1321:
145:
1311:
920:
133:
4507:
36:
1306:{\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}}}
4517:
427:
689:
140:, 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.
234:
507:
1692:
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.
1535:
1741:
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
833:
422:{\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}}}
2824:
2762:
684:{\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}}}
2697:
2645:
1719:
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.
1653:
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
1588:
1418:
1684:
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
744:
925:
749:
512:
239:
1700:, 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.
1750:, 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
737:
1754:
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
1617:
913:
875:
855:
1637:
2769:
2707:
3165:
1725:
488:, stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. For
2653:
100:
2607:
1688:
and compare these to the expected frequency: points that fall more than 3 standard deviations from the norm are likely outliers (unless the
72:
1711:. For illustration, if events are taken to occur daily, this would correspond to an event expected every 1.4 million years. This gives a
3294:
4520:
3777:
454:
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%
79:
3685:
3114:
3089:
3064:
1774:
Because of the exponentially decreasing tails of the normal distribution, odds of higher deviations decrease very quickly. From the
1415:. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding):
4472:
53:
4338:
3550:
3309:
3158:
1546:
86:
4233:
3997:
3671:
1320:
68:
3992:
3936:
3834:
3596:
3234:
1775:
1343:
4546:
4278:
4012:
3865:
3540:
3284:
493:
3742:
4510:
4182:
4158:
3737:
3151:
1670:
4379:
4256:
4217:
4189:
4163:
4081:
4007:
3430:
3178:
3007:
2983:
2951:
2925:
2894:
119:
1530:{\displaystyle \Pr(\mu -2\sigma \leq X\leq \mu +2\sigma )=\Phi (2)-\Phi (-2)\approx 0.9772-(1-0.9772)\approx 0.9545}
4367:
4333:
4199:
4194:
4039:
3847:
3545:
3299:
1325:
17:
4117:
4030:
4002:
3911:
3860:
3732:
3515:
3480:
701:
828:{\displaystyle {\begin{aligned}{\frac {1}{\sqrt {2\pi }}}\int _{-n}^{n}e^{-{\frac {z^{2}}{2}}}dz\end{aligned}},}
4131:
4048:
3885:
3809:
3632:
3510:
3485:
3349:
3344:
3339:
1759:
1696:
One can compute more precisely, approximating the number of extreme moves of a given magnitude or greater by a
57:
4447:
4313:
4021:
3870:
3802:
3787:
3680:
3654:
3586:
3425:
3319:
3314:
3256:
3241:
93:
4541:
4283:
4273:
3964:
3890:
3591:
3450:
489:
4343:
4328:
4323:
4268:
4204:
4148:
3969:
3956:
3747:
3692:
3644:
3435:
3364:
3229:
1712:
692:
4462:
4238:
4057:
3839:
3792:
3661:
3637:
3617:
3460:
3334:
3214:
1681:(dividing by an estimate of the standard deviation), if the parameters are unknown and only estimated.
1405:. This is not a symmetrical interval – this is merely the probability that an observation is less than
496:. There may be certain assumptions for a distribution that force this probability to be at least 98%.
4467:
4251:
4212:
4086:
3923:
3767:
3712:
3610:
3574:
3445:
3410:
485:
477:, there is a convention of a five-sigma effect (99.99994% confidence) being required to qualify as a
4153:
3941:
3707:
3666:
3581:
3535:
3475:
3440:
3329:
3224:
3174:
466:
204:
4551:
4452:
4394:
4065:
3852:
3762:
3717:
3702:
3520:
3470:
3465:
3266:
3246:
2886:
2399:
478:
46:
3622:
3133:
1593:
4318:
4306:
4295:
4177:
4073:
3880:
3324:
3304:
3209:
2943:
1758:
not just to the hypothesis itself but to all possible alternative hypotheses. For this reason,
1666:
2997:
4442:
4399:
4243:
3918:
3772:
3752:
3649:
3219:
2880:
1743:
1730:
880:
860:
2935:
1677:(dividing by the population standard deviation), if the population parameters are known, or
4492:
4487:
4482:
4477:
4414:
4384:
4263:
3906:
3797:
3400:
3359:
3354:
3251:
1747:
1697:
1685:
1673:
depending on whether one knows the population mean or only estimates it. The next step is
3697:
840:
431:
The usefulness of this heuristic especially depends on the question under consideration.
8:
4426:
3951:
3931:
3901:
3875:
3829:
3757:
3569:
3505:
1734:
1540:
185:
144:
137:
4457:
3946:
3727:
3722:
3627:
3564:
3559:
3415:
3405:
3289:
3034:
2914:
2819:{\displaystyle {\tfrac {1}{1-\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}}}
2757:{\displaystyle {\tfrac {1}{1-\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}}}
1622:
189:
4355:
3782:
3525:
3455:
3420:
3369:
3003:
2979:
2947:
2936:
2921:
2890:
1755:
1708:
435:
181:
3530:
3204:
3143:
3026:
2993:
2454:
2233:
474:
470:
2909:
This usage of "three-sigma rule" entered common usage in the 2000s, e.g. cited in
2846:
2973:
1763:
1751:
1665:
To pass from a sample to a number of standard deviations, one first computes the
462:
214:
193:
188:: approximately 68%, 95%, and 99.7% of the values lie within one, two, and three
160:). The y-axis is logarithmically scaled (but the values on it are not modified).
3603:
2851:
2650:
2604:
1659:
1648:
1350:
696:
153:
180:, is a shorthand used to remember the percentage of values that lie within an
4535:
4226:
3974:
3261:
1762:
works not so much by confirming a hypothesis considered to be likely, but by
1674:
2938:
Linear and
Nonlinear Models: Fixed Effects, Random Effects, and Mixed Models
2344:
2292:
1678:
132:
3104:
3079:
3054:
2856:
2692:{\displaystyle 1-\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}
2516:
1746:. In such discussions it is important to be aware of the problem of the
1689:
455:
199:
In mathematical notation, these facts can be expressed as follows, where
3038:
2640:{\displaystyle \operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)}
2177:
165:
492:, the probability of being within the interval is at least 95% by the
2582:
1403:(1 − (1 − 0.97725)·2) = 0.9545 = 95.45%
451:
3030:
35:
2578:
2574:
2512:
1655:
2838:
157:
149:
3108:
3083:
3058:
2999:
Statistical Case
Studies for Industrial Process Improvement
1583:{\displaystyle {\bar {X}}\pm 2{\frac {\sigma }{\sqrt {n}}}}
1344:
cumulative distribution function of the normal distribution
225:
219:
209:
2511:
Once every 43 billion years (never in the history of the
1733:
gives the example of risk models according to which the
877:. We only need to calculate each integral for the cases
1342:
These numerical values "68%, 95%, 99.7%" come from the
3103:
3078:
3053:
2774:
2712:
473:
is of the order of a two-sigma effect (95%), while in
2772:
2710:
2656:
2610:
1625:
1596:
1549:
1421:
923:
883:
863:
843:
747:
704:
510:
237:
3173:
2882:
2453:Every 1.07 billion years (four occurrences in
2343:Every 1.38 million years (twice in history of
1315:
60:. Unsourced material may be challenged and removed.
2913:
2818:
2756:
2691:
2639:
1631:
1611:
1582:
1529:
1305:
907:
869:
849:
827:
731:
683:
421:
4533:
1804:Approx. frequency outside range for daily event
1590:is approximately a 95% confidence interval when
1422:
1178:
1053:
928:
515:
360:
301:
242:
3017:Pukelsheim, F. (1994). "The Three Sigma Rule".
2971:
2920:. McGraw Hill Professional. 2003. p. 359.
1658:if the population is assumed normal, and as a
484:A weaker three-sigma rule can be derived from
213:is an observation from a normally distributed
3159:
2992:
2398:Every 34 million years (twice since the
1662:if the population is potentially not normal.
1769:
1401:, corresponding to a prediction interval of
1707:event corresponds to a chance of about two
732:{\displaystyle z={\frac {x-\mu }{\sigma }}}
3166:
3152:
3016:
2131:Every 43 years (twice in a lifetime)
223:(mu) is the mean of the distribution, and
3115:On-Line Encyclopedia of Integer Sequences
3090:On-Line Encyclopedia of Integer Sequences
3065:On-Line Encyclopedia of Integer Sequences
2975:Understanding Statistical Process Control
2933:
120:Learn how and when to remove this message
2972:Wheeler, D. J.; Chambers, D. S. (1992).
1319:
143:
131:
3134:Calculate percentage proportion within
2916:Schaum's Outline of Business Statistics
2077:4.653E-04 = 0.04653 % = 465.3 ppm
1764:refuting hypotheses considered unlikely
1328:for the normal distribution with mean (
14:
4534:
3147:
2878:
4516:
2378:8.032E-11 = 0.08032 ppb = 80.32 ppt
837:and this integral is independent of
58:adding citations to reliable sources
29:
2577:years (never in the history of the
2211:5.733E-07 = 0.5733 ppm = 573.3 ppb
2037:2.700E-03 = 0.270 % = 2.700 ‰
1776:rules for normally distributed data
1619:is the average of a sample of size
229:(sigma) is its standard deviation:
24:
2291: years (thrice in history of
2176:Every 403 years (once in the
1642:
1482:
1467:
412:
353:
294:
25:
4563:
3126:
4515:
4506:
4505:
1349:The prediction interval for any
1326:cumulative distribution function
1316:Cumulative distribution function
34:
1737:crash would correspond to a 36-
494:Vysochanskij–Petunin inequality
45:needs additional citations for
3097:
3072:
3047:
2962:
2903:
2872:
1760:statistical hypothesis testing
1603:
1556:
1518:
1506:
1494:
1485:
1476:
1470:
1461:
1425:
1217:
1181:
1092:
1056:
967:
931:
554:
518:
465:, a result may be considered "
399:
363:
340:
304:
281:
245:
13:
1:
2996:; Spagon, Patrick D. (1997).
2942:. Walter de Gruyter. p.
2865:
2847:Six Sigma § Sigma levels
2515:, twice in the future of the
2581:, once during the life of a
176:, and sometimes abbreviated
148:Prediction interval (on the
27:Shorthand used in statistics
7:
2934:Grafarend, Erik W. (2006).
2832:
1356:corresponds numerically to
450:) expresses a conventional
10:
4568:
4547:Statistical approximations
4339:Wrapped asymmetric Laplace
3310:Extended negative binomial
3105:Sloane, N. J. A.
3080:Sloane, N. J. A.
3055:Sloane, N. J. A.
1846:Four or five times a week
1646:
1612:{\displaystyle {\bar {X}}}
4501:
4435:
4393:
4294:
4130:
4108:
4099:
3998:Generalized extreme value
3983:
3818:
3778:Relativistic Breit–Wigner
3494:
3391:
3382:
3275:
3195:
3186:
3175:Probability distributions
1997:1.242E-02 = 1.242 %
1957:4.550E-02 = 4.550 %
1917:1.336E-01 = 13.36 %
1877:3.173E-01 = 31.73 %
1837:6.171E-01 = 61.71 %
1798:
1795:population outside range
1770:Table of numerical values
1752:undermines the hypothesis
440:three-sigma rule of thumb
1801:frequency outside range
1789:population inside range
499:
3993:Generalized chi-squared
3937:Normal-inverse Gaussian
3109:"Sequence A270712"
3084:"Sequence A110894"
3059:"Sequence A178647"
2887:Oxford University Press
2400:extinction of dinosaurs
1886:Twice or thrice a week
1543:as used in statistics:
908:{\displaystyle n=1,2,3}
870:{\displaystyle \sigma }
4305:Univariate (circular)
3866:Generalized hyperbolic
3295:Conway–Maxwell–Poisson
3285:Beta negative binomial
2820:
2758:
2693:
2641:
2550:1.244E-15 = 1.244 ppq
2488:6.382E-14 = 63.82 ppq
2433:2.560E-12 = 2.560 ppt
2326:1.973E-09 = 1.973 ppb
2267:3.798E-08 = 37.98 ppb
2162:6.795E-06 = 6.795 ppm
2117:6.334E-05 = 63.34 ppm
1715:: if one witnesses a 6
1633:
1613:
1584:
1531:
1339:
1307:
909:
871:
851:
829:
733:
685:
490:unimodal distributions
486:Chebyshev's inequality
423:
161:
141:
4350:Bivariate (spherical)
3848:Kaniadakis κ-Gaussian
3019:American Statistician
3002:. SIAM. p. 342.
2879:Huber, Franz (2018).
2821:
2759:
2694:
2642:
2232: years (once in
1793:Expected fraction of
1787:Expected fraction of
1744:stochastic volatility
1731:Nassim Nicholas Taleb
1713:simple normality test
1686:studentized residuals
1634:
1614:
1585:
1532:
1323:
1308:
910:
872:
852:
830:
734:
686:
424:
147:
136:For an approximately
135:
4415:Dirac delta function
4362:Bivariate (toroidal)
4319:Univariate von Mises
4190:Multivariate Laplace
4082:Shifted log-logistic
3431:Continuous Bernoulli
2770:
2708:
2654:
2608:
1698:Poisson distribution
1623:
1594:
1547:
1419:
1324:Diagram showing the
921:
881:
861:
850:{\displaystyle \mu }
841:
745:
702:
508:
235:
205:probability function
172:, also known as the
54:improve this article
4542:Normal distribution
4463:Natural exponential
4368:Bivariate von Mises
4334:Wrapped exponential
4200:Multivariate stable
4195:Multivariate normal
3516:Benktander 2nd kind
3511:Benktander 1st kind
3300:Discrete phase-type
2519:before its merger)
2086:Every 6 years
1778:for a daily event:
1541:confidence interval
1539:This is related to
1259:
1134:
1009:
784:
592:
458:as near certainty.
190:standard deviations
186:normal distribution
4118:Rectified Gaussian
4003:Generalized Pareto
3861:Generalized normal
3733:Matrix-exponential
3118:. OEIS Foundation.
3093:. OEIS Foundation.
3068:. OEIS Foundation.
2816:
2814:
2754:
2752:
2689:
2637:
1966:Every three weeks
1629:
1609:
1580:
1527:
1340:
1332:) 0 and variance (
1303:
1301:
1242:
1117:
992:
905:
867:
847:
825:
820:
767:
729:
693:change of variable
681:
679:
560:
436:empirical sciences
419:
417:
162:
142:
4529:
4528:
4126:
4125:
4095:
4094:
3986:whose type varies
3932:Normal (Gaussian)
3886:Hyperbolic secant
3835:Exponential power
3738:Maxwell–Boltzmann
3486:Wigner semicircle
3378:
3377:
3350:Parabolic fractal
3340:Negative binomial
2994:Czitrom, Veronica
2830:
2829:
2813:
2806:
2805:
2751:
2744:
2743:
2683:
2682:
2631:
2630:
1756:prior probability
1748:gambler's fallacy
1709:parts per billion
1671:error or residual
1632:{\displaystyle n}
1606:
1578:
1577:
1559:
1283:
1240:
1239:
1158:
1115:
1114:
1033:
990:
989:
808:
765:
764:
727:
654:
631:
613:
607:
438:, the so-called
182:interval estimate
152:) given from the
130:
129:
122:
104:
69:"68–95–99.7 rule"
16:(Redirected from
4559:
4519:
4518:
4509:
4508:
4448:Compound Poisson
4423:
4411:
4380:von Mises–Fisher
4376:
4364:
4352:
4314:Circular uniform
4310:
4230:
4174:
4145:
4106:
4105:
4008:Marchenko–Pastur
3871:Geometric stable
3788:Truncated normal
3681:Inverse Gaussian
3587:Hyperexponential
3426:Beta rectangular
3394:bounded interval
3389:
3388:
3257:Discrete uniform
3242:Poisson binomial
3193:
3192:
3168:
3161:
3154:
3145:
3144:
3120:
3119:
3101:
3095:
3094:
3076:
3070:
3069:
3051:
3045:
3042:
3013:
2989:
2966:
2960:
2957:
2941:
2930:
2919:
2907:
2901:
2900:
2876:
2825:
2823:
2822:
2817:
2815:
2812:
2811:
2807:
2801:
2797:
2775:
2763:
2761:
2760:
2755:
2753:
2750:
2749:
2745:
2739:
2735:
2713:
2702:1 in
2698:
2696:
2695:
2690:
2688:
2684:
2678:
2674:
2646:
2644:
2643:
2638:
2636:
2632:
2626:
2622:
2598:
2570:
2569:
2566:
2563:
2560:
2553:1 in
2547:
2546:
2543:
2540:
2537:
2508:
2507:
2504:
2501:
2498:
2491:1 in
2485:
2484:
2481:
2478:
2475:
2455:history of Earth
2450:
2449:
2446:
2443:
2436:1 in
2430:
2429:
2426:
2423:
2420:
2395:
2394:
2391:
2388:
2381:1 in
2375:
2374:
2371:
2368:
2365:
2340:
2339:
2336:
2329:1 in
2323:
2322:
2319:
2316:
2313:
2293:modern humankind
2290:
2289:
2281:
2280:
2277:
2270:1 in
2264:
2263:
2260:
2257:
2254:
2234:recorded history
2231:
2225:
2224:
2221:
2214:1 in
2208:
2207:
2204:
2201:
2198:
2173:
2172:
2165:1 in
2159:
2158:
2155:
2152:
2149:
2128:
2127:
2120:1 in
2114:
2113:
2110:
2107:
2104:
2080:1 in
2074:
2073:
2070:
2067:
2064:
2040:1 in
2034:
2033:
2030:
2027:
2024:
2000:1 in
1994:
1993:
1990:
1987:
1984:
1960:1 in
1954:
1953:
1950:
1947:
1944:
1920:2 in
1914:
1913:
1910:
1907:
1904:
1880:1 in
1874:
1873:
1870:
1867:
1864:
1840:3 in
1834:
1833:
1830:
1827:
1824:
1817:
1799:Approx. expected
1781:
1780:
1703:For example, a 6
1638:
1636:
1635:
1630:
1618:
1616:
1615:
1610:
1608:
1607:
1599:
1589:
1587:
1586:
1581:
1579:
1573:
1569:
1561:
1560:
1552:
1536:
1534:
1533:
1528:
1414:
1404:
1400:
1384:
1374:
1363:
1337:
1331:
1312:
1310:
1309:
1304:
1302:
1286:
1285:
1284:
1279:
1278:
1269:
1258:
1253:
1241:
1232:
1228:
1161:
1160:
1159:
1154:
1153:
1144:
1133:
1128:
1116:
1107:
1103:
1036:
1035:
1034:
1029:
1028:
1019:
1008:
1003:
991:
982:
978:
914:
912:
911:
906:
876:
874:
873:
868:
856:
854:
853:
848:
834:
832:
831:
826:
821:
811:
810:
809:
804:
803:
794:
783:
778:
766:
757:
753:
738:
736:
735:
730:
728:
723:
712:
695:in terms of the
690:
688:
687:
682:
680:
667:
666:
665:
664:
659:
655:
650:
639:
632:
624:
614:
612:
608:
600:
594:
591:
577:
475:particle physics
471:confidence level
447:
428:
426:
425:
420:
418:
228:
222:
212:
202:
196:, respectively.
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
18:Three sigma rule
4567:
4566:
4562:
4561:
4560:
4558:
4557:
4556:
4532:
4531:
4530:
4525:
4497:
4473:Maximum entropy
4431:
4419:
4407:
4397:
4389:
4372:
4360:
4348:
4303:
4290:
4227:Matrix-valued:
4224:
4170:
4141:
4133:
4122:
4110:
4101:
4091:
3985:
3979:
3896:
3822:
3820:
3814:
3743:Maxwell–Jüttner
3592:Hypoexponential
3498:
3496:
3495:supported on a
3490:
3451:Noncentral beta
3411:Balding–Nichols
3393:
3392:supported on a
3384:
3374:
3277:
3271:
3267:Zipf–Mandelbrot
3197:
3188:
3182:
3172:
3140:at WolframAlpha
3129:
3124:
3123:
3102:
3098:
3077:
3073:
3052:
3048:
3031:10.2307/2684253
3010:
2986:
2967:
2963:
2954:
2928:
2912:
2908:
2904:
2897:
2877:
2873:
2868:
2835:
2796:
2792:
2779:
2773:
2771:
2768:
2767:
2734:
2730:
2717:
2711:
2709:
2706:
2705:
2673:
2669:
2655:
2652:
2651:
2621:
2617:
2609:
2606:
2605:
2594:
2573:Once every 2.2
2567:
2564:
2561:
2558:
2556:
2544:
2541:
2538:
2535:
2533:
2505:
2502:
2499:
2496:
2494:
2482:
2479:
2476:
2473:
2471:
2447:
2444:
2441:
2439:
2427:
2424:
2421:
2418:
2416:
2392:
2389:
2386:
2384:
2372:
2369:
2366:
2363:
2361:
2337:
2334:
2332:
2320:
2317:
2314:
2311:
2309:
2287:
2285:
2278:
2275:
2273:
2261:
2258:
2255:
2252:
2250:
2229:
2222:
2219:
2217:
2205:
2202:
2199:
2196:
2194:
2170:
2168:
2156:
2153:
2150:
2147:
2145:
2125:
2123:
2111:
2108:
2105:
2102:
2100:
2071:
2068:
2065:
2062:
2060:
2031:
2028:
2025:
2022:
2020:
1991:
1988:
1985:
1982:
1980:
1951:
1948:
1945:
1942:
1940:
1911:
1908:
1905:
1902:
1900:
1871:
1868:
1865:
1862:
1860:
1831:
1828:
1825:
1822:
1820:
1809:
1800:
1772:
1651:
1645:
1643:Normality tests
1624:
1621:
1620:
1598:
1597:
1595:
1592:
1591:
1568:
1551:
1550:
1548:
1545:
1544:
1420:
1417:
1416:
1413:
1409:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1379:
1372:
1371:
1367:
1362:
1359:
1357:
1336:
1333:
1329:
1318:
1300:
1299:
1274:
1270:
1268:
1264:
1260:
1254:
1246:
1227:
1220:
1175:
1174:
1149:
1145:
1143:
1139:
1135:
1129:
1121:
1102:
1095:
1050:
1049:
1024:
1020:
1018:
1014:
1010:
1004:
996:
977:
970:
924:
922:
919:
918:
882:
879:
878:
862:
859:
858:
842:
839:
838:
819:
818:
799:
795:
793:
789:
785:
779:
771:
752:
748:
746:
743:
742:
713:
711:
703:
700:
699:
678:
677:
660:
640:
638:
634:
633:
623:
619:
615:
599:
598:
593:
578:
564:
511:
509:
506:
505:
502:
463:social sciences
445:
416:
415:
402:
357:
356:
343:
298:
297:
284:
238:
236:
233:
232:
224:
218:
215:random variable
208:
200:
170:68–95–99.7 rule
138:normal data set
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
4565:
4555:
4554:
4552:Rules of thumb
4549:
4544:
4527:
4526:
4524:
4523:
4513:
4502:
4499:
4498:
4496:
4495:
4490:
4485:
4480:
4475:
4470:
4468:Location–scale
4465:
4460:
4455:
4450:
4445:
4439:
4437:
4433:
4432:
4430:
4429:
4424:
4417:
4412:
4404:
4402:
4391:
4390:
4388:
4387:
4382:
4377:
4370:
4365:
4358:
4353:
4346:
4341:
4336:
4331:
4329:Wrapped Cauchy
4326:
4324:Wrapped normal
4321:
4316:
4311:
4300:
4298:
4292:
4291:
4289:
4288:
4287:
4286:
4281:
4279:Normal-inverse
4276:
4271:
4261:
4260:
4259:
4249:
4241:
4236:
4231:
4222:
4221:
4220:
4210:
4202:
4197:
4192:
4187:
4186:
4185:
4175:
4168:
4167:
4166:
4161:
4151:
4146:
4138:
4136:
4128:
4127:
4124:
4123:
4121:
4120:
4114:
4112:
4103:
4097:
4096:
4093:
4092:
4090:
4089:
4084:
4079:
4071:
4063:
4055:
4046:
4037:
4028:
4019:
4010:
4005:
4000:
3995:
3989:
3987:
3981:
3980:
3978:
3977:
3972:
3970:Variance-gamma
3967:
3962:
3954:
3949:
3944:
3939:
3934:
3929:
3921:
3916:
3915:
3914:
3904:
3899:
3894:
3888:
3883:
3878:
3873:
3868:
3863:
3858:
3850:
3845:
3837:
3832:
3826:
3824:
3816:
3815:
3813:
3812:
3810:Wilks's lambda
3807:
3806:
3805:
3795:
3790:
3785:
3780:
3775:
3770:
3765:
3760:
3755:
3750:
3748:Mittag-Leffler
3745:
3740:
3735:
3730:
3725:
3720:
3715:
3710:
3705:
3700:
3695:
3690:
3689:
3688:
3678:
3669:
3664:
3659:
3658:
3657:
3647:
3645:gamma/Gompertz
3642:
3641:
3640:
3635:
3625:
3620:
3615:
3614:
3613:
3601:
3600:
3599:
3594:
3589:
3579:
3578:
3577:
3567:
3562:
3557:
3556:
3555:
3554:
3553:
3543:
3533:
3528:
3523:
3518:
3513:
3508:
3502:
3500:
3497:semi-infinite
3492:
3491:
3489:
3488:
3483:
3478:
3473:
3468:
3463:
3458:
3453:
3448:
3443:
3438:
3433:
3428:
3423:
3418:
3413:
3408:
3403:
3397:
3395:
3386:
3380:
3379:
3376:
3375:
3373:
3372:
3367:
3362:
3357:
3352:
3347:
3342:
3337:
3332:
3327:
3322:
3317:
3312:
3307:
3302:
3297:
3292:
3287:
3281:
3279:
3276:with infinite
3273:
3272:
3270:
3269:
3264:
3259:
3254:
3249:
3244:
3239:
3238:
3237:
3230:Hypergeometric
3227:
3222:
3217:
3212:
3207:
3201:
3199:
3190:
3184:
3183:
3171:
3170:
3163:
3156:
3148:
3142:
3141:
3128:
3127:External links
3125:
3122:
3121:
3096:
3071:
3046:
3044:
3043:
3014:
3008:
2990:
2984:
2961:
2959:
2958:
2952:
2931:
2926:
2902:
2895:
2889:. p. 80.
2870:
2869:
2867:
2864:
2863:
2862:
2854:
2852:Standard score
2849:
2844:
2834:
2831:
2828:
2827:
2810:
2804:
2800:
2795:
2791:
2788:
2785:
2782:
2778:
2764:
2748:
2742:
2738:
2733:
2729:
2726:
2723:
2720:
2716:
2703:
2700:
2687:
2681:
2677:
2672:
2668:
2665:
2662:
2659:
2648:
2635:
2629:
2625:
2620:
2616:
2613:
2602:
2587:
2586:
2571:
2554:
2551:
2548:
2531:
2521:
2520:
2509:
2492:
2489:
2486:
2469:
2459:
2458:
2451:
2437:
2434:
2431:
2414:
2404:
2403:
2396:
2382:
2379:
2376:
2359:
2349:
2348:
2341:
2330:
2327:
2324:
2307:
2297:
2296:
2282:
2271:
2268:
2265:
2248:
2238:
2237:
2226:
2215:
2212:
2209:
2192:
2182:
2181:
2174:
2166:
2163:
2160:
2143:
2133:
2132:
2129:
2121:
2118:
2115:
2098:
2088:
2087:
2084:
2081:
2078:
2075:
2058:
2048:
2047:
2044:
2041:
2038:
2035:
2018:
2008:
2007:
2004:
2001:
1998:
1995:
1978:
1968:
1967:
1964:
1961:
1958:
1955:
1938:
1928:
1927:
1924:
1921:
1918:
1915:
1898:
1888:
1887:
1884:
1881:
1878:
1875:
1858:
1848:
1847:
1844:
1841:
1838:
1835:
1818:
1806:
1805:
1802:
1797:
1791:
1785:
1771:
1768:
1726:The Black Swan
1660:normality test
1649:Normality test
1647:Main article:
1644:
1641:
1628:
1605:
1602:
1576:
1572:
1567:
1564:
1558:
1555:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1411:
1407:
1396:
1392:
1388:
1380:
1369:
1365:
1364:
1360:
1351:standard score
1334:
1317:
1314:
1298:
1295:
1292:
1289:
1282:
1277:
1273:
1267:
1263:
1257:
1252:
1249:
1245:
1238:
1235:
1231:
1226:
1223:
1221:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1176:
1173:
1170:
1167:
1164:
1157:
1152:
1148:
1142:
1138:
1132:
1127:
1124:
1120:
1113:
1110:
1106:
1101:
1098:
1096:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1051:
1048:
1045:
1042:
1039:
1032:
1027:
1023:
1017:
1013:
1007:
1002:
999:
995:
988:
985:
981:
976:
973:
971:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
926:
904:
901:
898:
895:
892:
889:
886:
866:
846:
824:
817:
814:
807:
802:
798:
792:
788:
782:
777:
774:
770:
763:
760:
756:
751:
750:
726:
722:
719:
716:
710:
707:
697:standard score
676:
673:
670:
663:
658:
653:
649:
646:
643:
637:
630:
627:
622:
618:
611:
606:
603:
597:
590:
587:
584:
581:
576:
573:
570:
567:
563:
559:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
513:
504:We have that
501:
498:
414:
411:
408:
405:
403:
401:
398:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
358:
355:
352:
349:
346:
344:
342:
339:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
303:
300:
299:
296:
293:
290:
287:
285:
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
253:
250:
247:
244:
241:
240:
174:empirical rule
154:standard score
128:
127:
110:September 2023
42:
40:
33:
26:
9:
6:
4:
3:
2:
4564:
4553:
4550:
4548:
4545:
4543:
4540:
4539:
4537:
4522:
4514:
4512:
4504:
4503:
4500:
4494:
4491:
4489:
4486:
4484:
4481:
4479:
4476:
4474:
4471:
4469:
4466:
4464:
4461:
4459:
4456:
4454:
4451:
4449:
4446:
4444:
4441:
4440:
4438:
4434:
4428:
4425:
4422:
4418:
4416:
4413:
4410:
4406:
4405:
4403:
4401:
4396:
4392:
4386:
4383:
4381:
4378:
4375:
4371:
4369:
4366:
4363:
4359:
4357:
4354:
4351:
4347:
4345:
4342:
4340:
4337:
4335:
4332:
4330:
4327:
4325:
4322:
4320:
4317:
4315:
4312:
4309:
4308:
4302:
4301:
4299:
4297:
4293:
4285:
4282:
4280:
4277:
4275:
4272:
4270:
4267:
4266:
4265:
4262:
4258:
4255:
4254:
4253:
4250:
4248:
4247:
4242:
4240:
4239:Matrix normal
4237:
4235:
4232:
4229:
4228:
4223:
4219:
4216:
4215:
4214:
4211:
4209:
4208:
4205:Multivariate
4203:
4201:
4198:
4196:
4193:
4191:
4188:
4184:
4181:
4180:
4179:
4176:
4173:
4169:
4165:
4162:
4160:
4157:
4156:
4155:
4152:
4150:
4147:
4144:
4140:
4139:
4137:
4135:
4132:Multivariate
4129:
4119:
4116:
4115:
4113:
4107:
4104:
4098:
4088:
4085:
4083:
4080:
4078:
4076:
4072:
4070:
4068:
4064:
4062:
4060:
4056:
4054:
4052:
4047:
4045:
4043:
4038:
4036:
4034:
4029:
4027:
4025:
4020:
4018:
4016:
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3990:
3988:
3984:with support
3982:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3960:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3930:
3928:
3927:
3922:
3920:
3917:
3913:
3910:
3909:
3908:
3905:
3903:
3900:
3898:
3897:
3889:
3887:
3884:
3882:
3879:
3877:
3874:
3872:
3869:
3867:
3864:
3862:
3859:
3857:
3856:
3851:
3849:
3846:
3844:
3843:
3838:
3836:
3833:
3831:
3828:
3827:
3825:
3821:on the whole
3817:
3811:
3808:
3804:
3801:
3800:
3799:
3796:
3794:
3793:type-2 Gumbel
3791:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3769:
3766:
3764:
3761:
3759:
3756:
3754:
3751:
3749:
3746:
3744:
3741:
3739:
3736:
3734:
3731:
3729:
3726:
3724:
3721:
3719:
3716:
3714:
3711:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3691:
3687:
3684:
3683:
3682:
3679:
3677:
3675:
3670:
3668:
3665:
3663:
3662:Half-logistic
3660:
3656:
3653:
3652:
3651:
3648:
3646:
3643:
3639:
3636:
3634:
3631:
3630:
3629:
3626:
3624:
3621:
3619:
3618:Folded normal
3616:
3612:
3609:
3608:
3607:
3606:
3602:
3598:
3595:
3593:
3590:
3588:
3585:
3584:
3583:
3580:
3576:
3573:
3572:
3571:
3568:
3566:
3563:
3561:
3558:
3552:
3549:
3548:
3547:
3544:
3542:
3539:
3538:
3537:
3534:
3532:
3529:
3527:
3524:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3503:
3501:
3493:
3487:
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3467:
3464:
3462:
3461:Raised cosine
3459:
3457:
3454:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3427:
3424:
3422:
3419:
3417:
3414:
3412:
3409:
3407:
3404:
3402:
3399:
3398:
3396:
3390:
3387:
3381:
3371:
3368:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3335:Mixed Poisson
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3293:
3291:
3288:
3286:
3283:
3282:
3280:
3274:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3250:
3248:
3245:
3243:
3240:
3236:
3233:
3232:
3231:
3228:
3226:
3223:
3221:
3218:
3216:
3215:Beta-binomial
3213:
3211:
3208:
3206:
3203:
3202:
3200:
3194:
3191:
3185:
3180:
3176:
3169:
3164:
3162:
3157:
3155:
3150:
3149:
3146:
3139:
3137:
3131:
3130:
3117:
3116:
3110:
3106:
3100:
3092:
3091:
3085:
3081:
3075:
3067:
3066:
3060:
3056:
3050:
3040:
3036:
3032:
3028:
3024:
3020:
3015:
3011:
3009:9780898713947
3005:
3001:
3000:
2995:
2991:
2987:
2985:9780945320135
2981:
2978:. SPC Press.
2977:
2976:
2970:
2969:
2965:
2955:
2953:9783110162165
2949:
2945:
2940:
2939:
2932:
2929:
2927:9780071398763
2923:
2918:
2917:
2911:
2910:
2906:
2898:
2896:9780190845414
2892:
2888:
2884:
2883:
2875:
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2861:
2859:
2855:
2853:
2850:
2848:
2845:
2843:
2841:
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2808:
2802:
2798:
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2789:
2786:
2783:
2780:
2776:
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2740:
2736:
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2724:
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2714:
2704:
2701:
2699:
2685:
2679:
2675:
2670:
2666:
2663:
2660:
2657:
2649:
2647:
2633:
2627:
2623:
2618:
2614:
2611:
2603:
2601:
2597:
2592:
2589:
2588:
2584:
2580:
2576:
2572:
2555:
2552:
2549:
2532:
2530:
2526:
2523:
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2518:
2514:
2510:
2493:
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2456:
2452:
2438:
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2415:
2413:
2409:
2406:
2405:
2401:
2397:
2383:
2380:
2377:
2360:
2358:
2354:
2351:
2350:
2346:
2342:
2331:
2328:
2325:
2308:
2306:
2302:
2299:
2298:
2294:
2283:
2272:
2269:
2266:
2249:
2247:
2243:
2240:
2239:
2235:
2227:
2216:
2213:
2210:
2193:
2191:
2187:
2184:
2183:
2179:
2175:
2167:
2164:
2161:
2144:
2142:
2138:
2135:
2134:
2130:
2122:
2119:
2116:
2099:
2097:
2093:
2090:
2089:
2085:
2082:
2079:
2076:
2059:
2057:
2053:
2050:
2049:
2045:
2042:
2039:
2036:
2019:
2017:
2013:
2010:
2009:
2005:
2002:
1999:
1996:
1979:
1977:
1973:
1970:
1969:
1965:
1962:
1959:
1956:
1939:
1937:
1933:
1930:
1929:
1925:
1922:
1919:
1916:
1899:
1897:
1893:
1890:
1889:
1885:
1882:
1879:
1876:
1859:
1857:
1853:
1850:
1849:
1845:
1842:
1839:
1836:
1819:
1816:
1812:
1808:
1807:
1803:
1796:
1792:
1790:
1786:
1783:
1782:
1779:
1777:
1767:
1765:
1761:
1757:
1753:
1749:
1745:
1740:
1736:
1732:
1728:
1727:
1721:
1718:
1714:
1710:
1706:
1701:
1699:
1694:
1691:
1687:
1682:
1680:
1676:
1675:standardizing
1672:
1669:, either the
1668:
1663:
1661:
1657:
1650:
1640:
1626:
1600:
1574:
1570:
1565:
1562:
1553:
1542:
1537:
1524:
1521:
1515:
1512:
1509:
1503:
1500:
1497:
1491:
1488:
1479:
1473:
1464:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1378:For example,
1376:
1355:
1352:
1347:
1345:
1327:
1322:
1313:
1296:
1293:
1290:
1287:
1280:
1275:
1271:
1265:
1261:
1255:
1250:
1247:
1243:
1236:
1233:
1229:
1224:
1222:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1171:
1168:
1165:
1162:
1155:
1150:
1146:
1140:
1136:
1130:
1125:
1122:
1118:
1111:
1108:
1104:
1099:
1097:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1046:
1043:
1040:
1037:
1030:
1025:
1021:
1015:
1011:
1005:
1000:
997:
993:
986:
983:
979:
974:
972:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
916:
902:
899:
896:
893:
890:
887:
884:
864:
844:
835:
822:
815:
812:
805:
800:
796:
790:
786:
780:
775:
772:
768:
761:
758:
754:
740:
724:
720:
717:
714:
708:
705:
698:
694:
674:
671:
668:
661:
656:
651:
647:
644:
641:
635:
628:
625:
620:
616:
609:
604:
601:
595:
588:
585:
582:
579:
574:
571:
568:
565:
561:
557:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
497:
495:
491:
487:
482:
480:
476:
472:
468:
464:
459:
457:
453:
449:
441:
437:
432:
429:
409:
406:
404:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
350:
347:
345:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
291:
288:
286:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
230:
227:
221:
216:
211:
206:
197:
195:
191:
187:
183:
179:
175:
171:
167:
159:
155:
151:
146:
139:
134:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
4420:
4408:
4374:Multivariate
4373:
4361:
4349:
4344:Wrapped Lévy
4304:
4252:Matrix gamma
4245:
4225:
4213:Normal-gamma
4206:
4172:Continuous:
4171:
4142:
4087:Tukey lambda
4074:
4066:
4061:-exponential
4058:
4050:
4041:
4032:
4023:
4017:-exponential
4014:
3958:
3925:
3892:
3854:
3841:
3768:Poly-Weibull
3713:Log-logistic
3673:
3672:Hotelling's
3604:
3446:Logit-normal
3320:Gauss–Kuzmin
3315:Flory–Schulz
3196:with finite
3135:
3112:
3099:
3087:
3074:
3062:
3049:
3025:(2): 88–91.
3022:
3018:
2998:
2974:
2964:
2937:
2915:
2905:
2885:. New York:
2881:
2874:
2857:
2839:
2599:
2595:
2590:
2528:
2524:
2466:
2462:
2411:
2407:
2356:
2352:
2304:
2300:
2245:
2241:
2189:
2185:
2140:
2136:
2095:
2091:
2055:
2051:
2015:
2011:
1975:
1971:
1935:
1931:
1895:
1891:
1855:
1851:
1814:
1810:
1794:
1788:
1773:
1738:
1735:Black Monday
1724:
1722:
1716:
1704:
1702:
1695:
1683:
1679:studentizing
1664:
1652:
1538:
1383:(2) ≈ 0.9772
1377:
1353:
1348:
1341:
917:
836:
741:
503:
483:
460:
443:
439:
433:
430:
231:
198:
177:
173:
169:
163:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
4458:Exponential
4307:directional
4296:Directional
4183:Generalized
4154:Multinomial
4109:continuous-
4049:Kaniadakis
4040:Kaniadakis
4031:Kaniadakis
4022:Kaniadakis
4013:Kaniadakis
3965:Tracy–Widom
3942:Skew normal
3924:Noncentral
3708:Log-Laplace
3686:Generalized
3667:Half-normal
3633:Generalized
3597:Logarithmic
3582:Exponential
3536:Chi-squared
3476:U-quadratic
3441:Kumaraswamy
3383:Continuous
3330:Logarithmic
3225:Categorical
2517:Local Group
1690:sample size
467:significant
456:probability
4536:Categories
4453:Elliptical
4409:Degenerate
4395:Degenerate
4143:Discrete:
4102:univariate
3957:Student's
3912:Asymmetric
3891:Johnson's
3819:supported
3763:Phase-type
3718:Log-normal
3703:Log-Cauchy
3693:Kolmogorov
3611:Noncentral
3541:Noncentral
3521:Beta prime
3471:Triangular
3466:Reciprocal
3436:Irwin–Hall
3385:univariate
3365:Yule–Simon
3247:Rademacher
3189:univariate
2866:References
2860:-statistic
2178:modern era
2006:Quarterly
1399:) ≈ 0.9772
1358:(1 − (1 −
739:, we have
691:doing the
166:statistics
80:newspapers
4178:Dirichlet
4159:Dirichlet
4069:-Gaussian
4044:-Logistic
3881:Holtsmark
3853:Gaussian
3840:Fisher's
3823:real line
3325:Geometric
3305:Delaporte
3210:Bernoulli
3187:Discrete
2790:
2784:−
2728:
2722:−
2667:
2661:−
2615:
2583:red dwarf
2345:humankind
1667:deviation
1604:¯
1571:σ
1563:±
1557:¯
1522:≈
1513:−
1504:−
1498:≈
1489:−
1483:Φ
1480:−
1468:Φ
1459:σ
1450:μ
1447:≤
1441:≤
1438:σ
1432:−
1429:μ
1373:(z)) · 2)
1294:≈
1266:−
1248:−
1244:∫
1237:π
1215:σ
1206:μ
1203:≤
1197:≤
1194:σ
1188:−
1185:μ
1169:≈
1141:−
1123:−
1119:∫
1112:π
1090:σ
1081:μ
1078:≤
1072:≤
1069:σ
1063:−
1060:μ
1044:≈
1016:−
998:−
994:∫
987:π
965:σ
956:μ
953:≤
947:≤
944:σ
938:−
935:μ
865:σ
845:μ
791:−
773:−
769:∫
762:π
725:σ
721:μ
718:−
652:σ
648:μ
645:−
621:−
610:σ
605:π
589:σ
580:μ
575:σ
569:−
566:μ
562:∫
552:σ
543:μ
540:≤
534:≤
531:σ
525:−
522:μ
479:discovery
469:" if its
452:heuristic
413:%
407:≈
397:σ
388:μ
385:≤
379:≤
376:σ
370:−
367:μ
354:%
348:≈
338:σ
329:μ
326:≤
320:≤
317:σ
311:−
308:μ
295:%
289:≈
279:σ
270:μ
267:≤
261:≤
258:σ
252:−
249:μ
4511:Category
4443:Circular
4436:Families
4421:Singular
4400:singular
4164:Negative
4111:discrete
4077:-Weibull
4035:-Weibull
3919:Logistic
3803:Discrete
3773:Rayleigh
3753:Nakagami
3676:-squared
3650:Gompertz
3499:interval
3235:Negative
3220:Binomial
2833:See also
2579:Universe
2575:trillion
2513:Universe
1656:outliers
1338:) 1
156:(on the
4521:Commons
4493:Wrapped
4488:Tweedie
4483:Pearson
4478:Mixture
4385:Bingham
4284:Complex
4274:Inverse
4264:Wishart
4257:Inverse
4244:Matrix
4218:Inverse
4134:(joint)
4053:-Erlang
3907:Laplace
3798:Weibull
3655:Shifted
3638:Inverse
3623:Fréchet
3546:Inverse
3481:Uniform
3401:Arcsine
3360:Skellam
3355:Poisson
3278:support
3252:Soliton
3205:Benford
3198:support
3107:(ed.).
3082:(ed.).
3057:(ed.).
3039:2684253
2046:Yearly
1926:Weekly
1297:0.9973.
461:In the
434:In the
203:is the
192:of the
94:scholar
4427:Cantor
4269:Normal
4100:Mixed
4026:-Gamma
3952:Stable
3902:Landau
3876:Gumbel
3830:Cauchy
3758:Pareto
3570:Erlang
3551:Scaled
3506:Benini
3345:Panjer
3138:sigmas
3037:
3006:
2982:
2950:
2924:
2893:
2842:-value
2766:Every
2284:Every
2228:Every
1784:Range
1525:0.9545
1516:0.9772
1501:0.9772
1172:0.9545
1047:0.6827
168:, the
158:x-axis
150:y-axis
96:
89:
82:
75:
67:
4149:Ewens
3975:Voigt
3947:Slash
3728:Lomax
3723:Log-t
3628:Gamma
3575:Hyper
3565:Davis
3560:Dagum
3416:Bates
3406:ARGUS
3290:Borel
3035:JSTOR
2968:See:
2826:days
2534:0.999
2472:0.999
2465:± 7.5
2417:0.999
2362:0.999
2355:± 6.5
2310:0.999
2251:0.999
2244:± 5.5
2195:0.999
2146:0.999
2139:± 4.5
2101:0.999
2083:2149
2061:0.999
2054:± 3.5
2021:0.997
1981:0.987
1974:± 2.5
1941:0.954
1901:0.866
1894:± 1.5
1861:0.682
1821:0.382
1813:± 0.5
1385:, or
500:Proof
410:99.73
351:95.45
292:68.27
184:in a
101:JSTOR
87:books
4398:and
4356:Kent
3783:Rice
3698:Lévy
3526:Burr
3456:PERT
3421:Beta
3370:Zeta
3262:Zipf
3179:list
3113:The
3088:The
3063:The
3004:ISBN
2980:ISBN
2948:ISBN
2922:ISBN
2891:ISBN
2230:4776
2043:370
857:and
448:rule
442:(or
201:Pr()
194:mean
73:news
4234:LKJ
3531:Chi
3027:doi
2944:553
2787:erf
2725:erf
2664:erf
2612:erf
2568:348
2565:655
2562:397
2559:734
2557:803
2545:999
2542:999
2539:999
2536:999
2527:± 8
2506:101
2503:204
2500:601
2497:669
2483:936
2480:999
2477:999
2474:999
2448:445
2445:215
2442:682
2440:390
2428:440
2425:997
2422:999
2419:999
2410:± 7
2393:393
2390:197
2387:450
2373:680
2370:919
2367:999
2364:999
2338:346
2335:797
2333:506
2321:825
2318:026
2315:998
2312:999
2303:± 6
2288:090
2279:254
2276:330
2262:875
2259:020
2256:962
2253:999
2223:278
2220:744
2206:856
2203:696
2200:426
2197:999
2188:± 5
2171:160
2169:147
2157:751
2154:653
2151:204
2148:993
2126:787
2112:334
2109:516
2106:657
2103:936
2094:± 4
2072:929
2069:841
2066:741
2063:534
2032:740
2029:936
2026:203
2023:300
2014:± 3
2003:81
1992:448
1989:348
1986:669
1983:580
1963:22
1952:642
1949:103
1946:736
1943:499
1934:± 2
1923:15
1912:284
1909:462
1906:597
1903:385
1872:086
1869:137
1866:492
1863:689
1832:026
1829:548
1826:922
1823:924
1723:In
1410:+ 2
1395:+ 2
1387:Pr(
178:3sr
164:In
56:by
4538::
3111:.
3086:.
3061:.
3033:.
3023:48
3021:.
2946:.
2593:±
2585:)
2495:15
2457:)
2402:)
2385:12
2347:)
2295:)
2286:72
2274:26
2236:)
2180:)
2124:15
1883:3
1854:±
1843:5
1766:.
1729:,
1639:.
1423:Pr
1391:≤
1375:.
1346:.
1179:Pr
1054:Pr
929:Pr
915:.
516:Pr
481:.
361:Pr
302:Pr
243:Pr
217:,
207:,
4246:t
4207:t
4075:q
4067:q
4059:q
4051:κ
4042:κ
4033:κ
4024:κ
4015:κ
3959:t
3926:t
3895:U
3893:S
3855:q
3842:z
3674:T
3605:F
3181:)
3177:(
3167:e
3160:t
3153:v
3136:x
3132:"
3041:.
3029::
3012:.
2988:.
2956:.
2899:.
2858:t
2840:p
2809:)
2803:2
2799:x
2794:(
2781:1
2777:1
2747:)
2741:2
2737:x
2732:(
2719:1
2715:1
2686:)
2680:2
2676:x
2671:(
2658:1
2634:)
2628:2
2624:x
2619:(
2600:σ
2596:x
2591:μ
2529:σ
2525:μ
2467:σ
2463:μ
2412:σ
2408:μ
2357:σ
2353:μ
2305:σ
2301:μ
2246:σ
2242:μ
2218:1
2190:σ
2186:μ
2141:σ
2137:μ
2096:σ
2092:μ
2056:σ
2052:μ
2016:σ
2012:μ
1976:σ
1972:μ
1936:σ
1932:μ
1896:σ
1892:μ
1856:σ
1852:μ
1815:σ
1811:μ
1739:σ
1717:σ
1705:σ
1627:n
1601:X
1575:n
1566:2
1554:X
1519:)
1510:1
1507:(
1495:)
1492:2
1486:(
1477:)
1474:2
1471:(
1465:=
1462:)
1456:2
1453:+
1444:X
1435:2
1426:(
1412:σ
1408:μ
1397:σ
1393:μ
1389:X
1381:Φ
1370:σ
1368:,
1366:μ
1361:Φ
1354:z
1335:σ
1330:μ
1291:z
1288:d
1281:2
1276:2
1272:z
1262:e
1256:3
1251:3
1234:2
1230:1
1225:=
1218:)
1212:3
1209:+
1200:X
1191:3
1182:(
1166:z
1163:d
1156:2
1151:2
1147:z
1137:e
1131:2
1126:2
1109:2
1105:1
1100:=
1093:)
1087:2
1084:+
1075:X
1066:2
1057:(
1041:z
1038:d
1031:2
1026:2
1022:z
1012:e
1006:1
1001:1
984:2
980:1
975:=
968:)
962:1
959:+
950:X
941:1
932:(
903:3
900:,
897:2
894:,
891:1
888:=
885:n
823:,
816:z
813:d
806:2
801:2
797:z
787:e
781:n
776:n
759:2
755:1
715:x
709:=
706:z
675:,
672:x
669:d
662:2
657:)
642:x
636:(
629:2
626:1
617:e
602:2
596:1
586:n
583:+
572:n
558:=
555:)
549:n
546:+
537:X
528:n
519:(
446:σ
444:3
400:)
394:3
391:+
382:X
373:3
364:(
341:)
335:2
332:+
323:X
314:2
305:(
282:)
276:1
273:+
264:X
255:1
246:(
226:σ
220:μ
210:Χ
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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