |
| μ ± 5.5σ | 0.999999962020875 | 1 in 26,330,254 | Every 72,090 years |
| μ ± 6σ | 0.999999998026825 | 1 in 506,797,346 | Every 1.388 million years |
| μ ± 6.5σ | 0.999999999919680 | 1 in 12,450,197,393 | Every 34.087 million years |
| μ ± 7σ | 0.999999999997440 | 1 in 390,682,215,445 | Every 1.070×109 years |
| μ ± xσ | 1 in | Every days |
Thus for a daily process, a 6σ event is expected to happen less than once in a million years. This gives a simple normality test: if one witnesses a 6σ in daily data and significantly fewer than 1 million years have passed, then a normal distribution most likely does not provide a good model for the magnitude or frequency of large deviations in this respect. In The Black Swan, Nassim Nicholas Taleb gives the example of risk models for which the Black Monday crash was a 36-sigma event: the occurrence of such an event should instantly suggest a catastrophic flaw in a model.
Read more about this topic: 68-95-99.7 Rule
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