Extreme Tails of the Normal Distribution
notes
probability
Estimating extreme quantiles of the normal distribution.
Approximating Tails of the Normal Distribution
Feller vol 1. Chapter VII Lemma 2 page 175 gives
See also Abramowitz and Stegun asymptotic series 26.2.12
Using
The table below compares various estimates for
AS Table 26.2 | ||||
---|---|---|---|---|
2 | 1.64302 | 1.56871 | 1.57060 | 1.69365 |
3 | 2.86970 | 2.83054 | 2.83364 | 2.88169 |
4 | 4.49933 | 4.47551 | 4.48032 | 4.50353 |
5 | 6.54265 | 6.52674 | 6.53375 | 6.54447 |
6 | 9.00586 | 8.99454 | 9.00424 | 9.00678 |
7 | 11.89285 | 11.88440 | 11.89727 | 11.89336 |
8 | 15.20614 | 15.19960 | 15.21613 | 15.20644 |
9 | 18.94746 | 18.94226 | 18.96294 | 18.94765 |
10 | 23.11805 | 23.11381 | 23.13913 | 23.11818 |
15 | 50.43522 | 50.43331 | 50.48913 | 50.43524 |
20 | 88.56010 | 88.55902 | 88.65755 | 88.56010 |
25 | 137.51475 | 137.51406 | 137.66751 | 137.51475 |
50 | 544.96634 | 544.96616 | 545.57723 | 544.96634 |
100 | 2,173.87154 | 2,173.87150 | 2,176.31304 | 2,173.87154 |
200 | 8,688.58977 | 8,688.58976 | 8,698.35320 | 8,688.58977 |
500 | 54,289.90830 | 54,289.90830 | 54,350.92506 | 54,289.90830 |
Hence the famous Goldman Sachs CFO quote about 25 deviations from the mean “several days in a row” (Kevin Dowd et al “How Unlucky is 25-sigma?”) is ridiculous.