Intelligence quotient
An intelligence quotient, or IQ, is a score derived from one of several standardized tests designed to assess intelligence. The abbreviation "IQ" comes from the German term Intelligenz-Quotient, originally coined by psychologist William Stern. When modern IQ tests are devised, the mean (average) score within an age group is set to 100 and the standard deviation (SD) almost always to 15, although this was not always so historically.[1] Thus, the intention is that approximately 95% of the population scores within two SDs of the mean, i.e. has an IQ between 70 and 130.
中點是100， σ = 15. 160 = 100 + 4σ
IQ_curve.svg (SVG file, nominally 600 × 480 pixels, file size: 12 KB)
有文章說世界最聰明的十人 IQ 高到 230， 低到160。 感覺似乎不太對勁。
• [投資理財] 置頂: 當今在世智商最高十大天才
IQ 160 的人，３萬多人裏就有一個 (out of range of ± 4σ: 1 in 15,787(半個天才半個傻子)。 隻考慮天才： 1 in 15,787 x 2)！
Normal distribution: 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.
