24:
as a measure which seeks to evaluate differences between the statistical properties of random mechanisms where the outcome is two-valued — for example "success" or "failure", "win" or "lose". The idea is that the probability of success might vary between different sets of trials in different
32:
This measure compares the between-set variance of the sample proportions (evaluated for each set) with what the variance should be if there were no difference between in the true proportions of success across the different sets. Thus the measure is used to evaluate how data compares to a
391:
If the Lexis ratio is significantly below 1, the sampling is referred to as
Poissonian (or subnormal); it is equal to 1 the sampling is referred to as Bernoullian (or normal); and if it is above 1 it is referred to as Lexian (or supranormal).
386:
204:
This ratio ( Q ) is a measure that can be used to distinguish between three types of variation in sampling for attributes: Bernoullian, Lexian and
Poissonian. The Lexis ratio is sometimes also referred to as
121:
869:
564:
734:
492:
631:
187:
595:
431:
151:
189:
is the variance computed from the expected
Bernoulli distribution on the basis of the overall average proportion of success. Trials where
283:
51:
745:
633:
is the variance computed from the expected
Bernoulli distribution on the basis of the overall average proportion of success.
919:
introduced this statistic to test the then commonly held assumption that sampling data could be regarded as homogeneous.
517:
968:
963:
691:
441:
34:
604:
160:
504:() is the variance. The formula for the variance is approximate and holds only for large values of
25:
situations. This ratio is not much used currently having been largely replaced by the use of the
572:
401:
26:
129:
598:
154:
948:
905:
differ the sampling is said to be Lexian and the dispersion is said to be supranormal.
957:
916:
935:
Lexis W (1877) Zur
Theorie Der Massenerscheinungen in Der Menschlichen Gesellschaft.
601:
derived from the observed proportions of success in sets in "Lexis trials" and
157:
derived from the observed proportions of success in sets in "Lexis trials" and
21:
249:
and these samples have the proportion of the attribute being examined of
395:
Chuprov showed in 1922 that in the case of statistical homogeneity
381:{\displaystyle Q={\frac {\sum {n_{i}(p_{i}-p)^{2}}}{(k-1)p(1-p)}}}
908:
Lexian sampling occurs in sampling from non homogenous strata.
653:) be constant and let the actual probability of success in the
116:{\displaystyle L^{2}=Q^{2}={\frac {s^{2}}{\sigma _{0}^{2}}}.}
898:
are equal the sampling is said to be
Bernoullian; where the
864:{\displaystyle var(successes)=np(1-p)+n(n-1)var(p_{i})}
641:
A closely related concept is the Lexis variation. Let
748:
694:
607:
575:
520:
444:
404:
286:
163:
132:
54:
37:. The term "Lexis ratio" is sometimes referred to as
649:
be drawn at random. Let the probability of success (
863:
728:
625:
589:
559:{\displaystyle Q={\frac {s^{2}}{\sigma _{0}^{2}}}}
558:
486:
425:
380:
193:falls significantly above or below 1 are known as
181:
145:
115:
8:
729:{\displaystyle p={\frac {1}{k}}\sum {p_{i}}}
739:The variance in the number of successes is
29:in testing for the homogeneity of samples.
852:
747:
719:
714:
701:
693:
617:
612:
606:
586:
580:
574:
548:
543:
533:
527:
519:
466:
443:
403:
333:
317:
304:
299:
293:
285:
173:
168:
162:
137:
131:
102:
97:
87:
81:
72:
59:
53:
928:
487:{\displaystyle var(Q)={\frac {2}{n-1}}}
277:respectively. Then the Lexis ratio is
7:
681:The average probability of success (
14:
626:{\displaystyle \sigma _{0}^{2}}
182:{\displaystyle \sigma _{0}^{2}}
949:Overdispersion § Binomial
858:
845:
833:
821:
812:
800:
788:
758:
460:
454:
414:
408:
372:
360:
354:
342:
330:
310:
1:
511:An alternative definition is
33:fixed-probability-of-success
985:
500:() is the expectation and
881:) is the variance of the
590:{\displaystyle s^{2}\,}
865:
730:
627:
591:
560:
488:
427:
426:{\displaystyle E(Q)=1}
382:
183:
147:
117:
35:Bernoulli distribution
866:
731:
645:samples each of size
628:
592:
561:
489:
428:
383:
184:
148:
146:{\displaystyle s^{2}}
118:
746:
692:
605:
573:
518:
442:
402:
284:
161:
130:
52:
622:
553:
178:
107:
969:Statistical ratios
964:Summary statistics
861:
726:
623:
608:
597:is the (weighted)
587:
556:
539:
484:
423:
378:
179:
164:
153:is the (weighted)
143:
113:
93:
709:
554:
482:
376:
108:
976:
936:
933:
870:
868:
867:
862:
857:
856:
735:
733:
732:
727:
725:
724:
723:
710:
702:
632:
630:
629:
624:
621:
616:
596:
594:
593:
588:
585:
584:
565:
563:
562:
557:
555:
552:
547:
538:
537:
528:
493:
491:
490:
485:
483:
481:
467:
432:
430:
429:
424:
387:
385:
384:
379:
377:
375:
340:
339:
338:
337:
322:
321:
309:
308:
294:
221:samples of size
188:
186:
185:
180:
177:
172:
152:
150:
149:
144:
142:
141:
122:
120:
119:
114:
109:
106:
101:
92:
91:
82:
77:
76:
64:
63:
27:chi-squared test
984:
983:
979:
978:
977:
975:
974:
973:
954:
953:
945:
940:
939:
934:
930:
925:
914:
904:
897:
887:
880:
848:
744:
743:
715:
690:
689:
677:
670:
663:
639:
637:Lexis variation
603:
602:
599:sample variance
576:
571:
570:
529:
516:
515:
471:
440:
439:
400:
399:
341:
329:
313:
300:
295:
282:
281:
276:
269:
262:
255:
248:
241:
234:
227:
215:
159:
158:
155:sample variance
133:
128:
127:
83:
68:
55:
50:
49:
12:
11:
5:
982:
980:
972:
971:
966:
956:
955:
952:
951:
944:
941:
938:
937:
927:
926:
924:
921:
913:
910:
902:
895:
885:
878:
872:
871:
860:
855:
851:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
737:
736:
722:
718:
713:
708:
705:
700:
697:
675:
668:
661:
638:
635:
620:
615:
611:
583:
579:
567:
566:
551:
546:
542:
536:
532:
526:
523:
480:
477:
474:
470:
465:
462:
459:
456:
453:
450:
447:
434:
433:
422:
419:
416:
413:
410:
407:
389:
388:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
336:
332:
328:
325:
320:
316:
312:
307:
303:
298:
292:
289:
274:
267:
260:
253:
246:
239:
232:
225:
214:
211:
201:respectively.
176:
171:
167:
140:
136:
124:
123:
112:
105:
100:
96:
90:
86:
80:
75:
71:
67:
62:
58:
13:
10:
9:
6:
4:
3:
2:
981:
970:
967:
965:
962:
961:
959:
950:
947:
946:
942:
932:
929:
922:
920:
918:
917:Wilhelm Lexis
911:
909:
906:
901:
894:
889:
884:
877:
853:
849:
842:
839:
836:
830:
827:
824:
818:
815:
809:
806:
803:
797:
794:
791:
785:
782:
779:
776:
773:
770:
767:
764:
761:
755:
752:
749:
742:
741:
740:
720:
716:
711:
706:
703:
698:
695:
688:
687:
686:
684:
679:
674:
667:
660:
656:
652:
648:
644:
636:
634:
618:
613:
609:
600:
581:
577:
549:
544:
540:
534:
530:
524:
521:
514:
513:
512:
509:
507:
503:
499:
494:
478:
475:
472:
468:
463:
457:
451:
448:
445:
437:
420:
417:
411:
405:
398:
397:
396:
393:
369:
366:
363:
357:
351:
348:
345:
334:
326:
323:
318:
314:
305:
301:
296:
290:
287:
280:
279:
278:
273:
266:
259:
252:
245:
238:
231:
224:
220:
217:Let there be
212:
210:
208:
202:
200:
196:
192:
174:
169:
165:
156:
138:
134:
110:
103:
98:
94:
88:
84:
78:
73:
69:
65:
60:
56:
48:
47:
46:
44:
40:
36:
30:
28:
23:
19:
931:
915:
907:
899:
892:
890:
882:
875:
873:
738:
682:
680:
672:
665:
658:
654:
650:
646:
642:
640:
568:
510:
505:
501:
497:
495:
438:
435:
394:
390:
271:
264:
257:
250:
243:
236:
229:
222:
218:
216:
206:
203:
198:
194:
190:
125:
42:
38:
31:
17:
15:
891:If all the
874:where var(
195:supernormal
20:is used in
18:Lexis ratio
958:Categories
923:References
657:sample be
213:Definition
199:subnormal,
22:statistics
828:−
807:−
712:∑
610:σ
541:σ
476:−
367:−
349:−
324:−
297:∑
166:σ
95:σ
943:See also
671:, ... ,
242:, ... ,
45:, where
912:History
270:, ...,
496:where
126:Where
685:) is
569:here
436:and
197:and
16:The
502:var
41:or
960::
888:.
678:.
664:,
508:.
263:,
256:,
235:,
228:,
209:.
903:i
900:p
896:i
893:p
886:i
883:p
879:i
876:p
859:)
854:i
850:p
846:(
843:r
840:a
837:v
834:)
831:1
825:n
822:(
819:n
816:+
813:)
810:p
804:1
801:(
798:p
795:n
792:=
789:)
786:s
783:e
780:s
777:s
774:e
771:c
768:c
765:u
762:s
759:(
756:r
753:a
750:v
721:i
717:p
707:k
704:1
699:=
696:p
683:p
676:k
673:p
669:2
666:p
662:1
659:p
655:k
651:p
647:n
643:k
619:2
614:0
582:2
578:s
550:2
545:0
535:2
531:s
525:=
522:Q
506:n
498:E
479:1
473:n
469:2
464:=
461:)
458:Q
455:(
452:r
449:a
446:v
421:1
418:=
415:)
412:Q
409:(
406:E
373:)
370:p
364:1
361:(
358:p
355:)
352:1
346:k
343:(
335:2
331:)
327:p
319:i
315:p
311:(
306:i
302:n
291:=
288:Q
275:k
272:p
268:3
265:p
261:2
258:p
254:1
251:p
247:k
244:n
240:3
237:n
233:3
230:n
226:1
223:n
219:k
207:L
191:L
175:2
170:0
139:2
135:s
111:.
104:2
99:0
89:2
85:s
79:=
74:2
70:Q
66:=
61:2
57:L
43:Q
39:L
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