3503:
3067:
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1129:
might actually exceed the error in the least noisy measurement if different measurements have very different errors. Instead of discarding the noisy measurements that increase the final error, the experimenter can combine all the measurements with appropriate weights so as to give more importance to
3338:
903:
1505:
2804:
2691:
3062:{\displaystyle Var(Y)=\sum _{i}{\frac {\sigma _{0}^{4}}{\sigma _{i}^{4}}}\sigma _{i}^{2}=\sigma _{0}^{4}\sum _{i}{\frac {1}{\sigma _{i}^{2}}}=\sigma _{0}^{4}{\frac {1}{\sigma _{0}^{2}}}=\sigma _{0}^{2}={\frac {1}{\sum _{i}1/\sigma _{i}^{2}}}.}
185:
1610:
2414:
3423:
2272:
1691:
1322:
283:
1208:
730:
3219:
2492:
1993:
791:
2548:
418:. If they are all noisy but unbiased, i.e., the measuring device does not systematically overestimate or underestimate the true value and the errors are scattered symmetrically, then the
547:
1727:
1375:
1380:
1788:
416:
2702:
3211:
3182:
2591:
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1858:
1127:
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982:
667:
1074:
to be the same. Some instruments could be noisier than others. In the example of measuring the acceleration due to gravity, the different "instruments" could be measuring
1072:
930:
574:
483:
594:
74:
3145:
2301:
2070:
2031:
461:
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2143:
1885:
1815:
1228:
786:
638:
329:
1510:
1905:
1092:
1045:
1025:
1005:
950:
766:
614:
349:
2309:
3520:
3346:
2151:
3081:
random variables inverse-variance weighted averages can also be derived as the maximum likelihood estimate for the true value. Furthermore, from a
1617:
1237:
196:
3484:
3567:
1133:
3539:
3586:
3546:
3333:{\displaystyle \mathbf {\hat {x}} =\left(\sum _{i}\mathbf {C} _{i}^{-1}\right)^{-1}\sum _{i}\mathbf {C} _{i}^{-1}\mathbf {x} _{i}}
2813:
by noting that the variance is a quadratic function of the weights. Thus, the minimum variance of the estimator is then given by:
675:
2425:
1917:
788:
but also has a scatter. If the individual measurements are uncorrelated, the square of the error in the estimate is given by
3524:
2810:
898:{\displaystyle Var({\overline {X}})={\frac {1}{n^{2}}}\sum _{i}\sigma _{i}^{2}=\left({\frac {\sigma }{\sqrt {n}}}\right)^{2}}
3553:
288:
If the variances of the measurements are all equal, then the inverse-variance weighted average becomes the simple average.
3535:
3613:
3155:
For multivariate distributions an equivalent argument leads to an optimal weighting based on the covariance matrices
190:
The inverse-variance weighted average has the least variance among all weighted averages, which can be calculated as
2500:
3513:
1693:, the estimator has a scatter smaller than the scatter in any individual measurement. Furthermore, the scatter in
492:
1908:
1696:
1500:{\displaystyle Var({\hat {\mu }})={\frac {\sum _{i}w_{i}^{2}\sigma _{i}^{2}}{\left(\sum _{i}w_{i}\right)^{2}}}}
1327:
3112:
and a flat prior is a normal distribution with the inverse-variance weighted average as a mean and variance
1047:
different instruments with varying quality of measurements, then there is no reason to expect the different
1740:
356:
3608:
3437:
2799:{\displaystyle w_{k}={\frac {1}{\sigma _{k}^{2}}}\left(\sum _{i}{\frac {1}{\sigma _{i}^{2}}}\right)^{-1}.}
44:
3560:
2686:{\displaystyle {\frac {2}{w_{0}}}=\sum _{i}{\frac {1}{\sigma _{i}^{2}}}:={\frac {1}{\sigma _{0}^{2}}}.}
3187:
3158:
2553:
2075:
1820:
3476:
1105:
3447:
3078:
2116:
955:
643:
3085:
perspective the posterior distribution for the true value given normally distributed observations
3082:
1050:
908:
552:
40:
180:{\displaystyle {\hat {y}}={\frac {\sum _{i}y_{i}/\sigma _{i}^{2}}{\sum _{i}1/\sigma _{i}^{2}}}.}
465:
307:
Suppose an experimenter wishes to measure the value of a quantity, say the acceleration due to
3480:
1099:
419:
579:
3442:
3428:
For multivariate distributions the term "precision-weighted" average is more commonly used.
3115:
2280:
2040:
2001:
1605:{\displaystyle Var({\hat {\mu }}_{\text{opt}})=\left(\sum _{i}\sigma _{i}^{-2}\right)^{-1}.}
424:
351:
308:
36:
28:
3088:
2121:
1863:
1793:
1213:
1095:
771:
623:
314:
2409:{\displaystyle 0={\frac {\partial }{\partial w_{k}}}Var(Y)=2w_{k}\sigma _{k}^{2}-w_{0},}
3469:
1890:
1077:
1030:
1010:
990:
935:
735:
599:
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732:. Note that this empirical average is also a random variable, whose expectation value
3602:
1231:
296:
292:
3418:{\displaystyle \mathbf {\hat {C}} =\left(\sum _{i}\mathbf {C} _{i}^{-1}\right)^{-1}}
1729:
decreases with adding more measurements, however noisier those measurements may be.
2809:
It is easy to see that this extremum solution corresponds to the minimum from the
2267:{\displaystyle Var(Y)=\sum _{i}w_{i}^{2}\sigma _{i}^{2}-w_{0}(\sum _{i}w_{i}-1).}
3502:
549:, and if the measurements are performed under identical scenarios, then all the
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etc. The simple average is no longer an optimal estimator, since the error in
20:
617:
1686:{\displaystyle Var({\hat {\mu }}_{\text{opt}})<\min _{j}\sigma _{j}^{2}}
1317:{\displaystyle {\hat {\mu }}={\frac {\sum _{i}w_{i}X_{i}}{\sum _{i}w_{i}}}}
331:. A careful experimenter makes multiple measurements, which we denote with
2034:
486:
32:
278:{\displaystyle Var({\hat {y}})={\frac {1}{\sum _{i}1/\sigma _{i}^{2}}}.}
670:
1007:
repeated measurements with one instrument, if the experimenter makes
932:
are equal, then the error in the estimate decreases with increase in
1130:
the least noisy measurements and vice versa. Given the knowledge of
1203:{\displaystyle \sigma _{1}^{2},\sigma _{2}^{2},...,\sigma _{n}^{2}}
3496:
485:. The scatter in the measurement is then characterised by the
725:{\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i}X_{i}}
291:
Inverse-variance weighting is typically used in statistical
2487:{\displaystyle w_{k}={\frac {w_{0}/2}{\sigma _{k}^{2}}}.}
1988:{\displaystyle Var(Y)=\sum _{i}w_{i}^{2}\sigma _{i}^{2}.}
299:
to combine the results from independent measurements.
3467:
Joachim
Hartung; Guido Knapp; Bimal K. Sinha (2008).
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2004:
1920:
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1866:
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1743:
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1513:
1507:, which for the optimal choice of the weights become
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77:
2145:
to enforce the constraint, we express the variance:
68:, the inverse-variance weighted average is given by
3527:. Unsourced material may be challenged and removed.
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2072:to zero, while maintaining the constraint that
2543:{\displaystyle w_{k}\propto 1/\sigma _{k}^{2}}
50:Given a sequence of independent observations
8:
3471:Statistical meta-analysis with applications
1324:, for the particular choice of the weights
984:, thus making more observations preferred.
542:{\displaystyle Var(X_{i}):=\sigma _{i}^{2}}
43:to its variance (i.e., proportional to its
3184:of the individual vector-valued estimates
1722:{\displaystyle {\hat {\mu }}_{\text{opt}}}
3587:Learn how and when to remove this message
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581:
576:are the same, which we shall refer to by
560:
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110:
100:
93:
79:
78:
76:
3459:
2696:The individual normalised weights are:
1370:{\displaystyle w_{i}=1/\sigma _{i}^{2}}
27:is a method of aggregating two or more
39:. Each random variable is weighted in
1887:are all independent, the variance of
1783:{\displaystyle Y=\sum _{i}w_{i}X_{i}}
411:{\displaystyle X_{1},X_{2},...,X_{n}}
7:
3525:adding citations to reliable sources
1998:For optimality, we wish to minimise
2325:
2321:
2033:which can be done by equating the
1210:, an optimal estimator to measure
469:
14:
311:, whose true value happens to be
3501:
3383:
3353:
3320:
3300:
3256:
3226:
3206:{\displaystyle \mathbf {x} _{i}}
3193:
3177:{\displaystyle \mathbf {C} _{i}}
3164:
2586:{\displaystyle \sum _{i}w_{i}=1}
2108:{\displaystyle \sum _{i}w_{i}=1}
1853:{\displaystyle \sum _{i}w_{i}=1}
1737:Consider a generic weighted sum
1377:. The variance of the estimator
3512:needs additional citations for
2497:The main takeaway here is that
2037:with respect to the weights of
1122:{\displaystyle {\overline {X}}}
3134:
3128:
2838:
2832:
2811:second partial derivative test
2356:
2350:
2258:
2229:
2170:
2164:
2059:
2053:
2020:
2014:
1936:
1930:
1707:
1652:
1640:
1630:
1545:
1533:
1523:
1408:
1402:
1393:
1247:
817:
804:
755:
742:
653:
518:
505:
444:
431:
224:
218:
209:
84:
1:
977:{\displaystyle 1/{\sqrt {n}}}
662:{\displaystyle {\hat {\mu }}}
3536:"Inverse-variance weighting"
1114:
812:
750:
684:
1067:{\displaystyle \sigma _{i}}
925:{\displaystyle \sigma _{i}}
569:{\displaystyle \sigma _{i}}
3630:
1027:of the same quantity with
25:inverse-variance weighting
1817:are normalised such that
669:, is given by the simple
478:{\displaystyle \forall i}
616:measurements, a typical
489:of the random variables
589:{\displaystyle \sigma }
3438:Weighted least squares
3419:
3334:
3207:
3178:
3141:
3140:{\displaystyle Var(Y)}
3106:
3063:
2800:
2687:
2587:
2544:
2488:
2410:
2297:
2296:{\displaystyle k>0}
2268:
2139:
2109:
2066:
2065:{\displaystyle Var(Y)}
2027:
2026:{\displaystyle Var(Y)}
1989:
1901:
1881:
1854:
1811:
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1723:
1687:
1606:
1501:
1371:
1318:
1224:
1204:
1123:
1088:
1068:
1041:
1021:
1001:
978:
946:
926:
899:
782:
762:
726:
663:
634:
610:
590:
570:
543:
479:
457:
456:{\displaystyle E=\mu }
412:
345:
325:
279:
181:
3477:John Wiley & Sons
3420:
3335:
3208:
3179:
3142:
3107:
3105:{\displaystyle y_{i}}
3064:
2801:
2688:
2588:
2545:
2489:
2411:
2298:
2269:
2140:
2138:{\displaystyle w_{0}}
2110:
2067:
2028:
1990:
1902:
1882:
1880:{\displaystyle X_{i}}
1855:
1812:
1810:{\displaystyle w_{i}}
1785:
1724:
1688:
1607:
1502:
1372:
1319:
1225:
1205:
1124:
1089:
1069:
1042:
1022:
1002:
979:
947:
927:
900:
783:
763:
727:
664:
635:
611:
591:
571:
544:
480:
458:
413:
346:
326:
280:
182:
3521:improve this article
3347:
3220:
3188:
3159:
3116:
3089:
3079:normally distributed
3073:Normal distributions
2820:
2703:
2600:
2554:
2501:
2426:
2419:which implies that:
2310:
2281:
2152:
2122:
2076:
2041:
2002:
1918:
1891:
1864:
1821:
1794:
1790:, where the weights
1741:
1697:
1618:
1511:
1381:
1328:
1238:
1234:of the measurements
1223:{\displaystyle \mu }
1214:
1134:
1106:
1078:
1051:
1031:
1011:
991:
956:
936:
909:
905:. Hence, if all the
792:
781:{\displaystyle \mu }
772:
736:
676:
644:
633:{\displaystyle \mu }
624:
600:
580:
553:
493:
466:
425:
357:
335:
324:{\displaystyle \mu }
315:
197:
75:
3400:
3317:
3273:
3052:
3010:
2990:
2970:
2950:
2920:
2902:
2885:
2870:
2776:
2738:
2677:
2652:
2539:
2478:
2389:
2215:
2200:
2117:Lagrange multiplier
1981:
1966:
1909:Bienaymé's identity
1682:
1584:
1456:
1441:
1366:
1199:
1169:
1151:
1098:, from analysing a
864:
538:
268:
170:
135:
3614:Estimation methods
3415:
3381:
3380:
3330:
3298:
3297:
3254:
3253:
3203:
3174:
3137:
3102:
3059:
3038:
3029:
2996:
2976:
2956:
2936:
2930:
2906:
2888:
2871:
2856:
2853:
2796:
2762:
2756:
2724:
2683:
2663:
2638:
2632:
2583:
2566:
2540:
2525:
2484:
2464:
2406:
2375:
2293:
2264:
2241:
2201:
2186:
2185:
2135:
2105:
2088:
2062:
2023:
1985:
1967:
1952:
1951:
1897:
1877:
1850:
1833:
1807:
1780:
1759:
1719:
1683:
1668:
1667:
1602:
1567:
1566:
1497:
1473:
1442:
1427:
1426:
1367:
1352:
1314:
1300:
1268:
1220:
1200:
1185:
1155:
1137:
1119:
1084:
1064:
1037:
1017:
997:
974:
942:
922:
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849:
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758:
722:
711:
659:
630:
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566:
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524:
475:
453:
408:
341:
321:
275:
254:
245:
177:
156:
147:
121:
105:
41:inverse proportion
16:Statistical method
3597:
3596:
3589:
3571:
3486:978-0-470-29089-7
3371:
3359:
3288:
3244:
3232:
3151:Multivariate case
3054:
3020:
2991:
2951:
2921:
2886:
2844:
2777:
2747:
2739:
2678:
2653:
2623:
2618:
2557:
2479:
2339:
2232:
2176:
2079:
1942:
1907:is given by (see
1900:{\displaystyle Y}
1824:
1750:
1716:
1710:
1658:
1649:
1643:
1557:
1542:
1536:
1495:
1464:
1417:
1405:
1312:
1291:
1259:
1250:
1117:
1100:projectile motion
1087:{\displaystyle g}
1040:{\displaystyle n}
1020:{\displaystyle n}
1000:{\displaystyle n}
972:
945:{\displaystyle n}
883:
882:
840:
838:
815:
761:{\displaystyle E}
753:
702:
700:
687:
656:
609:{\displaystyle n}
420:expectation value
344:{\displaystyle n}
270:
236:
221:
172:
138:
96:
87:
3621:
3592:
3585:
3581:
3578:
3572:
3570:
3529:
3505:
3497:
3491:
3490:
3474:
3464:
3448:Cramér-Rao bound
3443:Portfolio theory
3424:
3422:
3421:
3416:
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3413:
3405:
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3399:
3391:
3386:
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3167:
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3060:
3055:
3053:
3051:
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3015:
3009:
3004:
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2989:
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2714:
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2512:
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2415:
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2407:
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2401:
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2383:
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2373:
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2302:
2300:
2299:
2294:
2273:
2271:
2270:
2265:
2251:
2250:
2240:
2228:
2227:
2214:
2209:
2199:
2194:
2184:
2144:
2142:
2141:
2136:
2134:
2133:
2114:
2112:
2111:
2106:
2098:
2097:
2087:
2071:
2069:
2068:
2063:
2032:
2030:
2029:
2024:
1994:
1992:
1991:
1986:
1980:
1975:
1965:
1960:
1950:
1906:
1904:
1903:
1898:
1886:
1884:
1883:
1878:
1876:
1875:
1859:
1857:
1856:
1851:
1843:
1842:
1832:
1816:
1814:
1813:
1808:
1806:
1805:
1789:
1787:
1786:
1781:
1779:
1778:
1769:
1768:
1758:
1728:
1726:
1725:
1720:
1718:
1717:
1714:
1712:
1711:
1703:
1692:
1690:
1689:
1684:
1681:
1676:
1666:
1651:
1650:
1647:
1645:
1644:
1636:
1614:Note that since
1611:
1609:
1608:
1603:
1598:
1597:
1589:
1585:
1583:
1575:
1565:
1544:
1543:
1540:
1538:
1537:
1529:
1506:
1504:
1503:
1498:
1496:
1494:
1493:
1488:
1484:
1483:
1482:
1472:
1457:
1455:
1450:
1440:
1435:
1425:
1415:
1407:
1406:
1398:
1376:
1374:
1373:
1368:
1365:
1360:
1351:
1340:
1339:
1323:
1321:
1320:
1315:
1313:
1311:
1310:
1309:
1299:
1289:
1288:
1287:
1278:
1277:
1267:
1257:
1252:
1251:
1243:
1229:
1227:
1226:
1221:
1209:
1207:
1206:
1201:
1198:
1193:
1168:
1163:
1150:
1145:
1128:
1126:
1125:
1120:
1118:
1110:
1093:
1091:
1090:
1085:
1073:
1071:
1070:
1065:
1063:
1062:
1046:
1044:
1043:
1038:
1026:
1024:
1023:
1018:
1006:
1004:
1003:
998:
983:
981:
980:
975:
973:
968:
966:
951:
949:
948:
943:
931:
929:
928:
923:
921:
920:
904:
902:
901:
896:
894:
893:
888:
884:
878:
874:
863:
858:
848:
839:
837:
836:
824:
816:
808:
787:
785:
784:
779:
767:
765:
764:
759:
754:
746:
731:
729:
728:
723:
721:
720:
710:
701:
693:
688:
680:
668:
666:
665:
660:
658:
657:
649:
639:
637:
636:
631:
615:
613:
612:
607:
595:
593:
592:
587:
575:
573:
572:
567:
565:
564:
548:
546:
545:
540:
537:
532:
517:
516:
484:
482:
481:
476:
462:
460:
459:
454:
443:
442:
417:
415:
414:
409:
407:
406:
382:
381:
369:
368:
352:random variables
350:
348:
347:
342:
330:
328:
327:
322:
309:gravity of Earth
284:
282:
281:
276:
271:
269:
267:
262:
253:
244:
231:
223:
222:
214:
186:
184:
183:
178:
173:
171:
169:
164:
155:
146:
136:
134:
129:
120:
115:
114:
104:
94:
89:
88:
80:
67:
58:
37:weighted average
31:to minimize the
29:random variables
3629:
3628:
3624:
3623:
3622:
3620:
3619:
3618:
3599:
3598:
3593:
3582:
3576:
3573:
3530:
3528:
3518:
3506:
3495:
3494:
3487:
3466:
3465:
3461:
3456:
3434:
3370:
3366:
3365:
3345:
3344:
3318:
3243:
3239:
3238:
3218:
3217:
3191:
3186:
3185:
3162:
3157:
3156:
3153:
3114:
3113:
3092:
3087:
3086:
3075:
3019:
2818:
2817:
2746:
2742:
2741:
2706:
2701:
2700:
2608:
2598:
2597:
2567:
2552:
2551:
2504:
2499:
2498:
2445:
2444:
2429:
2424:
2423:
2393:
2365:
2328:
2324:
2308:
2307:
2279:
2278:
2242:
2219:
2150:
2149:
2125:
2120:
2119:
2089:
2074:
2073:
2039:
2038:
2000:
1999:
1916:
1915:
1889:
1888:
1867:
1862:
1861:
1834:
1819:
1818:
1797:
1792:
1791:
1770:
1760:
1739:
1738:
1735:
1700:
1695:
1694:
1633:
1616:
1615:
1556:
1552:
1551:
1526:
1509:
1508:
1474:
1463:
1459:
1458:
1416:
1379:
1378:
1331:
1326:
1325:
1301:
1290:
1279:
1269:
1258:
1236:
1235:
1212:
1211:
1132:
1131:
1104:
1103:
1096:simple pendulum
1076:
1075:
1054:
1049:
1048:
1029:
1028:
1009:
1008:
989:
988:
954:
953:
934:
933:
912:
907:
906:
869:
868:
828:
790:
789:
770:
769:
734:
733:
712:
674:
673:
642:
641:
622:
621:
598:
597:
578:
577:
556:
551:
550:
508:
491:
490:
464:
463:
434:
423:
422:
398:
373:
360:
355:
354:
333:
332:
313:
312:
305:
235:
195:
194:
137:
106:
95:
73:
72:
65:
60:
59:with variances
56:
51:
17:
12:
11:
5:
3627:
3625:
3617:
3616:
3611:
3601:
3600:
3595:
3594:
3577:September 2012
3509:
3507:
3500:
3493:
3492:
3485:
3458:
3457:
3455:
3452:
3451:
3450:
3445:
3440:
3433:
3430:
3426:
3425:
3412:
3409:
3404:
3398:
3395:
3390:
3385:
3378:
3374:
3369:
3364:
3358:
3355:
3341:
3340:
3327:
3322:
3315:
3312:
3307:
3302:
3295:
3291:
3285:
3282:
3277:
3271:
3268:
3263:
3258:
3251:
3247:
3242:
3237:
3231:
3228:
3200:
3195:
3171:
3166:
3152:
3149:
3136:
3133:
3130:
3127:
3124:
3121:
3099:
3095:
3074:
3071:
3070:
3069:
3058:
3050:
3045:
3041:
3036:
3032:
3027:
3023:
3018:
3013:
3008:
3003:
2999:
2995:
2988:
2983:
2979:
2975:
2968:
2963:
2959:
2955:
2948:
2943:
2939:
2935:
2928:
2924:
2918:
2913:
2909:
2905:
2900:
2895:
2891:
2883:
2878:
2874:
2868:
2863:
2859:
2851:
2847:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2807:
2806:
2795:
2790:
2787:
2782:
2774:
2769:
2765:
2761:
2754:
2750:
2745:
2736:
2731:
2727:
2723:
2718:
2713:
2709:
2694:
2693:
2682:
2675:
2670:
2666:
2662:
2657:
2650:
2645:
2641:
2637:
2630:
2626:
2622:
2615:
2611:
2607:
2582:
2579:
2574:
2570:
2564:
2560:
2537:
2532:
2528:
2523:
2519:
2516:
2511:
2507:
2495:
2494:
2483:
2476:
2471:
2467:
2462:
2458:
2452:
2448:
2441:
2436:
2432:
2417:
2416:
2405:
2400:
2396:
2392:
2387:
2382:
2378:
2372:
2368:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2335:
2331:
2327:
2323:
2318:
2315:
2292:
2289:
2286:
2275:
2274:
2263:
2260:
2257:
2254:
2249:
2245:
2239:
2235:
2231:
2226:
2222:
2218:
2213:
2208:
2204:
2198:
2193:
2189:
2183:
2179:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2132:
2128:
2104:
2101:
2096:
2092:
2086:
2082:
2061:
2058:
2055:
2052:
2049:
2046:
2022:
2019:
2016:
2013:
2010:
2007:
1996:
1995:
1984:
1979:
1974:
1970:
1964:
1959:
1955:
1949:
1945:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1896:
1874:
1870:
1849:
1846:
1841:
1837:
1831:
1827:
1804:
1800:
1777:
1773:
1767:
1763:
1757:
1753:
1749:
1746:
1734:
1731:
1709:
1706:
1680:
1675:
1671:
1665:
1661:
1657:
1654:
1642:
1639:
1632:
1629:
1626:
1623:
1601:
1596:
1593:
1588:
1582:
1579:
1574:
1570:
1564:
1560:
1555:
1550:
1547:
1535:
1532:
1525:
1522:
1519:
1516:
1492:
1487:
1481:
1477:
1471:
1467:
1462:
1454:
1449:
1445:
1439:
1434:
1430:
1424:
1420:
1413:
1410:
1404:
1401:
1395:
1392:
1389:
1386:
1364:
1359:
1355:
1350:
1346:
1343:
1338:
1334:
1308:
1304:
1298:
1294:
1286:
1282:
1276:
1272:
1266:
1262:
1255:
1249:
1246:
1219:
1197:
1192:
1188:
1184:
1181:
1178:
1175:
1172:
1167:
1162:
1158:
1154:
1149:
1144:
1140:
1116:
1113:
1083:
1061:
1057:
1036:
1016:
996:
971:
965:
961:
941:
919:
915:
892:
887:
881:
877:
872:
867:
862:
857:
853:
847:
843:
835:
831:
827:
822:
819:
814:
811:
806:
803:
800:
797:
777:
757:
752:
749:
744:
741:
719:
715:
709:
705:
699:
696:
691:
686:
683:
655:
652:
629:
605:
585:
563:
559:
536:
531:
527:
523:
520:
515:
511:
507:
504:
501:
498:
474:
471:
452:
449:
446:
441:
437:
433:
430:
405:
401:
397:
394:
391:
388:
385:
380:
376:
372:
367:
363:
340:
320:
304:
301:
286:
285:
274:
266:
261:
257:
252:
248:
243:
239:
234:
229:
226:
220:
217:
211:
208:
205:
202:
188:
187:
176:
168:
163:
159:
154:
150:
145:
141:
133:
128:
124:
119:
113:
109:
103:
99:
92:
86:
83:
63:
54:
15:
13:
10:
9:
6:
4:
3:
2:
3626:
3615:
3612:
3610:
3609:Meta-analysis
3607:
3606:
3604:
3591:
3588:
3580:
3569:
3566:
3562:
3559:
3555:
3552:
3548:
3545:
3541:
3538: â
3537:
3533:
3532:Find sources:
3526:
3522:
3516:
3515:
3510:This article
3508:
3504:
3499:
3498:
3488:
3482:
3478:
3473:
3472:
3463:
3460:
3453:
3449:
3446:
3444:
3441:
3439:
3436:
3435:
3431:
3429:
3410:
3407:
3402:
3396:
3393:
3388:
3376:
3372:
3367:
3362:
3343:
3342:
3325:
3313:
3310:
3305:
3293:
3289:
3283:
3280:
3275:
3269:
3266:
3261:
3249:
3245:
3240:
3235:
3216:
3215:
3214:
3198:
3169:
3150:
3148:
3131:
3125:
3122:
3119:
3097:
3093:
3084:
3080:
3072:
3056:
3048:
3043:
3039:
3034:
3030:
3025:
3021:
3016:
3011:
3006:
3001:
2997:
2993:
2986:
2981:
2977:
2973:
2966:
2961:
2957:
2953:
2946:
2941:
2937:
2933:
2926:
2922:
2916:
2911:
2907:
2903:
2898:
2893:
2889:
2881:
2876:
2872:
2866:
2861:
2857:
2849:
2845:
2841:
2835:
2829:
2826:
2823:
2816:
2815:
2814:
2812:
2793:
2788:
2785:
2780:
2772:
2767:
2763:
2759:
2752:
2748:
2743:
2734:
2729:
2725:
2721:
2716:
2711:
2707:
2699:
2698:
2697:
2680:
2673:
2668:
2664:
2660:
2655:
2648:
2643:
2639:
2635:
2628:
2624:
2620:
2613:
2609:
2605:
2596:
2595:
2594:
2580:
2577:
2572:
2568:
2562:
2558:
2535:
2530:
2526:
2521:
2517:
2514:
2509:
2505:
2481:
2474:
2469:
2465:
2460:
2456:
2450:
2446:
2439:
2434:
2430:
2422:
2421:
2420:
2403:
2398:
2394:
2390:
2385:
2380:
2376:
2370:
2366:
2362:
2359:
2353:
2347:
2344:
2341:
2333:
2329:
2316:
2313:
2306:
2305:
2304:
2290:
2287:
2284:
2261:
2255:
2252:
2247:
2243:
2237:
2233:
2224:
2220:
2216:
2211:
2206:
2202:
2196:
2191:
2187:
2181:
2177:
2173:
2167:
2161:
2158:
2155:
2148:
2147:
2146:
2130:
2126:
2118:
2102:
2099:
2094:
2090:
2084:
2080:
2056:
2050:
2047:
2044:
2036:
2017:
2011:
2008:
2005:
1982:
1977:
1972:
1968:
1962:
1957:
1953:
1947:
1943:
1939:
1933:
1927:
1924:
1921:
1914:
1913:
1912:
1910:
1894:
1872:
1868:
1847:
1844:
1839:
1835:
1829:
1825:
1802:
1798:
1775:
1771:
1765:
1761:
1755:
1751:
1747:
1744:
1732:
1730:
1704:
1678:
1673:
1669:
1663:
1655:
1637:
1627:
1624:
1621:
1612:
1599:
1594:
1591:
1586:
1580:
1577:
1572:
1568:
1562:
1558:
1553:
1548:
1530:
1520:
1517:
1514:
1490:
1485:
1479:
1475:
1469:
1465:
1460:
1452:
1447:
1443:
1437:
1432:
1428:
1422:
1418:
1411:
1399:
1390:
1387:
1384:
1362:
1357:
1353:
1348:
1344:
1341:
1336:
1332:
1306:
1302:
1296:
1292:
1284:
1280:
1274:
1270:
1264:
1260:
1253:
1244:
1233:
1232:weighted mean
1217:
1195:
1190:
1186:
1182:
1179:
1176:
1173:
1170:
1165:
1160:
1156:
1152:
1147:
1142:
1138:
1111:
1101:
1097:
1081:
1059:
1055:
1034:
1014:
994:
985:
969:
963:
959:
939:
917:
913:
890:
885:
879:
875:
870:
865:
860:
855:
851:
845:
841:
833:
829:
825:
820:
809:
801:
798:
795:
775:
747:
739:
717:
713:
707:
703:
697:
694:
689:
681:
672:
650:
640:, denoted as
627:
619:
603:
583:
561:
557:
534:
529:
525:
521:
513:
509:
502:
499:
496:
488:
472:
450:
447:
439:
435:
428:
421:
403:
399:
395:
392:
389:
386:
383:
378:
374:
370:
365:
361:
353:
338:
318:
310:
302:
300:
298:
297:sensor fusion
294:
293:meta-analysis
289:
272:
264:
259:
255:
250:
246:
241:
237:
232:
227:
215:
206:
203:
200:
193:
192:
191:
174:
166:
161:
157:
152:
148:
143:
139:
131:
126:
122:
117:
111:
107:
101:
97:
90:
81:
71:
70:
69:
66:
57:
48:
46:
42:
38:
34:
30:
26:
22:
3583:
3574:
3564:
3557:
3550:
3543:
3531:
3519:Please help
3514:verification
3511:
3470:
3462:
3427:
3154:
3076:
2808:
2695:
2496:
2418:
2276:
1997:
1736:
1613:
986:
596:. Given the
306:
290:
287:
189:
61:
52:
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18:
1230:would be a
987:Instead of
3603:Categories
3547:newspapers
3454:References
2115:. Using a
1860:. If the
1733:Derivation
21:statistics
3408:−
3394:−
3373:∑
3357:^
3311:−
3290:∑
3281:−
3267:−
3246:∑
3230:^
3040:σ
3022:∑
2998:σ
2978:σ
2958:σ
2938:σ
2923:∑
2908:σ
2890:σ
2873:σ
2858:σ
2846:∑
2786:−
2764:σ
2749:∑
2726:σ
2665:σ
2640:σ
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2559:∑
2527:σ
2515:∝
2466:σ
2391:−
2377:σ
2326:∂
2322:∂
2253:−
2234:∑
2217:−
2203:σ
2178:∑
2081:∑
1969:σ
1944:∑
1826:∑
1752:∑
1708:^
1705:μ
1670:σ
1641:^
1638:μ
1592:−
1578:−
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1534:^
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1444:σ
1419:∑
1403:^
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1261:∑
1248:^
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852:σ
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319:μ
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158:σ
140:∑
123:σ
98:∑
85:^
45:precision
3432:See also
3083:Bayesian
2550:. Since
2035:gradient
487:variance
33:variance
3561:scholar
1094:from a
671:average
303:Context
35:of the
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3554:books
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3481:ISBN
3077:For
2288:>
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1656:<
620:for
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768:is
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