［也說 IQ 排名］230 可排世界第一, 160 三萬人就有一個！

湘西山民 · 2012-09-12 08:42:48Z

［也說 IQ 排名］230 可排世界第一, 160 三萬人就有一個！簡介

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Intelligence quotient

From Wikipedia, the free encyclopedia

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)

IQ = 160 ３萬人裏就有一個(out of range of ± 4σ: 1 in 15,787。隻考慮單邊： 1 in 15,787 x 2)！

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 ± $x$ σ	$textstyleoperatorname{erf}left(frac{x}{sqrt{2}}right)$	1 in $textstyle frac{1}{1-operatorname{erf}left(frac{x}{sqrt{2}}right)}$	Every $textstyle frac{1}{1-operatorname{erf}left(frac{x}{sqrt{2}}right)}$ 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.