815:
1361:
1070:
825:
for the population variance. In normal unweighted samples, the N in the denominator (corresponding to the sample size) is changed to N-1. While this is simple in unweighted samples, it becomes tedious for weighted samples. Thus, the unbiased estimator of weighted population variance is given by
548:
1082:
835:
477:
314:
810:{\displaystyle \sigma _{normal}^{2}\ ={\frac {\sum _{i=1}^{N}{\left(x_{i}-\mu \right)^{2}}}{N}}\;\;\;\sigma _{weighted}^{2}\ ={\frac {\sum _{i=1}^{N}{{w_{i}}\left(x_{i}-\mu \right)^{2}}}{\sum _{i=1}^{N}{w_{i}}}}}
1356:{\displaystyle s^{2}\ ={\frac {\sum _{i=1}^{N}{w_{i}{x_{i}}^{2}}\sum _{i=1}^{N}{w_{i}}-\left(\sum _{i=1}^{N}{w_{i}x_{i}}\right)^{2}}{\left(\sum _{i=1}^{N}{w_{i}}\right)^{2}-\sum _{i=1}^{N}{{w_{i}}^{2}}}}}
1065:{\displaystyle s^{2}\ ={\frac {\sum _{i=1}^{N}{w_{i}}}{\left(\sum _{i=1}^{N}{w_{i}}\right)^{2}-\sum _{i=1}^{N}{{w_{i}}^{2}}}}\ \sum _{i=1}^{N}{{w_{i}}\left(x_{i}-\mu \right)^{2}}}
325:
514:
In the special case, often encountered in practice, where the weights are normalized (i.e. are nonnegative and sum up to 1), the denominator of the fraction simplifies to 1.
193:
115:
489:. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counter-intuitive properties, as captured for instance in
204:
530:
of that mean. When a weighted mean is used, the variance of the weighted sample is different from the variance of the unweighted sample. The
482:
So data elements with a high weight contribute more to the weighted mean than do elements with a low weight.
497:
508:
501:
472:{\displaystyle {\bar {x}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}.}
496:
Weighted versions of other means can also be calculated. Examples of such weighted means include the
490:
822:
527:
827:
535:
486:
121:
29:
129:
51:
309:{\displaystyle {\bar {x}}={\frac {\sum _{i=1}^{n}w_{i}x_{i}}{\sum _{i=1}^{n}w_{i}}},}
17:
511:
and also occurs in a more general form in several other areas of mathematics.
523:
1366:
The standard deviation is simply the square root of the variance above.
485:
If all the weights are equal, then the weighted mean is the same as the
1076:
Which can also be written in terms of running sums for programming as:
198:
at least one of which is positive, is the quantity calculated by
522:
Typically when you calculate a mean it is important to know the
821:
For small sample of populations, it is customary to use an
1085:
838:
551:
328:
207:
132:
54:
1355:
1064:
809:
471:
308:
187:
109:
8:
507:The notion of weighted mean plays a role in
655:
654:
653:
1343:
1336:
1331:
1329:
1323:
1312:
1299:
1287:
1282:
1276:
1265:
1247:
1235:
1225:
1220:
1214:
1203:
1183:
1178:
1172:
1161:
1150:
1143:
1138:
1131:
1126:
1120:
1109:
1102:
1090:
1084:
1055:
1038:
1021:
1016:
1015:
1009:
998:
981:
974:
969:
967:
961:
950:
937:
925:
920:
914:
903:
884:
879:
873:
862:
855:
843:
837:
797:
792:
786:
775:
762:
745:
728:
723:
722:
716:
705:
698:
686:
660:
640:
623:
612:
606:
595:
588:
576:
556:
550:
457:
438:
425:
413:
403:
384:
374:
361:
351:
344:
330:
329:
327:
294:
284:
273:
261:
251:
241:
230:
223:
209:
208:
206:
181:
172:
153:
140:
131:
103:
94:
75:
62:
53:
7:
538:is defined similarly to the normal
24:
120:with corresponding non-negative
335:
214:
178:
133:
100:
55:
45:, of a non-empty list of data
1:
1380:
518:Weighted sample variance
32:for the continuous case.
498:weighted geometric mean
1357:
1328:
1281:
1219:
1177:
1125:
1066:
1014:
966:
919:
878:
811:
791:
721:
611:
509:descriptive statistics
502:weighted harmonic mean
473:
310:
289:
246:
189:
111:
1358:
1308:
1261:
1199:
1157:
1105:
1067:
994:
946:
899:
858:
812:
771:
701:
591:
474:
311:
269:
226:
190:
112:
1083:
836:
549:
326:
205:
130:
52:
691:
581:
188:{\displaystyle \,,}
110:{\displaystyle \,,}
1353:
1062:
823:unbiased estimator
807:
656:
552:
528:standard deviation
469:
306:
185:
107:
1351:
1098:
993:
989:
851:
805:
694:
651:
584:
542:sample variance:
491:Simpson's paradox
464:
338:
301:
217:
1371:
1362:
1360:
1359:
1354:
1352:
1350:
1349:
1348:
1347:
1342:
1341:
1340:
1327:
1322:
1304:
1303:
1298:
1294:
1293:
1292:
1291:
1280:
1275:
1253:
1252:
1251:
1246:
1242:
1241:
1240:
1239:
1230:
1229:
1218:
1213:
1189:
1188:
1187:
1176:
1171:
1156:
1155:
1154:
1149:
1148:
1147:
1136:
1135:
1124:
1119:
1103:
1096:
1095:
1094:
1071:
1069:
1068:
1063:
1061:
1060:
1059:
1054:
1050:
1043:
1042:
1027:
1026:
1025:
1013:
1008:
991:
990:
988:
987:
986:
985:
980:
979:
978:
965:
960:
942:
941:
936:
932:
931:
930:
929:
918:
913:
891:
890:
889:
888:
877:
872:
856:
849:
848:
847:
816:
814:
813:
808:
806:
804:
803:
802:
801:
790:
785:
769:
768:
767:
766:
761:
757:
750:
749:
734:
733:
732:
720:
715:
699:
692:
690:
685:
652:
647:
646:
645:
644:
639:
635:
628:
627:
610:
605:
589:
582:
580:
575:
478:
476:
475:
470:
465:
463:
462:
461:
443:
442:
430:
429:
419:
418:
417:
408:
407:
389:
388:
379:
378:
366:
365:
356:
355:
345:
340:
339:
331:
315:
313:
312:
307:
302:
300:
299:
298:
288:
283:
267:
266:
265:
256:
255:
245:
240:
224:
219:
218:
210:
194:
192:
191:
186:
177:
176:
158:
157:
145:
144:
116:
114:
113:
108:
99:
98:
80:
79:
67:
66:
43:weighted average
1379:
1378:
1374:
1373:
1372:
1370:
1369:
1368:
1332:
1330:
1283:
1260:
1256:
1255:
1254:
1231:
1221:
1198:
1194:
1193:
1179:
1139:
1137:
1127:
1104:
1086:
1081:
1080:
1075:
1034:
1033:
1029:
1028:
1017:
970:
968:
921:
898:
894:
893:
892:
880:
857:
839:
834:
833:
820:
793:
770:
741:
740:
736:
735:
724:
700:
619:
618:
614:
613:
590:
547:
546:
536:sample variance
520:
487:arithmetic mean
453:
434:
421:
420:
409:
399:
380:
370:
357:
347:
346:
324:
323:
290:
268:
257:
247:
225:
203:
202:
168:
149:
136:
128:
127:
90:
71:
58:
50:
49:
30:weight function
22:
21:
20:
12:
11:
5:
1377:
1375:
1364:
1363:
1346:
1339:
1335:
1326:
1321:
1318:
1315:
1311:
1307:
1302:
1297:
1290:
1286:
1279:
1274:
1271:
1268:
1264:
1259:
1250:
1245:
1238:
1234:
1228:
1224:
1217:
1212:
1209:
1206:
1202:
1197:
1192:
1186:
1182:
1175:
1170:
1167:
1164:
1160:
1153:
1146:
1142:
1134:
1130:
1123:
1118:
1115:
1112:
1108:
1101:
1093:
1089:
1073:
1072:
1058:
1053:
1049:
1046:
1041:
1037:
1032:
1024:
1020:
1012:
1007:
1004:
1001:
997:
984:
977:
973:
964:
959:
956:
953:
949:
945:
940:
935:
928:
924:
917:
912:
909:
906:
902:
897:
887:
883:
876:
871:
868:
865:
861:
854:
846:
842:
818:
817:
800:
796:
789:
784:
781:
778:
774:
765:
760:
756:
753:
748:
744:
739:
731:
727:
719:
714:
711:
708:
704:
697:
689:
684:
681:
678:
675:
672:
669:
666:
663:
659:
650:
643:
638:
634:
631:
626:
622:
617:
609:
604:
601:
598:
594:
587:
579:
574:
571:
568:
565:
562:
559:
555:
519:
516:
480:
479:
468:
460:
456:
452:
449:
446:
441:
437:
433:
428:
424:
416:
412:
406:
402:
398:
395:
392:
387:
383:
377:
373:
369:
364:
360:
354:
350:
343:
337:
334:
317:
316:
305:
297:
293:
287:
282:
279:
276:
272:
264:
260:
254:
250:
244:
239:
236:
233:
229:
222:
216:
213:
196:
195:
184:
180:
175:
171:
167:
164:
161:
156:
152:
148:
143:
139:
135:
118:
117:
106:
102:
97:
93:
89:
86:
83:
78:
74:
70:
65:
61:
57:
35:
34:
23:
15:
14:
13:
10:
9:
6:
4:
3:
2:
1376:
1367:
1344:
1337:
1333:
1324:
1319:
1316:
1313:
1309:
1305:
1300:
1295:
1288:
1284:
1277:
1272:
1269:
1266:
1262:
1257:
1248:
1243:
1236:
1232:
1226:
1222:
1215:
1210:
1207:
1204:
1200:
1195:
1190:
1184:
1180:
1173:
1168:
1165:
1162:
1158:
1151:
1144:
1140:
1132:
1128:
1121:
1116:
1113:
1110:
1106:
1099:
1091:
1087:
1079:
1078:
1077:
1056:
1051:
1047:
1044:
1039:
1035:
1030:
1022:
1018:
1010:
1005:
1002:
999:
995:
982:
975:
971:
962:
957:
954:
951:
947:
943:
938:
933:
926:
922:
915:
910:
907:
904:
900:
895:
885:
881:
874:
869:
866:
863:
859:
852:
844:
840:
832:
831:
830:
828:
824:
798:
794:
787:
782:
779:
776:
772:
763:
758:
754:
751:
746:
742:
737:
729:
725:
717:
712:
709:
706:
702:
695:
687:
682:
679:
676:
673:
670:
667:
664:
661:
657:
648:
641:
636:
632:
629:
624:
620:
615:
607:
602:
599:
596:
592:
585:
577:
572:
569:
566:
563:
560:
557:
553:
545:
544:
543:
541:
537:
533:
529:
525:
517:
515:
512:
510:
505:
503:
499:
494:
492:
488:
483:
466:
458:
454:
450:
447:
444:
439:
435:
431:
426:
422:
414:
410:
404:
400:
396:
393:
390:
385:
381:
375:
371:
367:
362:
358:
352:
348:
341:
332:
322:
321:
320:
319:which means:
303:
295:
291:
285:
280:
277:
274:
270:
262:
258:
252:
248:
242:
237:
234:
231:
227:
220:
211:
201:
200:
199:
182:
173:
169:
165:
162:
159:
154:
150:
146:
141:
137:
126:
125:
124:
123:
104:
95:
91:
87:
84:
81:
76:
72:
68:
63:
59:
48:
47:
46:
44:
40:
39:weighted mean
33:
31:
26:
25:
19:
1365:
1074:
819:
539:
531:
521:
513:
506:
495:
484:
481:
318:
197:
119:
42:
38:
36:
27:
18:User:Vossman
1310:∑
1306:−
1263:∑
1201:∑
1191:−
1159:∑
1107:∑
1048:μ
1045:−
996:∑
948:∑
944:−
901:∑
860:∑
773:∑
755:μ
752:−
703:∑
658:σ
633:μ
630:−
593:∑
554:σ
534:weighted
448:⋯
394:⋯
336:¯
271:∑
228:∑
215:¯
163:…
85:…
524:variance
500:and the
122:weights
1097:
992:
850:
693:
583:
540:biased
532:biased
41:, or
16:<
526:and
37:The
28:See
829::
504:.
493:.
1345:2
1338:i
1334:w
1325:N
1320:1
1317:=
1314:i
1301:2
1296:)
1289:i
1285:w
1278:N
1273:1
1270:=
1267:i
1258:(
1249:2
1244:)
1237:i
1233:x
1227:i
1223:w
1216:N
1211:1
1208:=
1205:i
1196:(
1185:i
1181:w
1174:N
1169:1
1166:=
1163:i
1152:2
1145:i
1141:x
1133:i
1129:w
1122:N
1117:1
1114:=
1111:i
1100:=
1092:2
1088:s
1057:2
1052:)
1040:i
1036:x
1031:(
1023:i
1019:w
1011:N
1006:1
1003:=
1000:i
983:2
976:i
972:w
963:N
958:1
955:=
952:i
939:2
934:)
927:i
923:w
916:N
911:1
908:=
905:i
896:(
886:i
882:w
875:N
870:1
867:=
864:i
853:=
845:2
841:s
799:i
795:w
788:N
783:1
780:=
777:i
764:2
759:)
747:i
743:x
738:(
730:i
726:w
718:N
713:1
710:=
707:i
696:=
688:2
683:d
680:e
677:t
674:h
671:g
668:i
665:e
662:w
649:N
642:2
637:)
625:i
621:x
616:(
608:N
603:1
600:=
597:i
586:=
578:2
573:l
570:a
567:m
564:r
561:o
558:n
467:.
459:n
455:w
451:+
445:+
440:2
436:w
432:+
427:1
423:w
415:n
411:x
405:n
401:w
397:+
391:+
386:2
382:x
376:2
372:w
368:+
363:1
359:x
353:1
349:w
342:=
333:x
304:,
296:i
292:w
286:n
281:1
278:=
275:i
263:i
259:x
253:i
249:w
243:n
238:1
235:=
232:i
221:=
212:x
183:,
179:]
174:n
170:w
166:,
160:,
155:2
151:w
147:,
142:1
138:w
134:[
105:,
101:]
96:n
92:x
88:,
82:,
77:2
73:x
69:,
64:1
60:x
56:[
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