1310:
134:
1300:
909:
122:
4496:
25:
1295:{\displaystyle {\begin{aligned}\Pr(\mu -1\sigma \leq X\leq \mu +1\sigma )&={\frac {1}{\sqrt {2\pi }}}\int _{-1}^{1}e^{-{\frac {z^{2}}{2}}}dz\approx 0.6827\\\Pr(\mu -2\sigma \leq X\leq \mu +2\sigma )&={\frac {1}{\sqrt {2\pi }}}\int _{-2}^{2}e^{-{\frac {z^{2}}{2}}}dz\approx 0.9545\\\Pr(\mu -3\sigma \leq X\leq \mu +3\sigma )&={\frac {1}{\sqrt {2\pi }}}\int _{-3}^{3}e^{-{\frac {z^{2}}{2}}}dz\approx 0.9973.\end{aligned}}}
4506:
416:
678:
129:, the values within one standard deviation of the mean account for about 68% of the set; while within two standard deviations account for about 95%; and within three standard deviations account for about 99.7%. Shown percentages are rounded theoretical probabilities intended only to approximate the empirical data derived from a normal population.
223:
496:
1681:
is significantly large, by which point one expects a sample this extreme), and if there are many points more than 3 standard deviations from the norm, one likely has reason to question the assumed normality of the distribution. This holds ever more strongly for moves of 4 or more standard deviations.
1524:
1730:
event: the occurrence of such an event should instantly suggest that the model is flawed, i.e. that the process under consideration is not satisfactorily modeled by a normal distribution. Refined models should then be considered, e.g. by the introduction of
822:
411:{\displaystyle {\begin{aligned}\Pr(\mu -1\sigma \leq X\leq \mu +1\sigma )&\approx 68.27\%\\\Pr(\mu -2\sigma \leq X\leq \mu +2\sigma )&\approx 95.45\%\\\Pr(\mu -3\sigma \leq X\leq \mu +3\sigma )&\approx 99.73\%\end{aligned}}}
2813:
2751:
673:{\displaystyle {\begin{aligned}\Pr(\mu -n\sigma \leq X\leq \mu +n\sigma )=\int _{\mu -n\sigma }^{\mu +n\sigma }{\frac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}dx,\end{aligned}}}
2686:
2634:
1708:
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.
1642:
The "68–95–99.7 rule" is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal. It is also used as a simple test for
1577:
1407:
1673:
To use as a test for outliers or a normality test, one computes the size of deviations in terms of standard deviations, and compares this to expected frequency. Given a sample set, one can compute the
733:
914:
738:
501:
228:
1689:, but simply, if one has multiple 4 standard deviation moves in a sample of size 1,000, one has strong reason to consider these outliers or question the assumed normality of the distribution.
1739:, which states that a single observation of a rare event does not contradict that the event is in fact rare. It is the observation of a plurality of purportedly rare events that increasingly
726:
1743:
that they are rare, i.e. the validity of the assumed model. A proper modelling of this process of gradual loss of confidence in a hypothesis would involve the designation of
1606:
902:
864:
844:
1626:
2758:
2696:
3154:
1714:
477:, stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. For
2642:
89:
2596:
1677:
and compare these to the expected frequency: points that fall more than 3 standard deviations from the norm are likely outliers (unless the
61:
1700:. For illustration, if events are taken to occur daily, this would correspond to an event expected every 1.4 million years. This gives a
3283:
4509:
3766:
443:
that nearly all values are taken to lie within three standard deviations of the mean, and thus it is empirically useful to treat 99.7%
68:
3674:
3103:
3078:
3053:
1763:
Because of the exponentially decreasing tails of the normal distribution, odds of higher deviations decrease very quickly. From the
1404:. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding):
4461:
42:
4327:
3539:
3298:
3147:
1535:
75:
4222:
3986:
3660:
1309:
57:
3981:
3925:
3823:
3585:
3223:
1764:
1332:
4535:
4267:
4001:
3854:
3529:
3273:
482:
3731:
4499:
4171:
4147:
3726:
3140:
1659:
4368:
4245:
4206:
4178:
4152:
4070:
3996:
3419:
3167:
2996:
2972:
2940:
2914:
2883:
108:
1519:{\displaystyle \Pr(\mu -2\sigma \leq X\leq \mu +2\sigma )=\Phi (2)-\Phi (-2)\approx 0.9772-(1-0.9772)\approx 0.9545}
4356:
4322:
4188:
4183:
4028:
3836:
3534:
3288:
1314:
4106:
4019:
3991:
3900:
3849:
3721:
3504:
3469:
690:
817:{\displaystyle {\begin{aligned}{\frac {1}{\sqrt {2\pi }}}\int _{-n}^{n}e^{-{\frac {z^{2}}{2}}}dz\end{aligned}},}
4120:
4037:
3874:
3798:
3621:
3499:
3474:
3338:
3333:
3328:
1748:
1685:
One can compute more precisely, approximating the number of extreme moves of a given magnitude or greater by a
46:
4436:
4302:
4010:
3859:
3791:
3776:
3669:
3643:
3575:
3414:
3308:
3303:
3245:
3230:
82:
4530:
4272:
4262:
3953:
3879:
3580:
3439:
478:
4332:
4317:
4312:
4257:
4193:
4137:
3958:
3945:
3736:
3681:
3633:
3424:
3353:
3218:
1701:
681:
4451:
4227:
4046:
3828:
3781:
3650:
3626:
3606:
3449:
3323:
3203:
1670:(dividing by an estimate of the standard deviation), if the parameters are unknown and only estimated.
1394:. This is not a symmetrical interval – this is merely the probability that an observation is less than
485:. There may be certain assumptions for a distribution that force this probability to be at least 98%.
4456:
4240:
4201:
4075:
3912:
3756:
3701:
3599:
3563:
3434:
3399:
474:
466:, there is a convention of a five-sigma effect (99.99994% confidence) being required to qualify as a
4142:
3930:
3696:
3655:
3570:
3524:
3464:
3429:
3318:
3213:
3163:
455:
193:
4540:
4441:
4383:
4054:
3841:
3751:
3706:
3691:
3509:
3459:
3454:
3255:
3235:
2875:
2388:
467:
35:
3611:
3122:
1582:
4307:
4295:
4284:
4166:
4062:
3869:
3313:
3293:
3198:
2932:
1747:
not just to the hypothesis itself but to all possible alternative hypotheses. For this reason,
1655:
2986:
4431:
4388:
4232:
3907:
3761:
3741:
3638:
3208:
2869:
1732:
1719:
869:
849:
2924:
1666:(dividing by the population standard deviation), if the population parameters are known, or
4481:
4476:
4471:
4466:
4403:
4373:
4252:
3895:
3786:
3389:
3348:
3343:
3240:
1736:
1686:
1674:
1662:
depending on whether one knows the population mean or only estimates it. The next step is
3686:
829:
420:
The usefulness of this heuristic especially depends on the question under consideration.
8:
4415:
3940:
3920:
3890:
3864:
3818:
3746:
3558:
3494:
1723:
1529:
174:
133:
126:
4446:
3935:
3716:
3711:
3616:
3553:
3548:
3404:
3394:
3278:
3023:
2903:
2808:{\displaystyle {\tfrac {1}{1-\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}}}
2746:{\displaystyle {\tfrac {1}{1-\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}}}
1611:
178:
4344:
3771:
3514:
3444:
3409:
3358:
2992:
2968:
2936:
2925:
2910:
2879:
1744:
1697:
424:
170:
3519:
3193:
3132:
3015:
2982:
2443:
2222:
463:
459:
2898:
This usage of "three-sigma rule" entered common usage in the 2000s, e.g. cited in
2835:
2962:
1752:
1740:
1654:
To pass from a sample to a number of standard deviations, one first computes the
451:
203:
182:
177:: approximately 68%, 95%, and 99.7% of the values lie within one, two, and three
149:). The y-axis is logarithmically scaled (but the values on it are not modified).
3592:
2840:
2639:
2593:
1648:
1637:
1339:
685:
142:
169:, is a shorthand used to remember the percentage of values that lie within an
4524:
4215:
3963:
3250:
1751:
works not so much by confirming a hypothesis considered to be likely, but by
1663:
2927:
Linear and
Nonlinear Models: Fixed Effects, Random Effects, and Mixed Models
2333:
2281:
1667:
121:
3093:
3068:
3043:
2845:
2681:{\displaystyle 1-\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}
2505:
1735:. In such discussions it is important to be aware of the problem of the
1678:
444:
188:
In mathematical notation, these facts can be expressed as follows, where
3027:
2629:{\displaystyle \operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}
2166:
154:
481:, the probability of being within the interval is at least 95% by the
2571:
1392:(1 − (1 − 0.97725)·2) = 0.9545 = 95.45%
440:
3019:
24:
2567:
2563:
2501:
1644:
2827:
146:
138:
3097:
3072:
3047:
2988:
Statistical Case
Studies for Industrial Process Improvement
1572:{\displaystyle {\bar {X}}\pm 2{\frac {\sigma }{\sqrt {n}}}}
1333:
cumulative distribution function of the normal distribution
214:
208:
198:
2500:
Once every 43 billion years (never in the history of the
1722:
gives the example of risk models according to which the
866:. We only need to calculate each integral for the cases
1331:
These numerical values "68%, 95%, 99.7%" come from the
3092:
3067:
3042:
2763:
2701:
462:
is of the order of a two-sigma effect (95%), while in
2761:
2699:
2645:
2599:
1614:
1585:
1538:
1410:
912:
872:
852:
832:
736:
693:
499:
226:
3162:
2871:
2442:Every 1.07 billion years (four occurrences in
2332:Every 1.38 million years (twice in history of
1304:
49:. Unsourced material may be challenged and removed.
2902:
2807:
2745:
2680:
2628:
1620:
1600:
1571:
1518:
1294:
896:
858:
838:
816:
720:
672:
410:
4522:
1793:Approx. frequency outside range for daily event
1579:is approximately a 95% confidence interval when
1411:
1167:
1042:
917:
504:
349:
290:
231:
3006:Pukelsheim, F. (1994). "The Three Sigma Rule".
2960:
2909:. McGraw Hill Professional. 2003. p. 359.
1647:if the population is assumed normal, and as a
473:A weaker three-sigma rule can be derived from
202:is an observation from a normally distributed
3148:
2981:
2387:Every 34 million years (twice since the
1651:if the population is potentially not normal.
1758:
1390:, corresponding to a prediction interval of
1696:event corresponds to a chance of about two
721:{\displaystyle z={\frac {x-\mu }{\sigma }}}
3155:
3141:
3005:
2120:Every 43 years (twice in a lifetime)
212:(mu) is the mean of the distribution, and
3104:On-Line Encyclopedia of Integer Sequences
3079:On-Line Encyclopedia of Integer Sequences
3054:On-Line Encyclopedia of Integer Sequences
2964:Understanding Statistical Process Control
2922:
109:Learn how and when to remove this message
2961:Wheeler, D. J.; Chambers, D. S. (1992).
1308:
132:
120:
3123:Calculate percentage proportion within
2905:Schaum's Outline of Business Statistics
2066:4.653E-04 = 0.04653 % = 465.3 ppm
1753:refuting hypotheses considered unlikely
1317:for the normal distribution with mean (
4523:
3136:
2867:
4505:
2367:8.032E-11 = 0.08032 ppb = 80.32 ppt
826:and this integral is independent of
47:adding citations to reliable sources
18:
2566:years (never in the history of the
2200:5.733E-07 = 0.5733 ppm = 573.3 ppb
2026:2.700E-03 = 0.270 % = 2.700 ‰
1765:rules for normally distributed data
1608:is the average of a sample of size
218:(sigma) is its standard deviation:
13:
2280: years (thrice in history of
2165:Every 403 years (once in the
1631:
1471:
1456:
401:
342:
283:
14:
4552:
3115:
4504:
4495:
4494:
1338:The prediction interval for any
1315:cumulative distribution function
1305:Cumulative distribution function
23:
1726:crash would correspond to a 36-
483:Vysochanskij–Petunin inequality
34:needs additional citations for
3086:
3061:
3036:
2951:
2892:
2861:
1749:statistical hypothesis testing
1592:
1545:
1507:
1495:
1483:
1474:
1465:
1459:
1450:
1414:
1206:
1170:
1081:
1045:
956:
920:
543:
507:
454:, a result may be considered "
388:
352:
329:
293:
270:
234:
1:
2985:; Spagon, Patrick D. (1997).
2931:. Walter de Gruyter. p.
2854:
2836:Six Sigma § Sigma levels
2504:, twice in the future of the
2570:, once during the life of a
165:, and sometimes abbreviated
137:Prediction interval (on the
16:Shorthand used in statistics
7:
2923:Grafarend, Erik W. (2006).
2821:
1345:corresponds numerically to
439:) expresses a conventional
10:
4557:
4536:Statistical approximations
4328:Wrapped asymmetric Laplace
3299:Extended negative binomial
3094:Sloane, N. J. A.
3069:Sloane, N. J. A.
3044:Sloane, N. J. A.
1835:Four or five times a week
1635:
1601:{\displaystyle {\bar {X}}}
4490:
4424:
4382:
4283:
4119:
4097:
4088:
3987:Generalized extreme value
3972:
3807:
3767:Relativistic Breit–Wigner
3483:
3380:
3371:
3264:
3184:
3175:
3164:Probability distributions
1986:1.242E-02 = 1.242 %
1946:4.550E-02 = 4.550 %
1906:1.336E-01 = 13.36 %
1866:3.173E-01 = 31.73 %
1826:6.171E-01 = 61.71 %
1787:
1784:population outside range
1759:Table of numerical values
1741:undermines the hypothesis
429:three-sigma rule of thumb
1790:frequency outside range
1778:population inside range
488:
3982:Generalized chi-squared
3926:Normal-inverse Gaussian
3098:"Sequence A270712"
3073:"Sequence A110894"
3048:"Sequence A178647"
2876:Oxford University Press
2389:extinction of dinosaurs
1875:Twice or thrice a week
1532:as used in statistics:
897:{\displaystyle n=1,2,3}
859:{\displaystyle \sigma }
4294:Univariate (circular)
3855:Generalized hyperbolic
3284:Conway–Maxwell–Poisson
3274:Beta negative binomial
2809:
2747:
2682:
2630:
2539:1.244E-15 = 1.244 ppq
2477:6.382E-14 = 63.82 ppq
2422:2.560E-12 = 2.560 ppt
2315:1.973E-09 = 1.973 ppb
2256:3.798E-08 = 37.98 ppb
2151:6.795E-06 = 6.795 ppm
2106:6.334E-05 = 63.34 ppm
1704:: if one witnesses a 6
1622:
1602:
1573:
1520:
1328:
1296:
898:
860:
840:
818:
722:
674:
479:unimodal distributions
475:Chebyshev's inequality
412:
150:
130:
4339:Bivariate (spherical)
3837:Kaniadakis κ-Gaussian
3008:American Statistician
2991:. SIAM. p. 342.
2868:Huber, Franz (2018).
2810:
2748:
2683:
2631:
2221: years (once in
1782:Expected fraction of
1776:Expected fraction of
1733:stochastic volatility
1720:Nassim Nicholas Taleb
1702:simple normality test
1675:studentized residuals
1623:
1603:
1574:
1521:
1312:
1297:
899:
861:
841:
819:
723:
675:
413:
136:
125:For an approximately
124:
4404:Dirac delta function
4351:Bivariate (toroidal)
4308:Univariate von Mises
4179:Multivariate Laplace
4071:Shifted log-logistic
3420:Continuous Bernoulli
2759:
2697:
2643:
2597:
1687:Poisson distribution
1612:
1583:
1536:
1408:
1313:Diagram showing the
910:
870:
850:
839:{\displaystyle \mu }
830:
734:
691:
497:
224:
194:probability function
161:, also known as the
43:improve this article
4531:Normal distribution
4452:Natural exponential
4357:Bivariate von Mises
4323:Wrapped exponential
4189:Multivariate stable
4184:Multivariate normal
3505:Benktander 2nd kind
3500:Benktander 1st kind
3289:Discrete phase-type
2508:before its merger)
2075:Every 6 years
1767:for a daily event:
1530:confidence interval
1528:This is related to
1248:
1123:
998:
773:
581:
447:as near certainty.
179:standard deviations
175:normal distribution
4107:Rectified Gaussian
3992:Generalized Pareto
3850:Generalized normal
3722:Matrix-exponential
3107:. OEIS Foundation.
3082:. OEIS Foundation.
3057:. OEIS Foundation.
2805:
2803:
2743:
2741:
2678:
2626:
1955:Every three weeks
1618:
1598:
1569:
1516:
1329:
1321:) 0 and variance (
1292:
1290:
1231:
1106:
981:
894:
856:
836:
814:
809:
756:
718:
682:change of variable
670:
668:
549:
425:empirical sciences
408:
406:
151:
131:
4518:
4517:
4115:
4114:
4084:
4083:
3975:whose type varies
3921:Normal (Gaussian)
3875:Hyperbolic secant
3824:Exponential power
3727:Maxwell–Boltzmann
3475:Wigner semicircle
3367:
3366:
3339:Parabolic fractal
3329:Negative binomial
2983:Czitrom, Veronica
2819:
2818:
2802:
2795:
2794:
2740:
2733:
2732:
2672:
2671:
2620:
2619:
1745:prior probability
1737:gambler's fallacy
1698:parts per billion
1660:error or residual
1621:{\displaystyle n}
1595:
1567:
1566:
1548:
1272:
1229:
1228:
1147:
1104:
1103:
1022:
979:
978:
797:
754:
753:
716:
643:
620:
602:
596:
427:, the so-called
171:interval estimate
141:) given from the
119:
118:
111:
93:
58:"68–95–99.7 rule"
4548:
4508:
4507:
4498:
4497:
4437:Compound Poisson
4412:
4400:
4369:von Mises–Fisher
4365:
4353:
4341:
4303:Circular uniform
4299:
4219:
4163:
4134:
4095:
4094:
3997:Marchenko–Pastur
3860:Geometric stable
3777:Truncated normal
3670:Inverse Gaussian
3576:Hyperexponential
3415:Beta rectangular
3383:bounded interval
3378:
3377:
3246:Discrete uniform
3231:Poisson binomial
3182:
3181:
3157:
3150:
3143:
3134:
3133:
3109:
3108:
3090:
3084:
3083:
3065:
3059:
3058:
3040:
3034:
3031:
3002:
2978:
2955:
2949:
2946:
2930:
2919:
2908:
2896:
2890:
2889:
2865:
2814:
2812:
2811:
2806:
2804:
2801:
2800:
2796:
2790:
2786:
2764:
2752:
2750:
2749:
2744:
2742:
2739:
2738:
2734:
2728:
2724:
2702:
2691:1 in
2687:
2685:
2684:
2679:
2677:
2673:
2667:
2663:
2635:
2633:
2632:
2627:
2625:
2621:
2615:
2611:
2587:
2559:
2558:
2555:
2552:
2549:
2542:1 in
2536:
2535:
2532:
2529:
2526:
2497:
2496:
2493:
2490:
2487:
2480:1 in
2474:
2473:
2470:
2467:
2464:
2444:history of Earth
2439:
2438:
2435:
2432:
2425:1 in
2419:
2418:
2415:
2412:
2409:
2384:
2383:
2380:
2377:
2370:1 in
2364:
2363:
2360:
2357:
2354:
2329:
2328:
2325:
2318:1 in
2312:
2311:
2308:
2305:
2302:
2282:modern humankind
2279:
2278:
2270:
2269:
2266:
2259:1 in
2253:
2252:
2249:
2246:
2243:
2223:recorded history
2220:
2214:
2213:
2210:
2203:1 in
2197:
2196:
2193:
2190:
2187:
2162:
2161:
2154:1 in
2148:
2147:
2144:
2141:
2138:
2117:
2116:
2109:1 in
2103:
2102:
2099:
2096:
2093:
2069:1 in
2063:
2062:
2059:
2056:
2053:
2029:1 in
2023:
2022:
2019:
2016:
2013:
1989:1 in
1983:
1982:
1979:
1976:
1973:
1949:1 in
1943:
1942:
1939:
1936:
1933:
1909:2 in
1903:
1902:
1899:
1896:
1893:
1869:1 in
1863:
1862:
1859:
1856:
1853:
1829:3 in
1823:
1822:
1819:
1816:
1813:
1806:
1788:Approx. expected
1770:
1769:
1692:For example, a 6
1627:
1625:
1624:
1619:
1607:
1605:
1604:
1599:
1597:
1596:
1588:
1578:
1576:
1575:
1570:
1568:
1562:
1558:
1550:
1549:
1541:
1525:
1523:
1522:
1517:
1403:
1393:
1389:
1373:
1363:
1352:
1326:
1320:
1301:
1299:
1298:
1293:
1291:
1275:
1274:
1273:
1268:
1267:
1258:
1247:
1242:
1230:
1221:
1217:
1150:
1149:
1148:
1143:
1142:
1133:
1122:
1117:
1105:
1096:
1092:
1025:
1024:
1023:
1018:
1017:
1008:
997:
992:
980:
971:
967:
903:
901:
900:
895:
865:
863:
862:
857:
845:
843:
842:
837:
823:
821:
820:
815:
810:
800:
799:
798:
793:
792:
783:
772:
767:
755:
746:
742:
727:
725:
724:
719:
717:
712:
701:
684:in terms of the
679:
677:
676:
671:
669:
656:
655:
654:
653:
648:
644:
639:
628:
621:
613:
603:
601:
597:
589:
583:
580:
566:
464:particle physics
460:confidence level
436:
417:
415:
414:
409:
407:
217:
211:
201:
191:
185:, respectively.
114:
107:
103:
100:
94:
92:
51:
27:
19:
4556:
4555:
4551:
4550:
4549:
4547:
4546:
4545:
4521:
4520:
4519:
4514:
4486:
4462:Maximum entropy
4420:
4408:
4396:
4386:
4378:
4361:
4349:
4337:
4292:
4279:
4216:Matrix-valued:
4213:
4159:
4130:
4122:
4111:
4099:
4090:
4080:
3974:
3968:
3885:
3811:
3809:
3803:
3732:Maxwell–Jüttner
3581:Hypoexponential
3487:
3485:
3484:supported on a
3479:
3440:Noncentral beta
3400:Balding–Nichols
3382:
3381:supported on a
3373:
3363:
3266:
3260:
3256:Zipf–Mandelbrot
3186:
3177:
3171:
3161:
3129:at WolframAlpha
3118:
3113:
3112:
3091:
3087:
3066:
3062:
3041:
3037:
3020:10.2307/2684253
2999:
2975:
2956:
2952:
2943:
2917:
2901:
2897:
2893:
2886:
2866:
2862:
2857:
2824:
2785:
2781:
2768:
2762:
2760:
2757:
2756:
2723:
2719:
2706:
2700:
2698:
2695:
2694:
2662:
2658:
2644:
2641:
2640:
2610:
2606:
2598:
2595:
2594:
2583:
2562:Once every 2.2
2556:
2553:
2550:
2547:
2545:
2533:
2530:
2527:
2524:
2522:
2494:
2491:
2488:
2485:
2483:
2471:
2468:
2465:
2462:
2460:
2436:
2433:
2430:
2428:
2416:
2413:
2410:
2407:
2405:
2381:
2378:
2375:
2373:
2361:
2358:
2355:
2352:
2350:
2326:
2323:
2321:
2309:
2306:
2303:
2300:
2298:
2276:
2274:
2267:
2264:
2262:
2250:
2247:
2244:
2241:
2239:
2218:
2211:
2208:
2206:
2194:
2191:
2188:
2185:
2183:
2159:
2157:
2145:
2142:
2139:
2136:
2134:
2114:
2112:
2100:
2097:
2094:
2091:
2089:
2060:
2057:
2054:
2051:
2049:
2020:
2017:
2014:
2011:
2009:
1980:
1977:
1974:
1971:
1969:
1940:
1937:
1934:
1931:
1929:
1900:
1897:
1894:
1891:
1889:
1860:
1857:
1854:
1851:
1849:
1820:
1817:
1814:
1811:
1809:
1798:
1789:
1761:
1640:
1634:
1632:Normality tests
1613:
1610:
1609:
1587:
1586:
1584:
1581:
1580:
1557:
1540:
1539:
1537:
1534:
1533:
1409:
1406:
1405:
1402:
1398:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1368:
1361:
1360:
1356:
1351:
1348:
1346:
1325:
1322:
1318:
1307:
1289:
1288:
1263:
1259:
1257:
1253:
1249:
1243:
1235:
1216:
1209:
1164:
1163:
1138:
1134:
1132:
1128:
1124:
1118:
1110:
1091:
1084:
1039:
1038:
1013:
1009:
1007:
1003:
999:
993:
985:
966:
959:
913:
911:
908:
907:
871:
868:
867:
851:
848:
847:
831:
828:
827:
808:
807:
788:
784:
782:
778:
774:
768:
760:
741:
737:
735:
732:
731:
702:
700:
692:
689:
688:
667:
666:
649:
629:
627:
623:
622:
612:
608:
604:
588:
587:
582:
567:
553:
500:
498:
495:
494:
491:
452:social sciences
434:
405:
404:
391:
346:
345:
332:
287:
286:
273:
227:
225:
222:
221:
213:
207:
204:random variable
197:
189:
159:68–95–99.7 rule
127:normal data set
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
4554:
4544:
4543:
4541:Rules of thumb
4538:
4533:
4516:
4515:
4513:
4512:
4502:
4491:
4488:
4487:
4485:
4484:
4479:
4474:
4469:
4464:
4459:
4457:Location–scale
4454:
4449:
4444:
4439:
4434:
4428:
4426:
4422:
4421:
4419:
4418:
4413:
4406:
4401:
4393:
4391:
4380:
4379:
4377:
4376:
4371:
4366:
4359:
4354:
4347:
4342:
4335:
4330:
4325:
4320:
4318:Wrapped Cauchy
4315:
4313:Wrapped normal
4310:
4305:
4300:
4289:
4287:
4281:
4280:
4278:
4277:
4276:
4275:
4270:
4268:Normal-inverse
4265:
4260:
4250:
4249:
4248:
4238:
4230:
4225:
4220:
4211:
4210:
4209:
4199:
4191:
4186:
4181:
4176:
4175:
4174:
4164:
4157:
4156:
4155:
4150:
4140:
4135:
4127:
4125:
4117:
4116:
4113:
4112:
4110:
4109:
4103:
4101:
4092:
4086:
4085:
4082:
4081:
4079:
4078:
4073:
4068:
4060:
4052:
4044:
4035:
4026:
4017:
4008:
3999:
3994:
3989:
3984:
3978:
3976:
3970:
3969:
3967:
3966:
3961:
3959:Variance-gamma
3956:
3951:
3943:
3938:
3933:
3928:
3923:
3918:
3910:
3905:
3904:
3903:
3893:
3888:
3883:
3877:
3872:
3867:
3862:
3857:
3852:
3847:
3839:
3834:
3826:
3821:
3815:
3813:
3805:
3804:
3802:
3801:
3799:Wilks's lambda
3796:
3795:
3794:
3784:
3779:
3774:
3769:
3764:
3759:
3754:
3749:
3744:
3739:
3737:Mittag-Leffler
3734:
3729:
3724:
3719:
3714:
3709:
3704:
3699:
3694:
3689:
3684:
3679:
3678:
3677:
3667:
3658:
3653:
3648:
3647:
3646:
3636:
3634:gamma/Gompertz
3631:
3630:
3629:
3624:
3614:
3609:
3604:
3603:
3602:
3590:
3589:
3588:
3583:
3578:
3568:
3567:
3566:
3556:
3551:
3546:
3545:
3544:
3543:
3542:
3532:
3522:
3517:
3512:
3507:
3502:
3497:
3491:
3489:
3486:semi-infinite
3481:
3480:
3478:
3477:
3472:
3467:
3462:
3457:
3452:
3447:
3442:
3437:
3432:
3427:
3422:
3417:
3412:
3407:
3402:
3397:
3392:
3386:
3384:
3375:
3369:
3368:
3365:
3364:
3362:
3361:
3356:
3351:
3346:
3341:
3336:
3331:
3326:
3321:
3316:
3311:
3306:
3301:
3296:
3291:
3286:
3281:
3276:
3270:
3268:
3265:with infinite
3262:
3261:
3259:
3258:
3253:
3248:
3243:
3238:
3233:
3228:
3227:
3226:
3219:Hypergeometric
3216:
3211:
3206:
3201:
3196:
3190:
3188:
3179:
3173:
3172:
3160:
3159:
3152:
3145:
3137:
3131:
3130:
3117:
3116:External links
3114:
3111:
3110:
3085:
3060:
3035:
3033:
3032:
3003:
2997:
2979:
2973:
2950:
2948:
2947:
2941:
2920:
2915:
2891:
2884:
2878:. p. 80.
2859:
2858:
2856:
2853:
2852:
2851:
2843:
2841:Standard score
2838:
2833:
2823:
2820:
2817:
2816:
2799:
2793:
2789:
2784:
2780:
2777:
2774:
2771:
2767:
2753:
2737:
2731:
2727:
2722:
2718:
2715:
2712:
2709:
2705:
2692:
2689:
2676:
2670:
2666:
2661:
2657:
2654:
2651:
2648:
2637:
2624:
2618:
2614:
2609:
2605:
2602:
2591:
2576:
2575:
2560:
2543:
2540:
2537:
2520:
2510:
2509:
2498:
2481:
2478:
2475:
2458:
2448:
2447:
2440:
2426:
2423:
2420:
2403:
2393:
2392:
2385:
2371:
2368:
2365:
2348:
2338:
2337:
2330:
2319:
2316:
2313:
2296:
2286:
2285:
2271:
2260:
2257:
2254:
2237:
2227:
2226:
2215:
2204:
2201:
2198:
2181:
2171:
2170:
2163:
2155:
2152:
2149:
2132:
2122:
2121:
2118:
2110:
2107:
2104:
2087:
2077:
2076:
2073:
2070:
2067:
2064:
2047:
2037:
2036:
2033:
2030:
2027:
2024:
2007:
1997:
1996:
1993:
1990:
1987:
1984:
1967:
1957:
1956:
1953:
1950:
1947:
1944:
1927:
1917:
1916:
1913:
1910:
1907:
1904:
1887:
1877:
1876:
1873:
1870:
1867:
1864:
1847:
1837:
1836:
1833:
1830:
1827:
1824:
1807:
1795:
1794:
1791:
1786:
1780:
1774:
1760:
1757:
1715:The Black Swan
1649:normality test
1638:Normality test
1636:Main article:
1633:
1630:
1617:
1594:
1591:
1565:
1561:
1556:
1553:
1547:
1544:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1400:
1396:
1385:
1381:
1377:
1369:
1358:
1354:
1353:
1349:
1340:standard score
1323:
1306:
1303:
1287:
1284:
1281:
1278:
1271:
1266:
1262:
1256:
1252:
1246:
1241:
1238:
1234:
1227:
1224:
1220:
1215:
1212:
1210:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1165:
1162:
1159:
1156:
1153:
1146:
1141:
1137:
1131:
1127:
1121:
1116:
1113:
1109:
1102:
1099:
1095:
1090:
1087:
1085:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1040:
1037:
1034:
1031:
1028:
1021:
1016:
1012:
1006:
1002:
996:
991:
988:
984:
977:
974:
970:
965:
962:
960:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
928:
925:
922:
919:
916:
915:
893:
890:
887:
884:
881:
878:
875:
855:
835:
813:
806:
803:
796:
791:
787:
781:
777:
771:
766:
763:
759:
752:
749:
745:
740:
739:
715:
711:
708:
705:
699:
696:
686:standard score
665:
662:
659:
652:
647:
642:
638:
635:
632:
626:
619:
616:
611:
607:
600:
595:
592:
586:
579:
576:
573:
570:
565:
562:
559:
556:
552:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
502:
493:We have that
490:
487:
403:
400:
397:
394:
392:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
347:
344:
341:
338:
335:
333:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
288:
285:
282:
279:
276:
274:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
233:
230:
229:
163:empirical rule
143:standard score
117:
116:
99:September 2023
31:
29:
22:
15:
9:
6:
4:
3:
2:
4553:
4542:
4539:
4537:
4534:
4532:
4529:
4528:
4526:
4511:
4503:
4501:
4493:
4492:
4489:
4483:
4480:
4478:
4475:
4473:
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4438:
4435:
4433:
4430:
4429:
4427:
4423:
4417:
4414:
4411:
4407:
4405:
4402:
4399:
4395:
4394:
4392:
4390:
4385:
4381:
4375:
4372:
4370:
4367:
4364:
4360:
4358:
4355:
4352:
4348:
4346:
4343:
4340:
4336:
4334:
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4298:
4297:
4291:
4290:
4288:
4286:
4282:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4255:
4254:
4251:
4247:
4244:
4243:
4242:
4239:
4237:
4236:
4231:
4229:
4228:Matrix normal
4226:
4224:
4221:
4218:
4217:
4212:
4208:
4205:
4204:
4203:
4200:
4198:
4197:
4194:Multivariate
4192:
4190:
4187:
4185:
4182:
4180:
4177:
4173:
4170:
4169:
4168:
4165:
4162:
4158:
4154:
4151:
4149:
4146:
4145:
4144:
4141:
4139:
4136:
4133:
4129:
4128:
4126:
4124:
4121:Multivariate
4118:
4108:
4105:
4104:
4102:
4096:
4093:
4087:
4077:
4074:
4072:
4069:
4067:
4065:
4061:
4059:
4057:
4053:
4051:
4049:
4045:
4043:
4041:
4036:
4034:
4032:
4027:
4025:
4023:
4018:
4016:
4014:
4009:
4007:
4005:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3979:
3977:
3973:with support
3971:
3965:
3962:
3960:
3957:
3955:
3952:
3950:
3949:
3944:
3942:
3939:
3937:
3934:
3932:
3929:
3927:
3924:
3922:
3919:
3917:
3916:
3911:
3909:
3906:
3902:
3899:
3898:
3897:
3894:
3892:
3889:
3887:
3886:
3878:
3876:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3845:
3840:
3838:
3835:
3833:
3832:
3827:
3825:
3822:
3820:
3817:
3816:
3814:
3810:on the whole
3806:
3800:
3797:
3793:
3790:
3789:
3788:
3785:
3783:
3782:type-2 Gumbel
3780:
3778:
3775:
3773:
3770:
3768:
3765:
3763:
3760:
3758:
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3723:
3720:
3718:
3715:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3676:
3673:
3672:
3671:
3668:
3666:
3664:
3659:
3657:
3654:
3652:
3651:Half-logistic
3649:
3645:
3642:
3641:
3640:
3637:
3635:
3632:
3628:
3625:
3623:
3620:
3619:
3618:
3615:
3613:
3610:
3608:
3607:Folded normal
3605:
3601:
3598:
3597:
3596:
3595:
3591:
3587:
3584:
3582:
3579:
3577:
3574:
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3569:
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3562:
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3528:
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3508:
3506:
3503:
3501:
3498:
3496:
3493:
3492:
3490:
3482:
3476:
3473:
3471:
3468:
3466:
3463:
3461:
3458:
3456:
3453:
3451:
3450:Raised cosine
3448:
3446:
3443:
3441:
3438:
3436:
3433:
3431:
3428:
3426:
3423:
3421:
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3387:
3385:
3379:
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3370:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3340:
3337:
3335:
3332:
3330:
3327:
3325:
3324:Mixed Poisson
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3295:
3292:
3290:
3287:
3285:
3282:
3280:
3277:
3275:
3272:
3271:
3269:
3263:
3257:
3254:
3252:
3249:
3247:
3244:
3242:
3239:
3237:
3234:
3232:
3229:
3225:
3222:
3221:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3204:Beta-binomial
3202:
3200:
3197:
3195:
3192:
3191:
3189:
3183:
3180:
3174:
3169:
3165:
3158:
3153:
3151:
3146:
3144:
3139:
3138:
3135:
3128:
3126:
3120:
3119:
3106:
3105:
3099:
3095:
3089:
3081:
3080:
3074:
3070:
3064:
3056:
3055:
3049:
3045:
3039:
3029:
3025:
3021:
3017:
3013:
3009:
3004:
3000:
2998:9780898713947
2994:
2990:
2989:
2984:
2980:
2976:
2974:9780945320135
2970:
2967:. SPC Press.
2966:
2965:
2959:
2958:
2954:
2944:
2942:9783110162165
2938:
2934:
2929:
2928:
2921:
2918:
2916:9780071398763
2912:
2907:
2906:
2900:
2899:
2895:
2887:
2885:9780190845414
2881:
2877:
2873:
2872:
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2649:
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2638:
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2622:
2616:
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2607:
2603:
2600:
2592:
2590:
2586:
2581:
2578:
2577:
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2569:
2565:
2561:
2544:
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2538:
2521:
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2515:
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2503:
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2317:
2314:
2297:
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2291:
2288:
2287:
2283:
2272:
2261:
2258:
2255:
2238:
2236:
2232:
2229:
2228:
2224:
2216:
2205:
2202:
2199:
2182:
2180:
2176:
2173:
2172:
2168:
2164:
2156:
2153:
2150:
2133:
2131:
2127:
2124:
2123:
2119:
2111:
2108:
2105:
2088:
2086:
2082:
2079:
2078:
2074:
2071:
2068:
2065:
2048:
2046:
2042:
2039:
2038:
2034:
2031:
2028:
2025:
2008:
2006:
2002:
1999:
1998:
1994:
1991:
1988:
1985:
1968:
1966:
1962:
1959:
1958:
1954:
1951:
1948:
1945:
1928:
1926:
1922:
1919:
1918:
1914:
1911:
1908:
1905:
1888:
1886:
1882:
1879:
1878:
1874:
1871:
1868:
1865:
1848:
1846:
1842:
1839:
1838:
1834:
1831:
1828:
1825:
1808:
1805:
1801:
1797:
1796:
1792:
1785:
1781:
1779:
1775:
1772:
1771:
1768:
1766:
1756:
1754:
1750:
1746:
1742:
1738:
1734:
1729:
1725:
1721:
1717:
1716:
1710:
1707:
1703:
1699:
1695:
1690:
1688:
1683:
1680:
1676:
1671:
1669:
1665:
1664:standardizing
1661:
1658:, either the
1657:
1652:
1650:
1646:
1639:
1629:
1615:
1589:
1563:
1559:
1554:
1551:
1542:
1531:
1526:
1513:
1510:
1504:
1501:
1498:
1492:
1489:
1486:
1480:
1477:
1468:
1462:
1453:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1367:For example,
1365:
1344:
1341:
1336:
1334:
1316:
1311:
1302:
1285:
1282:
1279:
1276:
1269:
1264:
1260:
1254:
1250:
1244:
1239:
1236:
1232:
1225:
1222:
1218:
1213:
1211:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1160:
1157:
1154:
1151:
1144:
1139:
1135:
1129:
1125:
1119:
1114:
1111:
1107:
1100:
1097:
1093:
1088:
1086:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1035:
1032:
1029:
1026:
1019:
1014:
1010:
1004:
1000:
994:
989:
986:
982:
975:
972:
968:
963:
961:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
905:
891:
888:
885:
882:
879:
876:
873:
853:
833:
824:
811:
804:
801:
794:
789:
785:
779:
775:
769:
764:
761:
757:
750:
747:
743:
729:
713:
709:
706:
703:
697:
694:
687:
683:
663:
660:
657:
650:
645:
640:
636:
633:
630:
624:
617:
614:
609:
605:
598:
593:
590:
584:
577:
574:
571:
568:
563:
560:
557:
554:
550:
546:
540:
537:
534:
531:
528:
525:
522:
519:
516:
513:
510:
486:
484:
480:
476:
471:
469:
465:
461:
457:
453:
448:
446:
442:
438:
430:
426:
421:
418:
398:
395:
393:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
339:
336:
334:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
280:
277:
275:
267:
264:
261:
258:
255:
252:
249:
246:
243:
240:
237:
219:
216:
210:
205:
200:
195:
186:
184:
180:
176:
172:
168:
164:
160:
156:
148:
144:
140:
135:
128:
123:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
4409:
4397:
4363:Multivariate
4362:
4350:
4338:
4333:Wrapped Lévy
4293:
4241:Matrix gamma
4234:
4214:
4202:Normal-gamma
4195:
4161:Continuous:
4160:
4131:
4076:Tukey lambda
4063:
4055:
4050:-exponential
4047:
4039:
4030:
4021:
4012:
4006:-exponential
4003:
3947:
3914:
3881:
3843:
3830:
3757:Poly-Weibull
3702:Log-logistic
3662:
3661:Hotelling's
3593:
3435:Logit-normal
3309:Gauss–Kuzmin
3304:Flory–Schulz
3185:with finite
3124:
3101:
3088:
3076:
3063:
3051:
3038:
3014:(2): 88–91.
3011:
3007:
2987:
2963:
2953:
2926:
2904:
2894:
2874:. New York:
2870:
2863:
2846:
2828:
2588:
2584:
2579:
2517:
2513:
2455:
2451:
2400:
2396:
2345:
2341:
2293:
2289:
2234:
2230:
2178:
2174:
2129:
2125:
2084:
2080:
2044:
2040:
2004:
2000:
1964:
1960:
1924:
1920:
1884:
1880:
1844:
1840:
1803:
1799:
1783:
1777:
1762:
1727:
1724:Black Monday
1713:
1711:
1705:
1693:
1691:
1684:
1672:
1668:studentizing
1653:
1641:
1527:
1372:(2) ≈ 0.9772
1366:
1342:
1337:
1330:
906:
825:
730:
492:
472:
449:
432:
428:
422:
419:
220:
187:
166:
162:
158:
152:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
4447:Exponential
4296:directional
4285:Directional
4172:Generalized
4143:Multinomial
4098:continuous-
4038:Kaniadakis
4029:Kaniadakis
4020:Kaniadakis
4011:Kaniadakis
4002:Kaniadakis
3954:Tracy–Widom
3931:Skew normal
3913:Noncentral
3697:Log-Laplace
3675:Generalized
3656:Half-normal
3622:Generalized
3586:Logarithmic
3571:Exponential
3525:Chi-squared
3465:U-quadratic
3430:Kumaraswamy
3372:Continuous
3319:Logarithmic
3214:Categorical
2506:Local Group
1679:sample size
456:significant
445:probability
4525:Categories
4442:Elliptical
4398:Degenerate
4384:Degenerate
4132:Discrete:
4091:univariate
3946:Student's
3901:Asymmetric
3880:Johnson's
3808:supported
3752:Phase-type
3707:Log-normal
3692:Log-Cauchy
3682:Kolmogorov
3600:Noncentral
3530:Noncentral
3510:Beta prime
3460:Triangular
3455:Reciprocal
3425:Irwin–Hall
3374:univariate
3354:Yule–Simon
3236:Rademacher
3178:univariate
2855:References
2849:-statistic
2167:modern era
1995:Quarterly
1388:) ≈ 0.9772
1347:(1 − (1 −
728:, we have
680:doing the
155:statistics
69:newspapers
4167:Dirichlet
4148:Dirichlet
4058:-Gaussian
4033:-Logistic
3870:Holtsmark
3842:Gaussian
3829:Fisher's
3812:real line
3314:Geometric
3294:Delaporte
3199:Bernoulli
3176:Discrete
2779:
2773:−
2717:
2711:−
2656:
2650:−
2604:
2572:red dwarf
2334:humankind
1656:deviation
1593:¯
1560:σ
1552:±
1546:¯
1511:≈
1502:−
1493:−
1487:≈
1478:−
1472:Φ
1469:−
1457:Φ
1448:σ
1439:μ
1436:≤
1430:≤
1427:σ
1421:−
1418:μ
1362:(z)) · 2)
1283:≈
1255:−
1237:−
1233:∫
1226:π
1204:σ
1195:μ
1192:≤
1186:≤
1183:σ
1177:−
1174:μ
1158:≈
1130:−
1112:−
1108:∫
1101:π
1079:σ
1070:μ
1067:≤
1061:≤
1058:σ
1052:−
1049:μ
1033:≈
1005:−
987:−
983:∫
976:π
954:σ
945:μ
942:≤
936:≤
933:σ
927:−
924:μ
854:σ
834:μ
780:−
762:−
758:∫
751:π
714:σ
710:μ
707:−
641:σ
637:μ
634:−
610:−
599:σ
594:π
578:σ
569:μ
564:σ
558:−
555:μ
551:∫
541:σ
532:μ
529:≤
523:≤
520:σ
514:−
511:μ
468:discovery
458:" if its
441:heuristic
402:%
396:≈
386:σ
377:μ
374:≤
368:≤
365:σ
359:−
356:μ
343:%
337:≈
327:σ
318:μ
315:≤
309:≤
306:σ
300:−
297:μ
284:%
278:≈
268:σ
259:μ
256:≤
250:≤
247:σ
241:−
238:μ
4500:Category
4432:Circular
4425:Families
4410:Singular
4389:singular
4153:Negative
4100:discrete
4066:-Weibull
4024:-Weibull
3908:Logistic
3792:Discrete
3762:Rayleigh
3742:Nakagami
3665:-squared
3639:Gompertz
3488:interval
3224:Negative
3209:Binomial
2822:See also
2568:Universe
2564:trillion
2502:Universe
1645:outliers
1327:) 1
145:(on the
4510:Commons
4482:Wrapped
4477:Tweedie
4472:Pearson
4467:Mixture
4374:Bingham
4273:Complex
4263:Inverse
4253:Wishart
4246:Inverse
4233:Matrix
4207:Inverse
4123:(joint)
4042:-Erlang
3896:Laplace
3787:Weibull
3644:Shifted
3627:Inverse
3612:Fréchet
3535:Inverse
3470:Uniform
3390:Arcsine
3349:Skellam
3344:Poisson
3267:support
3241:Soliton
3194:Benford
3187:support
3096:(ed.).
3071:(ed.).
3046:(ed.).
3028:2684253
2035:Yearly
1915:Weekly
1286:0.9973.
450:In the
423:In the
192:is the
181:of the
83:scholar
4416:Cantor
4258:Normal
4089:Mixed
4015:-Gamma
3941:Stable
3891:Landau
3865:Gumbel
3819:Cauchy
3747:Pareto
3559:Erlang
3540:Scaled
3495:Benini
3334:Panjer
3127:sigmas
3026:
2995:
2971:
2939:
2913:
2882:
2831:-value
2755:Every
2273:Every
2217:Every
1773:Range
1514:0.9545
1505:0.9772
1490:0.9772
1161:0.9545
1036:0.6827
157:, the
147:x-axis
139:y-axis
85:
78:
71:
64:
56:
4138:Ewens
3964:Voigt
3936:Slash
3717:Lomax
3712:Log-t
3617:Gamma
3564:Hyper
3554:Davis
3549:Dagum
3405:Bates
3395:ARGUS
3279:Borel
3024:JSTOR
2957:See:
2815:days
2523:0.999
2461:0.999
2454:± 7.5
2406:0.999
2351:0.999
2344:± 6.5
2299:0.999
2240:0.999
2233:± 5.5
2184:0.999
2135:0.999
2128:± 4.5
2090:0.999
2072:2149
2050:0.999
2043:± 3.5
2010:0.997
1970:0.987
1963:± 2.5
1930:0.954
1890:0.866
1883:± 1.5
1850:0.682
1810:0.382
1802:± 0.5
1374:, or
489:Proof
399:99.73
340:95.45
281:68.27
173:in a
90:JSTOR
76:books
4387:and
4345:Kent
3772:Rice
3687:Lévy
3515:Burr
3445:PERT
3410:Beta
3359:Zeta
3251:Zipf
3168:list
3102:The
3077:The
3052:The
2993:ISBN
2969:ISBN
2937:ISBN
2911:ISBN
2880:ISBN
2219:4776
2032:370
846:and
437:rule
431:(or
190:Pr()
183:mean
62:news
4223:LKJ
3520:Chi
3016:doi
2933:553
2776:erf
2714:erf
2653:erf
2601:erf
2557:348
2554:655
2551:397
2548:734
2546:803
2534:999
2531:999
2528:999
2525:999
2516:± 8
2495:101
2492:204
2489:601
2486:669
2472:936
2469:999
2466:999
2463:999
2437:445
2434:215
2431:682
2429:390
2417:440
2414:997
2411:999
2408:999
2399:± 7
2382:393
2379:197
2376:450
2362:680
2359:919
2356:999
2353:999
2327:346
2324:797
2322:506
2310:825
2307:026
2304:998
2301:999
2292:± 6
2277:090
2268:254
2265:330
2251:875
2248:020
2245:962
2242:999
2212:278
2209:744
2195:856
2192:696
2189:426
2186:999
2177:± 5
2160:160
2158:147
2146:751
2143:653
2140:204
2137:993
2115:787
2101:334
2098:516
2095:657
2092:936
2083:± 4
2061:929
2058:841
2055:741
2052:534
2021:740
2018:936
2015:203
2012:300
2003:± 3
1992:81
1981:448
1978:348
1975:669
1972:580
1952:22
1941:642
1938:103
1935:736
1932:499
1923:± 2
1912:15
1901:284
1898:462
1895:597
1892:385
1861:086
1858:137
1855:492
1852:689
1821:026
1818:548
1815:922
1812:924
1712:In
1399:+ 2
1384:+ 2
1376:Pr(
167:3sr
153:In
45:by
4527::
3100:.
3075:.
3050:.
3022:.
3012:48
3010:.
2935:.
2582:±
2574:)
2484:15
2446:)
2391:)
2374:12
2336:)
2284:)
2275:72
2263:26
2225:)
2169:)
2113:15
1872:3
1843:±
1832:5
1755:.
1718:,
1628:.
1412:Pr
1380:≤
1364:.
1335:.
1168:Pr
1043:Pr
918:Pr
904:.
505:Pr
470:.
350:Pr
291:Pr
232:Pr
206:,
196:,
4235:t
4196:t
4064:q
4056:q
4048:q
4040:κ
4031:κ
4022:κ
4013:κ
4004:κ
3948:t
3915:t
3884:U
3882:S
3844:q
3831:z
3663:T
3594:F
3170:)
3166:(
3156:e
3149:t
3142:v
3125:x
3121:"
3030:.
3018::
3001:.
2977:.
2945:.
2888:.
2847:t
2829:p
2798:)
2792:2
2788:x
2783:(
2770:1
2766:1
2736:)
2730:2
2726:x
2721:(
2708:1
2704:1
2675:)
2669:2
2665:x
2660:(
2647:1
2623:)
2617:2
2613:x
2608:(
2589:σ
2585:x
2580:μ
2518:σ
2514:μ
2456:σ
2452:μ
2401:σ
2397:μ
2346:σ
2342:μ
2294:σ
2290:μ
2235:σ
2231:μ
2207:1
2179:σ
2175:μ
2130:σ
2126:μ
2085:σ
2081:μ
2045:σ
2041:μ
2005:σ
2001:μ
1965:σ
1961:μ
1925:σ
1921:μ
1885:σ
1881:μ
1845:σ
1841:μ
1804:σ
1800:μ
1728:σ
1706:σ
1694:σ
1616:n
1590:X
1564:n
1555:2
1543:X
1508:)
1499:1
1496:(
1484:)
1481:2
1475:(
1466:)
1463:2
1460:(
1454:=
1451:)
1445:2
1442:+
1433:X
1424:2
1415:(
1401:σ
1397:μ
1386:σ
1382:μ
1378:X
1370:Φ
1359:σ
1357:,
1355:μ
1350:Φ
1343:z
1324:σ
1319:μ
1280:z
1277:d
1270:2
1265:2
1261:z
1251:e
1245:3
1240:3
1223:2
1219:1
1214:=
1207:)
1201:3
1198:+
1189:X
1180:3
1171:(
1155:z
1152:d
1145:2
1140:2
1136:z
1126:e
1120:2
1115:2
1098:2
1094:1
1089:=
1082:)
1076:2
1073:+
1064:X
1055:2
1046:(
1030:z
1027:d
1020:2
1015:2
1011:z
1001:e
995:1
990:1
973:2
969:1
964:=
957:)
951:1
948:+
939:X
930:1
921:(
892:3
889:,
886:2
883:,
880:1
877:=
874:n
812:,
805:z
802:d
795:2
790:2
786:z
776:e
770:n
765:n
748:2
744:1
704:x
698:=
695:z
664:,
661:x
658:d
651:2
646:)
631:x
625:(
618:2
615:1
606:e
591:2
585:1
575:n
572:+
561:n
547:=
544:)
538:n
535:+
526:X
517:n
508:(
435:σ
433:3
389:)
383:3
380:+
371:X
362:3
353:(
330:)
324:2
321:+
312:X
303:2
294:(
271:)
265:1
262:+
253:X
244:1
235:(
215:σ
209:μ
199:Χ
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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