406:
161:
135:
401:{\displaystyle \definecolor {Orange}{rgb}{1,0.5019607843137255,0}\definecolor {ChromeYellow}{rgb}{1,0.6549019607843137,0.011764705882352941}\definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\definecolor {Blue}{rgb}{0,0,1}\definecolor {Purple}{rgb}{0.5019607843137255,0,0.5019607843137255}H=({\color {Red}N}-1){\frac {\sum _{i=1}^{\color {Orange}g}{\color {ChromeYellow}n_{i}}({\color {Blue}{\bar {r}}_{i\cdot }}-{\color {Purple}{\bar {r}}})^{2}}{\sum _{i=1}^{\color {Orange}g}\sum _{j=1}^{\color {ChromeYellow}n_{i}}({\color {Green}r_{ij}}-{\color {Purple}{\bar {r}}})^{2}}},}
122:
alternative hypothesis is that at least one population median of one group is different from the population median of at least one other group. Otherwise, it is impossible to say, whether the rejection of the null hypothesis comes from the shift in locations or group dispersions. This is the same issue that happens also with the Mann-Whitney test. If the data contains potential outliers, if the population distributions have heavy tails, or if the population distributions are significantly skewed, the
Kruskal-Wallis test is more powerful at detecting differences among treatments than
5241:
31:
1933:
5227:
5265:
1211:
5253:
1873:
A large amount of computing resources is required to compute exact probabilities for the
Kruskal–Wallis test. Existing software only provides exact probabilities for sample sizes of less than about 30 participants. These software programs rely on the asymptotic approximation for larger sample sizes.
944:
116:
It is supposed that the treatments significantly affect the response level and then there is an order among the treatments: one tends to give the lowest response, another gives the next lowest response is second, and so forth. Since it is a nonparametric method, the
Kruskal–Wallis test does not
1859:
tests using Dunn's test, which (1) properly employs the same rankings as the
Kruskal–Wallis test, and (2) properly employs the pooled variance implied by the null hypothesis of the Kruskal–Wallis test in order to determine which of the sample pairs are significantly different. When performing
121:
of the residuals, unlike the analogous one-way analysis of variance. If the researcher can make the assumptions of an identically shaped and scaled distribution for all groups, except for any difference in medians, then the null hypothesis is that the medians of all groups are equal, and the
1854:
If the statistic is not significant, there is no evidence of stochastic dominance among the samples. However, if the test is significant then at least one sample stochastically dominates another sample. Then, a researcher might use sample contrasts between individual sample pairs, or
540:
802:
770:
686:
1874:
Exact probability values for larger sample sizes are available. Spurrier (2003) published exact probability tables for samples as large as 45 participants. Meyer and Seaman (2006) produced exact probability distributions for samples as large as 105 participants.
487:
126:. On the other hand, if the population distributions are normal or are light-tailed and symmetric, then ANOVA F-test will generally have greater power which is the probability of rejecting the null hypothesis when it indeed should be rejected.
104:
one other sample. The test does not identify where this stochastic dominance occurs or for how many pairs of groups stochastic dominance obtains. For analyzing the specific sample pairs for stochastic dominance, Dunn's test, pairwise
1206:{\displaystyle {\begin{aligned}H&={\frac {12}{N(N+1)}}\sum _{i=1}^{g}n_{i}\left({\bar {r}}_{i\cdot }-{\frac {N+1}{2}}\right)^{2}\\&={\frac {12}{N(N+1)}}\sum _{i=1}^{g}n_{i}{\bar {r}}_{i\cdot }^{2}-\ 3(N+1)\end{aligned}}}
456:
1339:
513:
775:
712:
586:
1481:
949:
937:
2062:
To determine which months differ, post-hoc tests may be performed using a
Wilcoxon test for each pair of months, with a Bonferroni (or other) correction for multiple hypothesis testing.
463:
1816:
85:
for testing whether samples originate from the same distribution. It is used for comparing two or more independent samples of equal or different sample sizes. It extends the
885:
1510:
432:
2807:
2768:
1530:
1550:
2390:
1744:
1647:
1600:
4362:
1920:
The following example uses data from
Chambers et al. on daily readings of ozone for May 1 to September 30, 1973, in New York City. The data are in the R data set
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4867:
1905:
1768:
1687:
1667:
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706:
580:
560:
507:
62:
1842:
1717:
1427:
1406:
1359:
5017:
4641:
3282:
2249:
The post-hoc tests indicate that, after
Bonferroni correction for multiple testing, the following differences are significant (adjusted p < 0.05).
4415:
439:
4854:
2743:
Won Choi, Jae Won Lee, Myung-Hoe Huh, and Seung-Ho Kang (2003). "An
Algorithm for Computing the Exact Distribution of the Kruskal–Wallis Test".
2673:
2639:
2604:
3277:
2977:
2546:
Divine; Norton; Barón; Juarez-Colunga (2018). "The
Wilcoxon–Mann–Whitney Procedure Fails as a Test of Medians". The American Statistician.
1242:
134:
3881:
3029:
535:{\displaystyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}r_{ij}}
2654:
Corder, G.W. & Foreman, D.I. (2010). Nonparametric
Statistics for Non-statisticians: A Step-by-Step Approach. Hoboken, NJ: Wiley.
5301:
4664:
4556:
2934:
2434:
797:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}r_{ij}}
5269:
4842:
4716:
765:{\textstyle \definecolor {Purple}{rgb}{0.5019607843137255,0,0.5019607843137255}{\color {Purple}{\bar {r}}}={\tfrac {1}{2}}(N+1)}
4900:
4561:
4306:
3677:
3267:
2893:
1844:
681:{\displaystyle \definecolor {blue}{rgb}{0,0,1}{\color {blue}{\bar {r}}_{i\cdot }}={\frac {\sum _{j=1}^{n_{i}}{r_{ij}}}{n_{i}}}}
3891:
4951:
4163:
3970:
3859:
3817:
2791:
2459:
3056:
2869:
153:
ignoring group membership. Assign any tied values the average of the ranks they would have received had they not been tied.
5296:
5194:
4153:
2274:
1444:
4203:
2726:
2584:
Bruin (2006). "FAQ: Why is the Mann-Whitney significant when the medians are equal?". UCLA: Statistical Consulting Group.
1938:
The Kruskal-Wallis test finds a significant difference (p = 6.901e-06) indicating that ozone differs among the 5 months.
4745:
4694:
4679:
4669:
4538:
4410:
4377:
4158:
3988:
2745:
2343:
2318:
1861:
4814:
4115:
5089:
4890:
3869:
3538:
3002:
2950:
2307:
4974:
4941:
5291:
4946:
4689:
4448:
4354:
4334:
4242:
3953:
3771:
3254:
3126:
2845:
2363:
4120:
3886:
3744:
890:
482:{\textstyle \definecolor {ChromeYellow}{rgb}{1,0.6549019607843137,0.011764705882352941}\color {ChromeYellow}n_{i}}
4706:
4474:
4195:
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3898:
3756:
3737:
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3166:
79:
4819:
2821:
5189:
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3440:
3131:
2970:
2297:
1848:
1771:
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1690:
4210:
2348:
86:
2565:
Hart (2001). "Mann-Whitney test is not just a test of medians: differences in spread can be important". BMJ.
1777:
1219:
A correction for ties if using the short-cut formula described in the previous point can be made by dividing
5098:
4711:
4651:
4588:
3948:
3810:
3800:
3650:
3564:
4859:
4796:
1860:
multiple sample contrasts or tests, the Type I error rate tends to become inflated, raising concerns about
93:, which is used for comparing only two groups. The parametric equivalent of the Kruskal–Wallis test is the
5136:
5066:
4551:
4438:
3435:
3332:
3239:
3118:
3017:
2265:
The Kruskal-Wallis test can be implemented in many programming tools and languages. We list here only the
1774:. If a table of the chi-squared probability distribution is available, the critical value of chi-squared,
30:
5257:
4135:
5161:
5103:
5046:
4872:
4765:
4674:
4400:
4284:
4143:
4025:
4017:
3832:
3728:
3706:
3665:
3630:
3597:
3543:
3518:
3473:
3412:
3372:
3174:
2997:
2762:
1438:
110:
5240:
4130:
2368:
1887:
Choi et al. made a review of two methods that had been developed to compute the exact distribution of
5084:
4659:
4608:
4584:
4546:
4464:
4443:
4395:
4274:
4252:
4221:
4007:
3958:
3876:
3849:
3805:
3761:
3523:
3299:
3179:
2718:
Paper presented at the annual meeting of the American Educational Research Association, San Francisco
2716:
Meyer; Seaman (April 2006). "Expanded tables of critical values for the Kruskal–Wallis H statistic".
1408:
that are tied at a particular value. This correction usually makes little difference in the value of
101:
832:
5231:
5156:
5079:
4760:
4524:
4517:
4479:
4387:
4367:
4339:
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3923:
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3733:
3694:
3584:
3574:
3483:
3262:
3218:
3136:
3061:
2963:
118:
4806:
2722:
Critical value tables and exact probabilities from Meyer and Seaman are available for download at
5245:
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4755:
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4266:
4000:
3983:
3943:
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3749:
3711:
3682:
3642:
3602:
3548:
3465:
3151:
3146:
2801:
1649:, the null hypothesis is rejected. If possible (no ties, sample not too big) one should compare
1486:
2426:
415:
5151:
5121:
5113:
4933:
4924:
4849:
4780:
4636:
4621:
4596:
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4425:
4291:
4279:
3905:
3822:
3766:
3689:
3533:
3455:
3234:
3108:
2930:
2922:
2787:
2669:
2635:
2600:
2455:
2430:
1515:
106:
5176:
5131:
4895:
4882:
4775:
4750:
4684:
4616:
4494:
4102:
3995:
3928:
3841:
3788:
3607:
3478:
3272:
3071:
3038:
2750:
2698:
2627:
2566:
2547:
2487:
2418:
2399:
1907:, proposed a new one, and compared the exact distribution to its chi-squared approximation.
1535:
82:
1722:
1625:
1578:
5093:
4837:
4699:
4626:
4301:
4175:
4148:
4125:
4094:
3721:
3716:
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3400:
3051:
2730:
2353:
71:
67:
1932:
5042:
5037:
3500:
3430:
3076:
2509:
2311:
1890:
1753:
1672:
1652:
1605:
1558:
1364:
1222:
812:
691:
565:
545:
492:
47:
27:
Non-parametric method for testing whether samples originate from the same distribution
5285:
5199:
5166:
5029:
4990:
4801:
4770:
4234:
4188:
3793:
3495:
3322:
3086:
3081:
2621:
2419:
2358:
1821:
1696:
94:
3352:
2689:
Spurrier, J. D. (2003). "On the null distribution of the Kruskal–Wallis statistic".
5141:
5074:
5051:
4966:
4296:
3592:
3490:
3425:
3367:
3289:
3244:
2403:
1555:
Finally, the decision to reject or accept the null hypothesis is made by comparing
1434:
1412:
1391:
1344:
17:
2551:
2782:
John M. Chambers, William S. Cleveland, Beat Kleiner, and Paul A. Tukey (1983).
2702:
113:, or the more powerful but less well known Conover–Iman test are sometimes used.
5184:
5146:
4829:
4730:
4592:
4405:
4372:
3864:
3781:
3776:
3420:
3377:
3357:
3337:
3327:
3096:
2266:
1602:(obtained from a table or software) for a given significance or alpha level. If
451:{\textstyle \definecolor {Orange}{rgb}{1,0.5019607843137255,0}\color {Orange}g}
4030:
3510:
3210:
3141:
3091:
3066:
2986:
2723:
2631:
2570:
4183:
4035:
3655:
3450:
3362:
3347:
3342:
3307:
2599:. Duxbury advanced series. Pacific Gove, CA: Brooks-Cole; Thomson Learning.
2388:
Kruskal; Wallis (1952). "Use of ranks in one-criterion variance analysis".
2754:
3699:
3317:
3194:
3189:
3184:
3156:
138:
An illustration of how to assign any tied values the average of the rank
5204:
4905:
2330:
2286:
1924:, and the analysis is included in the documentation for the R function
5126:
4107:
4081:
4061:
3312:
3103:
1334:{\displaystyle 1-{\frac {\sum _{i=1}^{G}(t_{i}^{3}-t_{i})}{N^{3}-N}}}
123:
100:
A significant Kruskal–Wallis test indicates that at least one sample
2491:
809:
If the data contain no ties, the denominator of the expression for
2278:
133:
29:
2478:
Dunn, Olive Jean (1964). "Multiple comparisons using rank sums".
143:
Rank all data from all groups together; i.e., rank the data from
3046:
2786:. Belmont, Calif: Wadsworth International Group, Duxbury Press.
1214:
The last formula contains only the squares of the average ranks.
5015:
4582:
4329:
3628:
3398:
3015:
2959:
2620:
Berger, Paul D.; Maurer, Robert E.; Celli, Giovana B. (2018).
1669:
to the critical value obtained from the exact distribution of
2955:
1928:. Boxplots of ozone values by month are shown in the figure.
1689:. Otherwise, the distribution of H can be approximated by a
34:
Difference between ANOVA and Kruskal–Wallis test with ranks
2746:
Communications in Statistics - Simulation and Computation
2733:. A paper describing their work may also be found there.
1361:
is the number of groupings of different tied ranks, and
2668:. EMEA edition (Seventh ed.). Hoboken, NJ: Wiley.
2327:
KruskalWallisTest(groups::AbstractVector{<:Real}...)
2923:"Kruskal–Wallis one-way analysis of variance by ranks"
1824:
1699:
1415:
1394:
1367:
1347:
910:
778:
736:
715:
466:
442:
418:
1893:
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1608:
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1518:
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1245:
1225:
947:
893:
835:
815:
694:
589:
568:
548:
516:
495:
434:
is the total number of observations across all groups
164:
50:
4868:
Autoregressive conditional heteroskedasticity (ARCH)
2929:(2nd ed.). Boston: PWS-Kent. pp. 226–234.
2452:
Nonparametric Statistics for the Behavioral Sciences
1476:{\displaystyle {\bar {a}}={\frac {\alpha }{\Bbbk }}}
542:
is the rank (among all observations) of observation
5175:
5112:
5065:
5028:
4983:
4965:
4932:
4923:
4881:
4828:
4789:
4738:
4729:
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4607:
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4503:
4457:
4424:
4386:
4353:
4265:
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4093:
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4016:
3969:
3914:
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3583:
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3464:
3411:
3298:
3253:
3227:
3209:
3165:
3117:
3037:
3028:
2534:
Nonparametrics: Statistical methods based on ranks.
1441:is used to adjust the significance level, that is,
2664:Montgomery, Douglas C.; Runger, George C. (2018).
2597:An introduction to modern nonparametric statistics
2595:Higgins, James J.; Jeffrey Higgins, James (2004).
1899:
1836:
1810:
1762:
1738:
1711:
1681:
1661:
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931:
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821:
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764:
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680:
574:
554:
534:
501:
481:
450:
426:
400:
56:
2870:"Math – The Commons Math User Guide - Statistics"
2300:base-package has an implement of this test using
1433:When performing multiple sample comparisons, the
688:is the average rank of all observations in group
2666:Applied statistics and probability for engineers
4416:Multivariate adaptive regression splines (MARS)
2391:Journal of the American Statistical Association
1746:values are small (i.e., less than 5) the exact
2532:Lehmann, E. L., & D'Abrera, H. J. (1975).
2421:Nonparametric Statistics for Non-Statisticians
2971:
2417:Corder, Gregory W.; Foreman, Dale I. (2009).
1916:Test for differences in ozone levels by month
8:
2822:"scipy.stats.kruskal — SciPy v1.11.4 Manual"
2806:: CS1 maint: multiple names: authors list (
2767:: CS1 maint: multiple names: authors list (
932:{\displaystyle {\bar {r}}={\tfrac {N+1}{2}}}
2724:http://faculty.virginia.edu/kruskal-wallis/
2626:. Cham: Springer International Publishing.
2425:. Hoboken: John Wiley & Sons. pp.
5025:
5012:
4929:
4735:
4604:
4579:
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4326:
4054:
3837:
3638:
3625:
3408:
3395:
3034:
3025:
3012:
2978:
2964:
2956:
2894:"Nonparametric tests · HypothesisTests.jl"
2515:(Report). Los Alamos Scientific Laboratory
2454:(Second ed.). New York: McGraw–Hill.
2310:has the implement provided by provided by
1388:is the number of tied values within group
2508:Conover, W. Jay; Iman, Ronald L. (1979).
2503:
2501:
1892:
1823:
1802:
1785:
1779:
1755:
1730:
1724:
1698:
1674:
1654:
1633:
1627:
1607:
1586:
1580:
1560:
1539:
1537:
1517:
1491:
1490:
1488:
1468:
1463:
1449:
1448:
1446:
1437:tends to become inflated. Therefore, the
1414:
1393:
1372:
1366:
1346:
1316:
1301:
1288:
1283:
1270:
1259:
1252:
1244:
1224:
1169:
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1149:
1142:
1132:
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1090:
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1051:
1039:
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993:
962:
948:
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895:
894:
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869:
834:
814:
784:
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735:
719:
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693:
670:
655:
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619:
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197:
190:
174:
163:
49:
2846:"kruskal.test function - RDocumentation"
1818:, can be found by entering the table at
1430:unless there are a large number of ties.
2380:
1811:{\displaystyle \chi _{\alpha :g-1}^{2}}
1532:is the initial significance level, and
489:is the number of observations in group
4942:Kaplan–Meier estimator (product limit)
2799:
2760:
2473:
2471:
779:
717:
591:
517:
467:
443:
419:
367:
347:
329:
306:
263:
234:
217:
209:
175:
7:
5252:
4952:Accelerated failure time (AFT) model
2510:"On multiple-comparisons procedures"
1512:is the adjusted significance level,
5264:
4547:Analysis of variance (ANOVA, anova)
2784:Graphical Methods for Data Analysis
2691:Journal of Nonparametric Statistics
4642:Cochran–Mantel–Haenszel statistics
3268:Pearson product-moment correlation
25:
1770:can be quite different from this
5263:
5251:
5239:
5226:
5225:
2927:Applied Nonparametric Statistics
1931:
4901:Least-squares spectral analysis
2285:can return the test result and
156:The test statistic is given by
3882:Mean-unbiased minimum-variance
2404:10.1080/01621459.1952.10483441
1847:and looking under the desired
1496:
1454:
1307:
1276:
1196:
1184:
1155:
1111:
1099:
1033:
983:
971:
900:
880:{\displaystyle (N-1)N(N+1)/12}
866:
854:
848:
836:
759:
747:
724:
599:
383:
374:
343:
279:
270:
242:
230:
187:
171:
1:
5195:Geographic information system
4411:Simultaneous equations models
2951:An online version of the test
2552:10.1080/00031305.2017.1305291
4378:Coefficient of determination
3989:Uniformly most powerful test
2703:10.1080/10485250310001634719
1719:degrees of freedom. If some
95:one-way analysis of variance
4947:Proportional hazards models
4891:Spectral density estimation
4873:Vector autoregression (VAR)
4307:Maximum posterior estimator
3539:Randomized controlled trial
2749:(32, number 4): 1029–1040.
1552:is the number of contrasts.
5318:
4707:Multivariate distributions
3127:Average absolute deviation
2450:Siegel; Castellan (1988).
1505:{\displaystyle {\bar {a}}}
772:is the average of all the
427:{\textstyle \color {Red}N}
5221:
5024:
5011:
4695:Structural equation model
4603:
4578:
4349:
4325:
4057:
4031:Score/Lagrange multiplier
3637:
3624:
3446:Sample size determination
3407:
3394:
3024:
3011:
2993:
2921:Daniel, Wayne W. (1990).
2632:10.1007/978-3-319-64583-4
2256:Month 9 vs Months 7 and 8
2253:Month 5 vs Months 7 and 8
5302:Nonparametric statistics
5190:Environmental statistics
4712:Elliptical distributions
4505:Generalized linear model
4434:Simple linear regression
4204:Hodges–Lehmann estimator
3661:Probability distribution
3570:Stochastic approximation
3132:Coefficient of variation
2571:10.1136/bmj.323.7309.391
2269:free software packages:
2064:
1940:
1869:Exact probability tables
1772:chi-squared distribution
1748:probability distribution
1691:chi-squared distribution
102:stochastically dominates
4850:Cross-correlation (XCF)
4458:Non-standard predictors
3892:Lehmann–Scheffé theorem
3565:Adaptive clinical trial
2364:Jonckheere's trend test
1525:{\displaystyle \alpha }
458:is the number of groups
5246:Mathematics portal
5067:Engineering statistics
4975:Nelson–Aalen estimator
4552:Analysis of covariance
4439:Ordinary least squares
4363:Pearson product-moment
3767:Statistical functional
3678:Empirical distribution
3511:Controlled experiments
3240:Frequency distribution
3018:Descriptive statistics
2850:www.rdocumentation.org
2281:package, the function
2103:"bonferroni"
1901:
1838:
1812:
1764:
1740:
1713:
1683:
1663:
1643:
1616:
1596:
1569:
1546:
1545:{\displaystyle \Bbbk }
1526:
1506:
1477:
1423:
1402:
1382:
1355:
1335:
1275:
1233:
1207:
1137:
1009:
933:
881:
823:
798:
766:
702:
682:
649:
576:
556:
536:
503:
483:
452:
428:
402:
342:
312:
215:
139:
76:one-way ANOVA on ranks
58:
35:
5162:Population statistics
5104:System identification
4838:Autocorrelation (ACF)
4766:Exponential smoothing
4680:Discriminant analysis
4675:Canonical correlation
4539:Partition of variance
4401:Regression validation
4245:(Jonckheere–Terpstra)
4144:Likelihood-ratio test
3833:Frequentist inference
3745:Location–scale family
3666:Sampling distribution
3631:Statistical inference
3598:Cross-sectional study
3585:Observational studies
3544:Randomized experiment
3373:Stem-and-leaf display
3175:Central limit theorem
2755:10.1081/SAC-120023876
1902:
1878:Exact distribution of
1839:
1813:
1765:
1741:
1739:{\displaystyle n_{i}}
1714:
1684:
1664:
1644:
1642:{\displaystyle H_{c}}
1617:
1597:
1595:{\displaystyle H_{c}}
1570:
1547:
1527:
1507:
1478:
1424:
1403:
1383:
1356:
1336:
1255:
1234:
1208:
1117:
989:
934:
882:
824:
799:
767:
703:
683:
622:
577:
557:
537:
504:
484:
453:
429:
403:
313:
290:
193:
137:
111:Bonferroni correction
59:
33:
5297:Analysis of variance
5085:Probabilistic design
4670:Principal components
4513:Exponential families
4465:Nonlinear regression
4444:General linear model
4406:Mixed effects models
4396:Errors and residuals
4373:Confounding variable
4275:Bayesian probability
4253:Van der Waerden test
4243:Ordered alternative
4008:Multiple comparisons
3887:Rao–Blackwellization
3850:Estimating equations
3806:Statistical distance
3524:Factorial experiment
3057:Arithmetic-Geometric
2349:Mann–Whitney U tests
2067:pairwise.wilcox.test
1891:
1862:multiple comparisons
1822:
1778:
1754:
1723:
1697:
1673:
1653:
1626:
1606:
1579:
1575:to a critical value
1559:
1536:
1516:
1487:
1445:
1439:Bonferroni procedure
1413:
1392:
1365:
1345:
1243:
1223:
945:
891:
833:
813:
776:
713:
692:
587:
566:
546:
514:
493:
464:
440:
416:
162:
48:
5157:Official statistics
5080:Methods engineering
4761:Seasonal adjustment
4529:Poisson regressions
4449:Bayesian regression
4388:Regression analysis
4368:Partial correlation
4340:Regression analysis
3939:Prediction interval
3934:Likelihood interval
3924:Confidence interval
3916:Interval estimation
3877:Unbiased estimators
3695:Model specification
3575:Up-and-down designs
3263:Partial correlation
3219:Index of dispersion
3137:Interquartile range
2623:Experimental Design
2283:scipy.stats.kruskal
1807:
1293:
1174:
119:normal distribution
40:Kruskal–Wallis test
18:Kruskal-Wallis test
5177:Spatial statistics
5057:Medical statistics
4957:First hitting time
4911:Whittle likelihood
4562:Degrees of freedom
4557:Multivariate ANOVA
4490:Heteroscedasticity
4302:Bayesian estimator
4267:Bayesian inference
4116:Kolmogorov–Smirnov
4001:Randomization test
3971:Testing hypotheses
3944:Tolerance interval
3855:Maximum likelihood
3750:Exponential family
3683:Density estimation
3643:Statistical theory
3603:Natural experiment
3549:Scientific control
3466:Survey methodology
3152:Standard deviation
2874:commons.apache.org
2729:2018-10-17 at the
2369:Mood's Median test
2323:HypothesisTests.jl
1897:
1845:degrees of freedom
1834:
1808:
1781:
1760:
1736:
1709:
1679:
1659:
1639:
1612:
1592:
1565:
1542:
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1473:
1419:
1398:
1381:{\textstyle t_{i}}
1378:
1351:
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1279:
1229:
1203:
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1148:
929:
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877:
819:
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762:
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730:
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424:
423:
398:
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361:
340:
310:
276:
257:
228:
213:
179:
140:
54:
36:
5292:Statistical tests
5279:
5278:
5217:
5216:
5213:
5212:
5152:National accounts
5122:Actuarial science
5114:Social statistics
5007:
5006:
5003:
5002:
4999:
4998:
4934:Survival function
4919:
4918:
4781:Granger causality
4622:Contingency table
4597:Survival analysis
4574:
4573:
4570:
4569:
4426:Linear regression
4321:
4320:
4317:
4316:
4292:Credible interval
4261:
4260:
4044:
4043:
3860:Method of moments
3729:Parametric family
3690:Statistical model
3620:
3619:
3616:
3615:
3534:Random assignment
3456:Statistical power
3390:
3389:
3386:
3385:
3235:Contingency table
3205:
3204:
3072:Generalized/power
2675:978-1-119-40036-3
2641:978-3-319-64582-7
2606:978-0-534-38775-4
2325:has the function
1900:{\displaystyle H}
1763:{\displaystyle H}
1682:{\displaystyle H}
1662:{\displaystyle H}
1615:{\displaystyle H}
1568:{\displaystyle H}
1499:
1471:
1457:
1329:
1232:{\displaystyle H}
1180:
1158:
1115:
1067:
1036:
987:
926:
903:
822:{\displaystyle H}
744:
727:
701:{\displaystyle i}
676:
602:
575:{\displaystyle i}
555:{\displaystyle j}
502:{\displaystyle i}
393:
377:
273:
245:
57:{\displaystyle H}
16:(Redirected from
5309:
5267:
5266:
5255:
5254:
5244:
5243:
5229:
5228:
5132:Crime statistics
5026:
5013:
4930:
4896:Fourier analysis
4883:Frequency domain
4863:
4810:
4776:Structural break
4736:
4685:Cluster analysis
4632:Log-linear model
4605:
4580:
4521:
4495:Homoscedasticity
4351:
4327:
4246:
4238:
4230:
4229:(Kruskal–Wallis)
4214:
4199:
4154:Cross validation
4139:
4121:Anderson–Darling
4068:
4055:
4026:Likelihood-ratio
4018:Parametric tests
3996:Permutation test
3979:1- & 2-tails
3870:Minimum distance
3842:Point estimation
3838:
3789:Optimal decision
3740:
3639:
3626:
3608:Quasi-experiment
3558:Adaptive designs
3409:
3396:
3273:Rank correlation
3035:
3026:
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2398:(260): 583–621.
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1935:
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1837:{\textstyle g-1}
1835:
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1769:
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1712:{\textstyle g-1}
1710:
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83:statistical test
63:
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60:
55:
21:
5317:
5316:
5312:
5311:
5310:
5308:
5307:
5306:
5282:
5281:
5280:
5275:
5238:
5209:
5171:
5108:
5094:quality control
5061:
5043:Clinical trials
5020:
4995:
4979:
4967:Hazard function
4961:
4915:
4877:
4861:
4824:
4820:Breusch–Godfrey
4808:
4785:
4725:
4700:Factor analysis
4646:
4627:Graphical model
4599:
4566:
4533:
4519:
4499:
4453:
4420:
4382:
4345:
4344:
4313:
4257:
4244:
4236:
4228:
4212:
4197:
4176:Rank statistics
4170:
4149:Model selection
4137:
4095:Goodness of fit
4089:
4066:
4040:
4012:
3965:
3910:
3899:Median unbiased
3827:
3738:
3671:Order statistic
3633:
3612:
3579:
3553:
3505:
3460:
3403:
3401:Data collection
3382:
3294:
3249:
3223:
3201:
3161:
3113:
3030:Continuous data
3020:
3007:
2989:
2984:
2947:
2937:
2920:
2917:
2915:Further reading
2912:
2911:
2902:
2900:
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2891:
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2878:
2876:
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2867:
2863:
2854:
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2815:
2798:
2794:
2781:
2780:
2776:
2759:
2742:
2741:
2737:
2731:Wayback Machine
2715:
2714:
2710:
2688:
2687:
2683:
2676:
2663:
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2492:10.2307/1266041
2477:
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2449:
2448:
2444:
2437:
2416:
2415:
2411:
2387:
2386:
2382:
2377:
2354:Bonferroni test
2340:
2329:to compute the
2326:
2322:
2301:
2287:
2282:
2263:
2247:
2246:
2243:
2240:
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2129:
2126:
2123:
2120:
2117:
2114:
2111:
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2105:
2102:
2099:
2097:p.adjust.method
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2060:
2059:
2056:
2053:
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2047:
2044:
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2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1925:
1921:
1918:
1913:
1889:
1888:
1885:
1879:
1871:
1851:or alpha level.
1820:
1819:
1776:
1775:
1752:
1751:
1726:
1721:
1720:
1695:
1694:
1671:
1670:
1651:
1650:
1629:
1624:
1623:
1622:is bigger than
1604:
1603:
1582:
1577:
1576:
1557:
1556:
1534:
1533:
1514:
1513:
1485:
1484:
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1363:
1362:
1343:
1342:
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1311:
1297:
1254:
1241:
1240:
1221:
1220:
1213:
1200:
1199:
1138:
1095:
1081:
1080:
1053:
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1021:
1020:
1010:
967:
955:
943:
942:
912:
889:
888:
831:
830:
811:
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780:
774:
773:
711:
710:
690:
689:
666:
651:
638:
621:
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468:
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382:
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289:
278:
235:
218:
192:
160:
159:
148:
144:
132:
72:W. Allen Wallis
68:William Kruskal
46:
45:
44:Kruskal–Wallis
28:
23:
22:
15:
12:
11:
5:
5315:
5313:
5305:
5304:
5299:
5294:
5284:
5283:
5277:
5276:
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5261:
5249:
5235:
5222:
5219:
5218:
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5214:
5211:
5210:
5208:
5207:
5202:
5197:
5192:
5187:
5181:
5179:
5173:
5172:
5170:
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5159:
5154:
5149:
5144:
5139:
5134:
5129:
5124:
5118:
5116:
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5109:
5107:
5106:
5101:
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5087:
5082:
5077:
5071:
5069:
5063:
5062:
5060:
5059:
5054:
5049:
5040:
5038:Bioinformatics
5034:
5032:
5022:
5021:
5016:
5009:
5008:
5005:
5004:
5001:
5000:
4997:
4996:
4994:
4993:
4987:
4985:
4981:
4980:
4978:
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4969:
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4954:
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4921:
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4917:
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4914:
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4908:
4903:
4898:
4893:
4887:
4885:
4879:
4878:
4876:
4875:
4870:
4865:
4857:
4852:
4847:
4846:
4845:
4843:partial (PACF)
4834:
4832:
4826:
4825:
4823:
4822:
4817:
4812:
4804:
4799:
4793:
4791:
4790:Specific tests
4787:
4786:
4784:
4783:
4778:
4773:
4768:
4763:
4758:
4753:
4748:
4742:
4740:
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4722:
4721:
4720:
4719:
4704:
4703:
4702:
4692:
4690:Classification
4687:
4682:
4677:
4672:
4667:
4662:
4656:
4654:
4648:
4647:
4645:
4644:
4639:
4637:McNemar's test
4634:
4629:
4624:
4619:
4613:
4611:
4601:
4600:
4583:
4576:
4575:
4572:
4571:
4568:
4567:
4565:
4564:
4559:
4554:
4549:
4543:
4541:
4535:
4534:
4532:
4531:
4515:
4509:
4507:
4501:
4500:
4498:
4497:
4492:
4487:
4482:
4477:
4475:Semiparametric
4472:
4467:
4461:
4459:
4455:
4454:
4452:
4451:
4446:
4441:
4436:
4430:
4428:
4422:
4421:
4419:
4418:
4413:
4408:
4403:
4398:
4392:
4390:
4384:
4383:
4381:
4380:
4375:
4370:
4365:
4359:
4357:
4347:
4346:
4343:
4342:
4337:
4331:
4330:
4323:
4322:
4319:
4318:
4315:
4314:
4312:
4311:
4310:
4309:
4299:
4294:
4289:
4288:
4287:
4282:
4271:
4269:
4263:
4262:
4259:
4258:
4256:
4255:
4250:
4249:
4248:
4240:
4232:
4216:
4213:(Mann–Whitney)
4208:
4207:
4206:
4193:
4192:
4191:
4180:
4178:
4172:
4171:
4169:
4168:
4167:
4166:
4161:
4156:
4146:
4141:
4138:(Shapiro–Wilk)
4133:
4128:
4123:
4118:
4113:
4105:
4099:
4097:
4091:
4090:
4088:
4087:
4079:
4070:
4058:
4052:
4050:Specific tests
4046:
4045:
4042:
4041:
4039:
4038:
4033:
4028:
4022:
4020:
4014:
4013:
4011:
4010:
4005:
4004:
4003:
3993:
3992:
3991:
3981:
3975:
3973:
3967:
3966:
3964:
3963:
3962:
3961:
3956:
3946:
3941:
3936:
3931:
3926:
3920:
3918:
3912:
3911:
3909:
3908:
3903:
3902:
3901:
3896:
3895:
3894:
3889:
3874:
3873:
3872:
3867:
3862:
3857:
3846:
3844:
3835:
3829:
3828:
3826:
3825:
3820:
3815:
3814:
3813:
3803:
3798:
3797:
3796:
3786:
3785:
3784:
3779:
3774:
3764:
3759:
3754:
3753:
3752:
3747:
3742:
3726:
3725:
3724:
3719:
3714:
3704:
3703:
3702:
3697:
3687:
3686:
3685:
3675:
3674:
3673:
3663:
3658:
3653:
3647:
3645:
3635:
3634:
3629:
3622:
3621:
3618:
3617:
3614:
3613:
3611:
3610:
3605:
3600:
3595:
3589:
3587:
3581:
3580:
3578:
3577:
3572:
3567:
3561:
3559:
3555:
3554:
3552:
3551:
3546:
3541:
3536:
3531:
3526:
3521:
3515:
3513:
3507:
3506:
3504:
3503:
3501:Standard error
3498:
3493:
3488:
3487:
3486:
3481:
3470:
3468:
3462:
3461:
3459:
3458:
3453:
3448:
3443:
3438:
3433:
3431:Optimal design
3428:
3423:
3417:
3415:
3405:
3404:
3399:
3392:
3391:
3388:
3387:
3384:
3383:
3381:
3380:
3375:
3370:
3365:
3360:
3355:
3350:
3345:
3340:
3335:
3330:
3325:
3320:
3315:
3310:
3304:
3302:
3296:
3295:
3293:
3292:
3287:
3286:
3285:
3280:
3270:
3265:
3259:
3257:
3251:
3250:
3248:
3247:
3242:
3237:
3231:
3229:
3228:Summary tables
3225:
3224:
3222:
3221:
3215:
3213:
3207:
3206:
3203:
3202:
3200:
3199:
3198:
3197:
3192:
3187:
3177:
3171:
3169:
3163:
3162:
3160:
3159:
3154:
3149:
3144:
3139:
3134:
3129:
3123:
3121:
3115:
3114:
3112:
3111:
3106:
3101:
3100:
3099:
3094:
3089:
3084:
3079:
3074:
3069:
3064:
3062:Contraharmonic
3059:
3054:
3043:
3041:
3032:
3022:
3021:
3016:
3009:
3008:
3006:
3005:
3000:
2994:
2991:
2990:
2985:
2983:
2982:
2975:
2968:
2960:
2954:
2953:
2946:
2945:External links
2943:
2942:
2941:
2935:
2916:
2913:
2910:
2909:
2898:juliastats.org
2885:
2861:
2837:
2826:docs.scipy.org
2813:
2792:
2774:
2735:
2708:
2697:(6): 685–691.
2681:
2674:
2656:
2647:
2640:
2612:
2605:
2587:
2576:
2557:
2538:
2525:
2497:
2486:(3): 241–252.
2467:
2460:
2442:
2435:
2409:
2379:
2378:
2376:
2373:
2372:
2371:
2366:
2361:
2356:
2351:
2346:
2339:
2336:
2335:
2334:
2321:, the package
2315:
2312:Apache Commons
2305:
2295:
2262:
2261:Implementation
2259:
2258:
2257:
2254:
2065:
1941:
1917:
1914:
1912:
1909:
1896:
1884:
1876:
1870:
1867:
1866:
1865:
1852:
1833:
1830:
1827:
1805:
1800:
1797:
1794:
1791:
1788:
1784:
1759:
1733:
1729:
1708:
1705:
1702:
1678:
1658:
1636:
1632:
1611:
1589:
1585:
1564:
1553:
1541:
1521:
1498:
1495:
1470:
1467:
1462:
1456:
1453:
1431:
1422:{\textstyle H}
1418:
1401:{\textstyle i}
1397:
1375:
1371:
1354:{\textstyle G}
1350:
1327:
1324:
1319:
1315:
1309:
1304:
1300:
1296:
1291:
1286:
1282:
1278:
1273:
1268:
1265:
1262:
1258:
1251:
1248:
1228:
1217:
1216:
1215:
1198:
1195:
1192:
1189:
1186:
1183:
1177:
1172:
1167:
1164:
1157:
1154:
1145:
1141:
1135:
1130:
1127:
1124:
1120:
1113:
1110:
1107:
1104:
1101:
1098:
1094:
1089:
1086:
1084:
1082:
1077:
1072:
1066:
1062:
1059:
1056:
1050:
1045:
1042:
1035:
1032:
1024:
1017:
1013:
1007:
1002:
999:
996:
992:
985:
982:
979:
976:
973:
970:
966:
961:
958:
956:
954:
951:
950:
925:
921:
918:
915:
908:
902:
899:
876:
872:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
818:
807:
806:
805:
790:
787:
783:
761:
758:
755:
752:
749:
743:
740:
734:
726:
723:
708:
697:
673:
669:
661:
658:
654:
645:
641:
635:
632:
629:
625:
618:
611:
608:
601:
598:
582:
571:
551:
528:
525:
521:
509:
498:
475:
471:
459:
446:
435:
422:
410:
409:
397:
389:
385:
376:
373:
365:
358:
355:
351:
345:
337:
333:
326:
323:
320:
316:
309:
303:
300:
297:
293:
285:
281:
272:
269:
261:
254:
251:
244:
241:
232:
225:
221:
212:
206:
203:
200:
196:
189:
186:
183:
178:
173:
170:
167:
154:
131:
128:
80:non-parametric
53:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5314:
5303:
5300:
5298:
5295:
5293:
5290:
5289:
5287:
5272:
5271:
5262:
5260:
5259:
5250:
5248:
5247:
5242:
5236:
5234:
5233:
5224:
5223:
5220:
5206:
5203:
5201:
5200:Geostatistics
5198:
5196:
5193:
5191:
5188:
5186:
5183:
5182:
5180:
5178:
5174:
5168:
5167:Psychometrics
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5123:
5120:
5119:
5117:
5115:
5111:
5105:
5102:
5100:
5097:
5095:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5072:
5070:
5068:
5064:
5058:
5055:
5053:
5050:
5048:
5044:
5041:
5039:
5036:
5035:
5033:
5031:
5030:Biostatistics
5027:
5023:
5019:
5014:
5010:
4992:
4991:Log-rank test
4989:
4988:
4986:
4982:
4976:
4973:
4972:
4970:
4968:
4964:
4958:
4955:
4953:
4950:
4948:
4945:
4943:
4940:
4939:
4937:
4935:
4931:
4928:
4926:
4922:
4912:
4909:
4907:
4904:
4902:
4899:
4897:
4894:
4892:
4889:
4888:
4886:
4884:
4880:
4874:
4871:
4869:
4866:
4864:
4862:(Box–Jenkins)
4858:
4856:
4853:
4851:
4848:
4844:
4841:
4840:
4839:
4836:
4835:
4833:
4831:
4827:
4821:
4818:
4816:
4815:Durbin–Watson
4813:
4811:
4805:
4803:
4800:
4798:
4797:Dickey–Fuller
4795:
4794:
4792:
4788:
4782:
4779:
4777:
4774:
4772:
4771:Cointegration
4769:
4767:
4764:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4746:Decomposition
4744:
4743:
4741:
4737:
4734:
4732:
4728:
4718:
4715:
4714:
4713:
4710:
4709:
4708:
4705:
4701:
4698:
4697:
4696:
4693:
4691:
4688:
4686:
4683:
4681:
4678:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4657:
4655:
4653:
4649:
4643:
4640:
4638:
4635:
4633:
4630:
4628:
4625:
4623:
4620:
4618:
4617:Cohen's kappa
4615:
4614:
4612:
4610:
4606:
4602:
4598:
4594:
4590:
4586:
4581:
4577:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4544:
4542:
4540:
4536:
4530:
4526:
4522:
4516:
4514:
4511:
4510:
4508:
4506:
4502:
4496:
4493:
4491:
4488:
4486:
4483:
4481:
4478:
4476:
4473:
4471:
4470:Nonparametric
4468:
4466:
4463:
4462:
4460:
4456:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4431:
4429:
4427:
4423:
4417:
4414:
4412:
4409:
4407:
4404:
4402:
4399:
4397:
4394:
4393:
4391:
4389:
4385:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4360:
4358:
4356:
4352:
4348:
4341:
4338:
4336:
4333:
4332:
4328:
4324:
4308:
4305:
4304:
4303:
4300:
4298:
4295:
4293:
4290:
4286:
4283:
4281:
4278:
4277:
4276:
4273:
4272:
4270:
4268:
4264:
4254:
4251:
4247:
4241:
4239:
4233:
4231:
4225:
4224:
4223:
4220:
4219:Nonparametric
4217:
4215:
4209:
4205:
4202:
4201:
4200:
4194:
4190:
4189:Sample median
4187:
4186:
4185:
4182:
4181:
4179:
4177:
4173:
4165:
4162:
4160:
4157:
4155:
4152:
4151:
4150:
4147:
4145:
4142:
4140:
4134:
4132:
4129:
4127:
4124:
4122:
4119:
4117:
4114:
4112:
4110:
4106:
4104:
4101:
4100:
4098:
4096:
4092:
4086:
4084:
4080:
4078:
4076:
4071:
4069:
4064:
4060:
4059:
4056:
4053:
4051:
4047:
4037:
4034:
4032:
4029:
4027:
4024:
4023:
4021:
4019:
4015:
4009:
4006:
4002:
3999:
3998:
3997:
3994:
3990:
3987:
3986:
3985:
3982:
3980:
3977:
3976:
3974:
3972:
3968:
3960:
3957:
3955:
3952:
3951:
3950:
3947:
3945:
3942:
3940:
3937:
3935:
3932:
3930:
3927:
3925:
3922:
3921:
3919:
3917:
3913:
3907:
3904:
3900:
3897:
3893:
3890:
3888:
3885:
3884:
3883:
3880:
3879:
3878:
3875:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3852:
3851:
3848:
3847:
3845:
3843:
3839:
3836:
3834:
3830:
3824:
3821:
3819:
3816:
3812:
3809:
3808:
3807:
3804:
3802:
3799:
3795:
3794:loss function
3792:
3791:
3790:
3787:
3783:
3780:
3778:
3775:
3773:
3770:
3769:
3768:
3765:
3763:
3760:
3758:
3755:
3751:
3748:
3746:
3743:
3741:
3735:
3732:
3731:
3730:
3727:
3723:
3720:
3718:
3715:
3713:
3710:
3709:
3708:
3705:
3701:
3698:
3696:
3693:
3692:
3691:
3688:
3684:
3681:
3680:
3679:
3676:
3672:
3669:
3668:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3648:
3646:
3644:
3640:
3636:
3632:
3627:
3623:
3609:
3606:
3604:
3601:
3599:
3596:
3594:
3591:
3590:
3588:
3586:
3582:
3576:
3573:
3571:
3568:
3566:
3563:
3562:
3560:
3556:
3550:
3547:
3545:
3542:
3540:
3537:
3535:
3532:
3530:
3527:
3525:
3522:
3520:
3517:
3516:
3514:
3512:
3508:
3502:
3499:
3497:
3496:Questionnaire
3494:
3492:
3489:
3485:
3482:
3480:
3477:
3476:
3475:
3472:
3471:
3469:
3467:
3463:
3457:
3454:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3427:
3424:
3422:
3419:
3418:
3416:
3414:
3410:
3406:
3402:
3397:
3393:
3379:
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3323:Control chart
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3305:
3303:
3301:
3297:
3291:
3288:
3284:
3281:
3279:
3276:
3275:
3274:
3271:
3269:
3266:
3264:
3261:
3260:
3258:
3256:
3252:
3246:
3243:
3241:
3238:
3236:
3233:
3232:
3230:
3226:
3220:
3217:
3216:
3214:
3212:
3208:
3196:
3193:
3191:
3188:
3186:
3183:
3182:
3181:
3178:
3176:
3173:
3172:
3170:
3168:
3164:
3158:
3155:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3124:
3122:
3120:
3116:
3110:
3107:
3105:
3102:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3065:
3063:
3060:
3058:
3055:
3053:
3050:
3049:
3048:
3045:
3044:
3042:
3040:
3036:
3033:
3031:
3027:
3023:
3019:
3014:
3010:
3004:
3001:
2999:
2996:
2995:
2992:
2988:
2981:
2976:
2974:
2969:
2967:
2962:
2961:
2958:
2952:
2949:
2948:
2944:
2938:
2936:0-534-91976-6
2932:
2928:
2924:
2919:
2918:
2914:
2899:
2895:
2889:
2886:
2875:
2871:
2865:
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5142:Econometrics
5092: /
5075:Chemometrics
5052:Epidemiology
5045: /
5018:Applications
4860:ARIMA model
4807:Q-statistic
4756:Stationarity
4652:Multivariate
4595: /
4591: /
4589:Multivariate
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4527: /
4523: /
4297:Bayes factor
4226:
4196:Signed rank
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3757:Completeness
3593:Cohort study
3491:Opinion poll
3426:Missing data
3413:Study design
3368:Scatter plot
3290:Scatter plot
3283:Spearman's ρ
3245:Grouped data
2926:
2901:. Retrieved
2897:
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2873:
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2849:
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2483:
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2302:kruskal.test
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1937:
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1926:kruskal.test
1919:
1886:
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1849:significance
1435:type I error
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149:
124:ANOVA F-test
115:
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37:
5270:WikiProject
5185:Cartography
5147:Jurimetrics
5099:Reliability
4830:Time domain
4809:(Ljung–Box)
4731:Time-series
4609:Categorical
4593:Time-series
4585:Categorical
4520:(Bernoulli)
4355:Correlation
4335:Correlation
4131:Jarque–Bera
4103:Chi-squared
3865:M-estimator
3818:Asymptotics
3762:Sufficiency
3529:Interaction
3441:Replication
3421:Effect size
3378:Violin plot
3358:Radar chart
3338:Forest plot
3328:Correlogram
3278:Kendall's τ
2536:Holden-Day.
2267:open source
2112:comparisons
829:is exactly
562:from group
109:tests with
5286:Categories
5137:Demography
4855:ARMA model
4660:Regression
4237:(Friedman)
4198:(Wilcoxon)
4136:Normality
4126:Lilliefors
4073:Student's
3949:Resampling
3823:Robustness
3811:divergence
3801:Efficiency
3739:(monotone)
3734:Likelihood
3651:Population
3484:Stratified
3436:Population
3255:Dependence
3211:Count data
3142:Percentile
3119:Dispersion
3052:Arithmetic
2987:Statistics
2903:2023-12-06
2879:2023-12-06
2855:2023-12-06
2831:2023-12-06
2793:053498052X
2519:2016-10-28
2461:0070573573
2375:References
2244:bonferroni
2235:adjustment
2148:airquality
2136:airquality
2085:airquality
2073:airquality
1967:airquality
1922:airquality
42:by ranks,
4518:Logistic
4285:posterior
4211:Rank sum
3959:Jackknife
3954:Bootstrap
3772:Bootstrap
3707:Parameter
3656:Statistic
3451:Statistic
3363:Run chart
3348:Pie chart
3343:Histogram
3333:Fan chart
3308:Bar chart
3190:L-moments
3077:Geometric
2802:cite book
2057:6.901e-06
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4802:Johansen
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4480:Isotonic
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3712:location
3519:Blocking
3474:Sampling
3353:Q–Q plot
3318:Box plot
3300:Graphics
3195:Skewness
3185:Kurtosis
3157:Variance
3087:Heronian
3082:Harmonic
2727:Archived
2338:See also
2118:Wilcoxon
2109:Pairwise
1857:post hoc
1483:, where
1341:, where
5258:Commons
5205:Kriging
5090:Process
5047:studies
4906:Wavelet
4739:General
3906:Plug-in
3700:L space
3479:Cluster
3180:Moments
2998:Outline
2331:p-value
2021:squared
2006:Kruskal
1973:Kruskal
1911:Example
939:. Thus
5127:Census
4717:Normal
4665:Manova
4485:Robust
4235:2-way
4227:1-way
4065:-test
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3104:Median
3097:Lehmer
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2275:Python
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2208:1.0000
2205:0.2591
2202:0.0012
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2187:0.0003
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2012:Wallis
1979:Wallis
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4751:Trend
4280:prior
4222:anova
4111:-test
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4077:-test
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3092:Heinz
3067:Cubic
3003:Index
2513:(PDF)
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2279:SciPy
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1997:Ozone
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