The table below gives the probability of a sample from a normal random variable being more than k standard deviations from its mean in either direction, k standard deviations above or below the mean. If you only want the one sided probability, divide by two.
As I wrote about here, the normal distribution assumption breaks down for large deviations. If your model produces astronomically large odds against an event that you’ve seen, there’s good reason to question your model. My main purpose in making a table of the probabilities below is to show that one should be very skeptical of claims such as “25 sigma events.”
For details on how the probabilities in the table were computed, see this post.
The probabilities are given in scientific notation, with the exponents zero-padded to line up vertically. So, for example, 6.33e-0005 means 6.33 × 10-5.
|--------+------------------------| | Sigmas | Two-tailed probability | |--------+------------------------| | 1 | 3.17310e-0001 | | 2 | 4.55002e-0002 | | 3 | 2.69979e-0003 | | 4 | 6.33424e-0005 | | 5 | 5.73303e-0007 | | 6 | 1.97317e-0009 | | 7 | 2.55962e-0012 | | 8 | 1.24419e-0015 | | 9 | 2.25717e-0019 | | 10 | 1.52397e-0023 | | 11 | 3.82131e-0028 | | 12 | 3.55296e-0033 | | 13 | 1.22343e-0038 | | 14 | 1.55870e-0044 | | 15 | 7.34193e-0051 | | 16 | 1.27775e-0057 | | 17 | 8.21199e-0065 | | 18 | 1.94818e-0072 | | 19 | 1.70544e-0080 | | 20 | 5.50724e-0089 | | 21 | 6.55855e-0098 | | 22 | 2.87978e-0107 | | 23 | 4.66127e-0117 | | 24 | 2.78078e-0127 | | 25 | 6.11339e-0138 | | 26 | 4.95212e-0149 | | 27 | 1.47789e-0160 | | 28 | 1.62477e-0172 | | 29 | 6.57957e-0185 | | 30 | 9.81342e-0198 | | 31 | 5.39050e-0211 | | 32 | 1.09041e-0224 | | 33 | 8.12237e-0239 | | 34 | 2.22779e-0253 | | 35 | 2.24982e-0268 | | 36 | 8.36524e-0284 | | 37 | 1.14511e-0299 | | 38 | 2.88615e-0316 | | 39 | 5.35439e-0333 | | 40 | 3.65672e-0350 | | 41 | 9.19284e-0368 | | 42 | 8.50689e-0386 | | 43 | 2.89763e-0404 | | 44 | 3.63292e-0423 | | 45 | 1.67647e-0442 | | 46 | 2.84747e-0462 | | 47 | 1.78004e-0482 | | 48 | 4.09548e-0503 | | 49 | 3.46795e-0524 | | 50 | 1.08075e-0545 | | 51 | 1.23953e-0567 | | 52 | 5.23196e-0590 | | 53 | 8.12710e-0613 | | 54 | 4.64586e-0636 | | 55 | 9.77354e-0660 | | 56 | 7.56634e-0684 | | 57 | 2.15558e-0708 | | 58 | 2.25985e-0733 | | 59 | 8.71832e-0759 | | 60 | 1.23769e-0784 | | 61 | 6.46580e-0811 | | 62 | 1.24294e-0837 | | 63 | 8.79226e-0865 | | 64 | 2.28856e-0892 | | 65 | 2.19198e-0920 | | 66 | 7.72540e-0949 | | 67 | 1.00186e-0977 | | 68 | 4.78078e-1007 | | 69 | 8.39438e-1037 | | 70 | 5.42344e-1067 | | 71 | 1.28930e-1097 | | 72 | 1.12778e-1128 | | 73 | 3.62984e-1160 | | 74 | 4.29869e-1192 | | 75 | 1.87313e-1224 | | 76 | 3.00320e-1257 | | 77 | 1.77166e-1290 | | 78 | 3.84552e-1324 | | 79 | 3.07119e-1358 | | 80 | 9.02474e-1393 | | 81 | 9.75742e-1428 | | 82 | 3.88156e-1463 | | 83 | 5.68131e-1499 | | 84 | 3.05956e-1535 | | 85 | 6.06229e-1572 | | 86 | 4.41957e-1609 | | 87 | 1.18546e-1646 | | 88 | 1.16992e-1684 | | 89 | 4.24804e-1723 | | 90 | 5.67520e-1762 | | 91 | 2.78954e-1801 | | 92 | 5.04477e-1841 | | 93 | 3.35666e-1881 | | 94 | 8.21731e-1922 | | 95 | 7.40126e-1963 | | 96 | 2.45265e-2004 | | 97 | 2.99032e-2046 | | 98 | 1.34138e-2088 | | 99 | 2.21379e-2131 | | 100 | 1.34422e-2174 | |--------+------------------------|