1534:
1031:
1971:
1529:{\displaystyle {\begin{aligned}{\overline {x}}&={\tfrac {1}{n}}\sum x_{i}&{\overline {y}}&={\tfrac {1}{n}}\sum y_{i},\\s_{xx}&={\tfrac {1}{n}}\sum (x_{i}-{\overline {x}})^{2}&&={\overline {x^{2}}}-{\overline {x}}^{2},\\s_{xy}&={\tfrac {1}{n}}\sum (x_{i}-{\overline {x}})(y_{i}-{\overline {y}})&&={\overline {xy}}-{\overline {x}}\,{\overline {y}},\\s_{yy}&={\tfrac {1}{n}}\sum (y_{i}-{\overline {y}})^{2}&&={\overline {y^{2}}}-{\overline {y}}^{2}.\end{aligned}}\,}
1000:
1545:
609:
31:
1966:{\displaystyle {\begin{aligned}&{\hat {\beta }}_{1}={\frac {s_{yy}-\delta s_{xx}+{\sqrt {(s_{yy}-\delta s_{xx})^{2}+4\delta s_{xy}^{2}}}}{2s_{xy}}},\\&{\hat {\beta }}_{0}={\overline {y}}-{\hat {\beta }}_{1}{\overline {x}},\\&{\hat {x}}_{i}^{*}=x_{i}+{\frac {{\hat {\beta }}_{1}}{{\hat {\beta }}_{1}^{2}+\delta }}(y_{i}-{\hat {\beta }}_{0}-{\hat {\beta }}_{1}x_{i}).\end{aligned}}}
995:{\displaystyle SSR=\sum _{i=1}^{n}{\bigg (}{\frac {\varepsilon _{i}^{2}}{\sigma _{\varepsilon }^{2}}}+{\frac {\eta _{i}^{2}}{\sigma _{\eta }^{2}}}{\bigg )}={\frac {1}{\sigma _{\epsilon }^{2}}}\sum _{i=1}^{n}{\Big (}(y_{i}-\beta _{0}-\beta _{1}x_{i}^{*})^{2}+\delta (x_{i}-x_{i}^{*})^{2}{\Big )}\ \to \ \min _{\beta _{0},\beta _{1},x_{1}^{*},\ldots ,x_{n}^{*}}SSR}
323:
397:
2261:
2203:
1550:
1036:
208:
598:
203:
2561:
York, D., Evensen, N. M., Martınez, M. L., and
Delgado, J. D. B.: Unified equations for the slope, intercept, and standard errors of the best straight line, Am. J. Phys., 72, 367–375,
117:, is known. In practice, this ratio might be estimated from related data-sources; however the regression procedure takes no account for possible errors in estimating this ratio.
2069:
2393:
When humans are asked to draw a linear regression on a scatterplot by guessing, their answers are closer to orthogonal regression than to ordinary least squares regression.
2115:
2344:
2005:
526:
2320:
460:
2291:
2135:
500:
480:
440:
420:
2535:
Ciccione, Lorenzo; Dehaene, Stanislas (August 2021). "Can humans perform mental regression on a graph? Accuracy and bias in the perception of scatterplots".
342:
2263:(also denoted in complex coordinates), which is the point whose horizontal and vertical locations are the averages of those of the data points. Then:
1017:
The solution can be expressed in terms of the second-degree sample moments. That is, we first calculate the following quantities (all sums go from
83:
2710:
2691:
2211:
2012:
2857:
2144:
2953:
2387:
537:
318:{\displaystyle {\begin{aligned}y_{i}&=y_{i}^{*}+\varepsilon _{i},\\x_{i}&=x_{i}^{*}+\eta _{i},\end{aligned}}}
2382:
falls on the orthogonal regression line for the three vertices. The quantification of a biological cell's intrinsic
106:
71:
2722:
121:
79:
2322:, the orthogonal regression line goes through the centroid and is parallel to the vector from the origin to
2948:
2018:
143:. However their ideas remained largely unnoticed for more than 50 years, until they were revived by
42:. This is different from the traditional least squares method, which measures error parallel to the
2896:"Uncoupling gene expression noise along the central dogma using genome engineered human cell lines"
2415:
2074:
110:
95:
2882:
2803:
2747:
2739:
2632:
2601:
2325:
152:
102:
1984:
505:
2927:
2840:
2706:
2687:
2679:
2667:
2375:
2371:
67:
2386:
can be quantified upon applying Deming regression to the observed behavior of a two reporter
2299:
445:
2917:
2907:
2866:
2830:
2793:
2731:
2657:
2624:
2615:
Coolidge, J. L. (1913). "Two geometrical applications of the mathematics of least squares".
2591:
2544:
75:
2878:
2874:
2363:
2270:
124:. Most statistical software packages used in clinical chemistry offer Deming regression.
2760:
2401:
The York regression extends Deming regression by allowing correlated errors in x and y.
17:
2922:
2895:
2383:
2138:
2120:
485:
465:
425:
405:
46:
axis. The case shown, with deviations measured perpendicularly, arises when errors in
2942:
2751:
2886:
98:, which allows for any number of predictors and a more complicated error structure.
2870:
2548:
2410:
2351:
30:
2782:"Reduction of observation equations which contain more than one observed quantity"
392:{\displaystyle \delta ={\frac {\sigma _{\varepsilon }^{2}}{\sigma _{\eta }^{2}}}.}
2852:
2835:
2662:
2645:
2354:
representation of the orthogonal regression line was given by
Coolidge in 1913.
2818:
59:
109:
in which the errors for the two variables are assumed to be independent and
2931:
603:
such that the weighted sum of squared residuals of the model is minimized:
2912:
2844:
120:
The Deming regression is only slightly more difficult to compute than the
2671:
2367:
2206:
336:
are independent and the ratio of their variances is assumed to be known:
2293:, then every line through the centroid is a line of best orthogonal fit.
2808:
2743:
2636:
2606:
2379:
2798:
2781:
2735:
2628:
2596:
2579:
2562:
2855:; Phelps, S. (2008). "Triangles, ellipses, and cubic polynomials".
2819:"Evaluation of regression procedures for method comparison studies"
2013:
perpendicular distances from the data points to the regression line
1539:
Finally, the least-squares estimates of model's parameters will be
29:
2378:
that is tangent to the triangle's sides at their midpoints. The
442:
parameters are often unknown, which complicates the estimate of
27:
Algorithm for the line of best fit for a two-dimensional dataset
2205:
the sum of the squared differences of the data points from the
2256:{\displaystyle {\overline {z}}={\tfrac {1}{n}}\sum z_{j}}
502:
is the same, these variances are likely to be equal, so
34:
Deming regression. The red lines show the error in both
2229:
1423:
1276:
1153:
1101:
1057:
2328:
2302:
2273:
2214:
2147:
2123:
2077:
2021:
1987:
1548:
1034:
612:
540:
508:
488:
468:
448:
428:
408:
345:
206:
2198:{\displaystyle S=\sum {(z_{j}-{\overline {z}})^{2}}}
78:
for a two-dimensional data set. It differs from the
2015:. In this case, denote each observation as a point
155:and related fields that the method was even dubbed
2773:Linear regression analysis of economic time series
2338:
2314:
2285:
2255:
2197:
2129:
2109:
2063:
1999:
1981:For the case of equal error variances, i.e., when
1965:
1528:
994:
592:
520:
494:
474:
454:
434:
414:
391:
317:
181:) are measured observations of the "true" values (
2646:"Incorrect Least–Squares Regression Coefficients"
897:
779:
726:
648:
593:{\displaystyle y^{*}=\beta _{0}+\beta _{1}x^{*},}
2720:Glaister, P. (2001). "Least squares revisited".
2686:. Wiley, NY (Dover Publications edition, 1985).
2450:
912:
8:
462:. Note that when the measurement method for
113:, and the ratio of their variances, denoted
2766:. Gentofte, Denmark: Steno Diabetes Center.
2510:
2486:
135: = 1, and then more generally by
2921:
2911:
2834:
2807:
2797:
2661:
2605:
2595:
2329:
2327:
2301:
2272:
2247:
2228:
2215:
2213:
2188:
2174:
2165:
2157:
2146:
2122:
2098:
2085:
2076:
2055:
2039:
2026:
2020:
1986:
1947:
1937:
1926:
1925:
1915:
1904:
1903:
1893:
1871:
1866:
1855:
1854:
1846:
1835:
1834:
1831:
1822:
1809:
1804:
1793:
1792:
1773:
1767:
1756:
1755:
1741:
1732:
1721:
1720:
1699:
1682:
1674:
1655:
1642:
1623:
1614:
1602:
1583:
1576:
1567:
1556:
1555:
1549:
1547:
1525:
1512:
1502:
1487:
1481:
1467:
1453:
1444:
1422:
1406:
1385:
1384:
1374:
1356:
1335:
1326:
1306:
1297:
1275:
1259:
1242:
1232:
1217:
1211:
1197:
1183:
1174:
1152:
1136:
1119:
1100:
1083:
1075:
1056:
1039:
1035:
1033:
975:
970:
951:
946:
933:
920:
915:
896:
895:
889:
879:
874:
861:
842:
832:
827:
817:
804:
791:
778:
777:
771:
760:
748:
743:
734:
725:
724:
716:
711:
701:
696:
690:
679:
674:
664:
659:
653:
647:
646:
640:
629:
611:
581:
571:
558:
545:
539:
531:We seek to find the line of "best fit"
507:
487:
467:
447:
427:
407:
378:
373:
363:
358:
352:
344:
302:
289:
284:
267:
250:
237:
232:
215:
207:
205:
2775:. DeErven F. Bohn, Haarlem, Netherlands.
2498:
2474:
151:. The latter book became so popular in
144:
2522:
2431:
136:
127:The model was originally introduced by
101:Deming regression is equivalent to the
2462:
2438:
2071:in the complex plane (i.e., the point
1006:
148:
128:
2761:"Deming regression, MethComp package"
2644:Cornbleet, P.J.; Gochman, N. (1979).
197:), which lie on the regression line:
7:
2064:{\displaystyle z_{j}=x_{j}+iy_{j}}
2011:: it minimizes the sum of squared
402:In practice, the variances of the
147:and later propagated even more by
25:
2759:Jensen, Anders Christian (2007).
2617:The American Mathematical Monthly
2563:https://doi.org/10.1119/1.1632486
167:Assume that the available data (
94:- axis. It is a special case of
2871:10.1080/00029890.2008.11920581
2705:. John Wiley & Sons, Inc.
2684:Statistical adjustment of data
2549:10.1016/j.cogpsych.2021.101406
2185:
2158:
2104:
2078:
1953:
1931:
1909:
1886:
1860:
1840:
1798:
1761:
1726:
1652:
1616:
1561:
1464:
1437:
1345:
1319:
1316:
1290:
1194:
1167:
905:
886:
854:
839:
784:
1:
2858:American Mathematical Monthly
2110:{\displaystyle (x_{j},y_{j})}
2580:"A problem in least squares"
2451:Cornbleet & Gochman 1979
2388:synthetic biological circuit
2220:
2179:
2007:, Deming regression becomes
1778:
1746:
1507:
1493:
1458:
1390:
1379:
1366:
1340:
1311:
1237:
1223:
1188:
1088:
1044:
86:in observations on both the
2339:{\displaystyle {\sqrt {S}}}
2970:
2380:major axis of this ellipse
2836:10.1093/clinchem/39.3.424
2701:Fuller, Wayne A. (1987).
2663:10.1093/clinchem/25.3.432
2370:with these points as its
2366:points in the plane, the
2000:{\displaystyle \delta =1}
521:{\displaystyle \delta =1}
107:errors-in-variables model
72:errors-in-variables model
2771:Koopmans, T. C. (1936).
2723:The Mathematical Gazette
2703:Measurement error models
1009:for a full derivation.
131:who considered the case
122:simple linear regression
82:in that it accounts for
80:simple linear regression
18:Perpendicular regression
2894:Quarton, T. G. (2020).
2780:Kummell, C. H. (1879).
2511:Minda & Phelps 2008
2487:Minda & Phelps 2008
2315:{\displaystyle S\neq 0}
455:{\displaystyle \delta }
74:that tries to find the
2900:Nucleic Acids Research
2578:Adcock, R. J. (1878).
2340:
2316:
2287:
2257:
2199:
2131:
2111:
2065:
2001:
1967:
1530:
996:
776:
645:
594:
522:
496:
476:
456:
436:
416:
393:
319:
55:
2362:In the case of three
2341:
2317:
2288:
2258:
2200:
2132:
2112:
2066:
2009:orthogonal regression
2002:
1977:Orthogonal regression
1968:
1531:
997:
756:
625:
595:
523:
497:
477:
457:
437:
417:
394:
320:
54:have equal variances.
33:
2537:Cognitive Psychology
2326:
2300:
2271:
2212:
2145:
2121:
2075:
2019:
1985:
1546:
1032:
610:
538:
506:
486:
466:
446:
426:
406:
343:
204:
111:normally distributed
2954:Regression analysis
2913:10.1093/nar/gkaa668
2817:Linnet, K. (1993).
2416:Regression dilution
2286:{\displaystyle S=0}
1876:
1814:
1687:
980:
956:
884:
837:
753:
721:
706:
684:
669:
383:
368:
294:
242:
96:total least squares
2823:Clinical Chemistry
2650:Clinical Chemistry
2336:
2312:
2283:
2253:
2238:
2195:
2127:
2107:
2061:
1997:
1963:
1961:
1853:
1791:
1670:
1526:
1523:
1432:
1285:
1162:
1110:
1066:
1021: = 1 to
992:
982:
966:
942:
870:
823:
739:
707:
692:
670:
655:
590:
518:
492:
472:
452:
432:
412:
389:
369:
354:
315:
313:
280:
228:
153:clinical chemistry
103:maximum likelihood
56:
2906:(16): 9406–9413.
2376:Steiner inellipse
2334:
2237:
2223:
2182:
2130:{\displaystyle i}
1934:
1912:
1884:
1863:
1843:
1801:
1781:
1764:
1749:
1729:
1709:
1688:
1564:
1510:
1496:
1461:
1431:
1393:
1382:
1369:
1343:
1314:
1284:
1240:
1226:
1191:
1161:
1109:
1091:
1065:
1047:
911:
910:
904:
754:
722:
685:
495:{\displaystyle y}
475:{\displaystyle x}
435:{\displaystyle y}
415:{\displaystyle x}
384:
159:in those fields.
157:Deming regression
105:estimation of an
68:W. Edwards Deming
64:Deming regression
16:(Redirected from
2961:
2935:
2925:
2915:
2890:
2848:
2838:
2813:
2811:
2801:
2776:
2767:
2765:
2755:
2716:
2697:
2675:
2665:
2640:
2611:
2609:
2599:
2566:
2559:
2553:
2552:
2532:
2526:
2520:
2514:
2513:, Corollary 2.4.
2508:
2502:
2496:
2490:
2484:
2478:
2472:
2466:
2460:
2454:
2448:
2442:
2436:
2345:
2343:
2342:
2337:
2335:
2330:
2321:
2319:
2318:
2313:
2292:
2290:
2289:
2284:
2262:
2260:
2259:
2254:
2252:
2251:
2239:
2230:
2224:
2216:
2204:
2202:
2201:
2196:
2194:
2193:
2192:
2183:
2175:
2170:
2169:
2136:
2134:
2133:
2128:
2116:
2114:
2113:
2108:
2103:
2102:
2090:
2089:
2070:
2068:
2067:
2062:
2060:
2059:
2044:
2043:
2031:
2030:
2006:
2004:
2003:
1998:
1972:
1970:
1969:
1964:
1962:
1952:
1951:
1942:
1941:
1936:
1935:
1927:
1920:
1919:
1914:
1913:
1905:
1898:
1897:
1885:
1883:
1875:
1870:
1865:
1864:
1856:
1851:
1850:
1845:
1844:
1836:
1832:
1827:
1826:
1813:
1808:
1803:
1802:
1794:
1789:
1782:
1774:
1772:
1771:
1766:
1765:
1757:
1750:
1742:
1737:
1736:
1731:
1730:
1722:
1717:
1710:
1708:
1707:
1706:
1690:
1689:
1686:
1681:
1660:
1659:
1650:
1649:
1631:
1630:
1615:
1610:
1609:
1591:
1590:
1577:
1572:
1571:
1566:
1565:
1557:
1552:
1535:
1533:
1532:
1527:
1524:
1517:
1516:
1511:
1503:
1497:
1492:
1491:
1482:
1474:
1472:
1471:
1462:
1454:
1449:
1448:
1433:
1424:
1414:
1413:
1394:
1386:
1383:
1375:
1370:
1365:
1357:
1349:
1344:
1336:
1331:
1330:
1315:
1307:
1302:
1301:
1286:
1277:
1267:
1266:
1247:
1246:
1241:
1233:
1227:
1222:
1221:
1212:
1204:
1202:
1201:
1192:
1184:
1179:
1178:
1163:
1154:
1144:
1143:
1124:
1123:
1111:
1102:
1092:
1084:
1080:
1079:
1067:
1058:
1048:
1040:
1001:
999:
998:
993:
981:
979:
974:
955:
950:
938:
937:
925:
924:
908:
902:
901:
900:
894:
893:
883:
878:
866:
865:
847:
846:
836:
831:
822:
821:
809:
808:
796:
795:
783:
782:
775:
770:
755:
752:
747:
735:
730:
729:
723:
720:
715:
705:
700:
691:
686:
683:
678:
668:
663:
654:
652:
651:
644:
639:
599:
597:
596:
591:
586:
585:
576:
575:
563:
562:
550:
549:
527:
525:
524:
519:
501:
499:
498:
493:
481:
479:
478:
473:
461:
459:
458:
453:
441:
439:
438:
433:
421:
419:
418:
413:
398:
396:
395:
390:
385:
382:
377:
367:
362:
353:
324:
322:
321:
316:
314:
307:
306:
293:
288:
272:
271:
255:
254:
241:
236:
220:
219:
76:line of best fit
21:
2969:
2968:
2964:
2963:
2962:
2960:
2959:
2958:
2939:
2938:
2893:
2851:
2816:
2799:10.2307/2635646
2779:
2770:
2763:
2758:
2736:10.2307/3620485
2719:
2713:
2700:
2694:
2678:
2643:
2629:10.2307/2973072
2614:
2597:10.2307/2635758
2577:
2569:
2560:
2556:
2534:
2533:
2529:
2521:
2517:
2509:
2505:
2497:
2493:
2485:
2481:
2473:
2469:
2461:
2457:
2449:
2445:
2437:
2433:
2424:
2407:
2399:
2397:York regression
2360:
2324:
2323:
2298:
2297:
2269:
2268:
2243:
2210:
2209:
2184:
2161:
2143:
2142:
2119:
2118:
2094:
2081:
2073:
2072:
2051:
2035:
2022:
2017:
2016:
1983:
1982:
1979:
1960:
1959:
1943:
1924:
1902:
1889:
1852:
1833:
1818:
1787:
1786:
1754:
1719:
1715:
1714:
1695:
1691:
1651:
1638:
1619:
1598:
1579:
1578:
1554:
1544:
1543:
1522:
1521:
1501:
1483:
1473:
1463:
1440:
1415:
1402:
1399:
1398:
1358:
1348:
1322:
1293:
1268:
1255:
1252:
1251:
1231:
1213:
1203:
1193:
1170:
1145:
1132:
1129:
1128:
1115:
1093:
1081:
1071:
1049:
1030:
1029:
1015:
929:
916:
885:
857:
838:
813:
800:
787:
608:
607:
577:
567:
554:
541:
536:
535:
528:for this case.
504:
503:
484:
483:
464:
463:
444:
443:
424:
423:
404:
403:
341:
340:
312:
311:
298:
273:
263:
260:
259:
246:
221:
211:
202:
201:
194:
186:
179:
172:
165:
145:Koopmans (1936)
139:with arbitrary
28:
23:
22:
15:
12:
11:
5:
2967:
2965:
2957:
2956:
2951:
2941:
2940:
2937:
2936:
2891:
2865:(8): 679–689.
2849:
2829:(3): 424–432.
2814:
2777:
2768:
2756:
2717:
2711:
2698:
2692:
2676:
2656:(3): 432–438.
2641:
2623:(6): 187–190.
2612:
2574:
2573:
2568:
2567:
2554:
2527:
2515:
2503:
2491:
2489:, Theorem 2.3.
2479:
2467:
2455:
2443:
2430:
2429:
2428:
2423:
2420:
2419:
2418:
2413:
2406:
2403:
2398:
2395:
2384:cellular noise
2359:
2356:
2348:
2347:
2333:
2311:
2308:
2305:
2294:
2282:
2279:
2276:
2250:
2246:
2242:
2236:
2233:
2227:
2222:
2219:
2191:
2187:
2181:
2178:
2173:
2168:
2164:
2160:
2156:
2153:
2150:
2141:). Denote as
2139:imaginary unit
2126:
2106:
2101:
2097:
2093:
2088:
2084:
2080:
2058:
2054:
2050:
2047:
2042:
2038:
2034:
2029:
2025:
1996:
1993:
1990:
1978:
1975:
1974:
1973:
1958:
1955:
1950:
1946:
1940:
1933:
1930:
1923:
1918:
1911:
1908:
1901:
1896:
1892:
1888:
1882:
1879:
1874:
1869:
1862:
1859:
1849:
1842:
1839:
1830:
1825:
1821:
1817:
1812:
1807:
1800:
1797:
1790:
1788:
1785:
1780:
1777:
1770:
1763:
1760:
1753:
1748:
1745:
1740:
1735:
1728:
1725:
1718:
1716:
1713:
1705:
1702:
1698:
1694:
1685:
1680:
1677:
1673:
1669:
1666:
1663:
1658:
1654:
1648:
1645:
1641:
1637:
1634:
1629:
1626:
1622:
1618:
1613:
1608:
1605:
1601:
1597:
1594:
1589:
1586:
1582:
1575:
1570:
1563:
1560:
1553:
1551:
1537:
1536:
1520:
1515:
1509:
1506:
1500:
1495:
1490:
1486:
1480:
1477:
1475:
1470:
1466:
1460:
1457:
1452:
1447:
1443:
1439:
1436:
1430:
1427:
1421:
1418:
1416:
1412:
1409:
1405:
1401:
1400:
1397:
1392:
1389:
1381:
1378:
1373:
1368:
1364:
1361:
1355:
1352:
1350:
1347:
1342:
1339:
1334:
1329:
1325:
1321:
1318:
1313:
1310:
1305:
1300:
1296:
1292:
1289:
1283:
1280:
1274:
1271:
1269:
1265:
1262:
1258:
1254:
1253:
1250:
1245:
1239:
1236:
1230:
1225:
1220:
1216:
1210:
1207:
1205:
1200:
1196:
1190:
1187:
1182:
1177:
1173:
1169:
1166:
1160:
1157:
1151:
1148:
1146:
1142:
1139:
1135:
1131:
1130:
1127:
1122:
1118:
1114:
1108:
1105:
1099:
1096:
1094:
1090:
1087:
1082:
1078:
1074:
1070:
1064:
1061:
1055:
1052:
1050:
1046:
1043:
1038:
1037:
1014:
1011:
1003:
1002:
991:
988:
985:
978:
973:
969:
965:
962:
959:
954:
949:
945:
941:
936:
932:
928:
923:
919:
914:
907:
899:
892:
888:
882:
877:
873:
869:
864:
860:
856:
853:
850:
845:
841:
835:
830:
826:
820:
816:
812:
807:
803:
799:
794:
790:
786:
781:
774:
769:
766:
763:
759:
751:
746:
742:
738:
733:
728:
719:
714:
710:
704:
699:
695:
689:
682:
677:
673:
667:
662:
658:
650:
643:
638:
635:
632:
628:
624:
621:
618:
615:
601:
600:
589:
584:
580:
574:
570:
566:
561:
557:
553:
548:
544:
517:
514:
511:
491:
471:
451:
431:
411:
400:
399:
388:
381:
376:
372:
366:
361:
357:
351:
348:
326:
325:
310:
305:
301:
297:
292:
287:
283:
279:
276:
274:
270:
266:
262:
261:
258:
253:
249:
245:
240:
235:
231:
227:
224:
222:
218:
214:
210:
209:
192:
184:
177:
170:
164:
161:
137:Kummell (1879)
66:, named after
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2966:
2955:
2952:
2950:
2949:Curve fitting
2947:
2946:
2944:
2933:
2929:
2924:
2919:
2914:
2909:
2905:
2901:
2897:
2892:
2888:
2884:
2880:
2876:
2872:
2868:
2864:
2860:
2859:
2854:
2850:
2846:
2842:
2837:
2832:
2828:
2824:
2820:
2815:
2810:
2805:
2800:
2795:
2792:(4): 97–105.
2791:
2787:
2783:
2778:
2774:
2769:
2762:
2757:
2753:
2749:
2745:
2741:
2737:
2733:
2729:
2725:
2724:
2718:
2714:
2712:0-471-86187-1
2708:
2704:
2699:
2695:
2693:0-486-64685-8
2689:
2685:
2681:
2680:Deming, W. E.
2677:
2673:
2669:
2664:
2659:
2655:
2651:
2647:
2642:
2638:
2634:
2630:
2626:
2622:
2618:
2613:
2608:
2603:
2598:
2593:
2589:
2585:
2581:
2576:
2575:
2571:
2570:
2564:
2558:
2555:
2550:
2546:
2542:
2538:
2531:
2528:
2524:
2519:
2516:
2512:
2507:
2504:
2500:
2499:Coolidge 1913
2495:
2492:
2488:
2483:
2480:
2476:
2475:Glaister 2001
2471:
2468:
2464:
2459:
2456:
2452:
2447:
2444:
2440:
2435:
2432:
2426:
2425:
2421:
2417:
2414:
2412:
2409:
2408:
2404:
2402:
2396:
2394:
2391:
2389:
2385:
2381:
2377:
2374:has a unique
2373:
2369:
2365:
2364:non-collinear
2357:
2355:
2353:
2352:trigonometric
2331:
2309:
2306:
2303:
2295:
2280:
2277:
2274:
2266:
2265:
2264:
2248:
2244:
2240:
2234:
2231:
2225:
2217:
2208:
2189:
2176:
2171:
2166:
2162:
2154:
2151:
2148:
2140:
2124:
2099:
2095:
2091:
2086:
2082:
2056:
2052:
2048:
2045:
2040:
2036:
2032:
2027:
2023:
2014:
2010:
1994:
1991:
1988:
1976:
1956:
1948:
1944:
1938:
1928:
1921:
1916:
1906:
1899:
1894:
1890:
1880:
1877:
1872:
1867:
1857:
1847:
1837:
1828:
1823:
1819:
1815:
1810:
1805:
1795:
1783:
1775:
1768:
1758:
1751:
1743:
1738:
1733:
1723:
1711:
1703:
1700:
1696:
1692:
1683:
1678:
1675:
1671:
1667:
1664:
1661:
1656:
1646:
1643:
1639:
1635:
1632:
1627:
1624:
1620:
1611:
1606:
1603:
1599:
1595:
1592:
1587:
1584:
1580:
1573:
1568:
1558:
1542:
1541:
1540:
1518:
1513:
1504:
1498:
1488:
1484:
1478:
1476:
1468:
1455:
1450:
1445:
1441:
1434:
1428:
1425:
1419:
1417:
1410:
1407:
1403:
1395:
1387:
1376:
1371:
1362:
1359:
1353:
1351:
1337:
1332:
1327:
1323:
1308:
1303:
1298:
1294:
1287:
1281:
1278:
1272:
1270:
1263:
1260:
1256:
1248:
1243:
1234:
1228:
1218:
1214:
1208:
1206:
1198:
1185:
1180:
1175:
1171:
1164:
1158:
1155:
1149:
1147:
1140:
1137:
1133:
1125:
1120:
1116:
1112:
1106:
1103:
1097:
1095:
1085:
1076:
1072:
1068:
1062:
1059:
1053:
1051:
1041:
1028:
1027:
1026:
1024:
1020:
1012:
1010:
1008:
1007:Jensen (2007)
989:
986:
983:
976:
971:
967:
963:
960:
957:
952:
947:
943:
939:
934:
930:
926:
921:
917:
890:
880:
875:
871:
867:
862:
858:
851:
848:
843:
833:
828:
824:
818:
814:
810:
805:
801:
797:
792:
788:
772:
767:
764:
761:
757:
749:
744:
740:
736:
731:
717:
712:
708:
702:
697:
693:
687:
680:
675:
671:
665:
660:
656:
641:
636:
633:
630:
626:
622:
619:
616:
613:
606:
605:
604:
587:
582:
578:
572:
568:
564:
559:
555:
551:
546:
542:
534:
533:
532:
529:
515:
512:
509:
489:
469:
449:
429:
409:
386:
379:
374:
370:
364:
359:
355:
349:
346:
339:
338:
337:
335:
331:
328:where errors
308:
303:
299:
295:
290:
285:
281:
277:
275:
268:
264:
256:
251:
247:
243:
238:
233:
229:
225:
223:
216:
212:
200:
199:
198:
196:
188:
180:
173:
163:Specification
162:
160:
158:
154:
150:
149:Deming (1943)
146:
142:
138:
134:
130:
129:Adcock (1878)
125:
123:
118:
116:
112:
108:
104:
99:
97:
93:
89:
85:
81:
77:
73:
69:
65:
61:
53:
49:
45:
41:
37:
32:
19:
2903:
2899:
2862:
2856:
2826:
2822:
2789:
2785:
2772:
2727:
2721:
2702:
2683:
2653:
2649:
2620:
2616:
2590:(2): 53–54.
2587:
2583:
2572:Bibliography
2557:
2540:
2536:
2530:
2523:Quarton 2020
2518:
2506:
2494:
2482:
2470:
2465:, Ch. 1.3.3.
2458:
2446:
2434:
2411:Line fitting
2400:
2392:
2361:
2349:
2008:
1980:
1538:
1022:
1018:
1016:
1004:
602:
530:
401:
333:
329:
327:
190:
182:
175:
168:
166:
156:
140:
132:
126:
119:
114:
100:
91:
87:
63:
57:
51:
47:
43:
39:
35:
2786:The Analyst
2730:: 104–107.
2584:The Analyst
2463:Fuller 1987
2439:Linnet 1993
2358:Application
2943:Categories
2543:: 101406.
2422:References
90:- and the
60:statistics
2853:Minda, D.
2752:125949467
2307:≠
2241:∑
2221:¯
2180:¯
2172:−
2155:∑
1989:δ
1932:^
1929:β
1922:−
1910:^
1907:β
1900:−
1881:δ
1861:^
1858:β
1841:^
1838:β
1811:∗
1799:^
1779:¯
1762:^
1759:β
1752:−
1747:¯
1727:^
1724:β
1668:δ
1636:δ
1633:−
1596:δ
1593:−
1562:^
1559:β
1508:¯
1499:−
1494:¯
1459:¯
1451:−
1435:∑
1391:¯
1380:¯
1372:−
1367:¯
1341:¯
1333:−
1312:¯
1304:−
1288:∑
1238:¯
1229:−
1224:¯
1189:¯
1181:−
1165:∑
1113:∑
1089:¯
1069:∑
1045:¯
977:∗
961:…
953:∗
931:β
918:β
906:→
881:∗
868:−
852:δ
834:∗
815:β
811:−
802:β
798:−
758:∑
745:ϵ
741:σ
713:η
709:σ
694:η
676:ε
672:σ
657:ε
627:∑
583:∗
569:β
556:β
547:∗
510:δ
450:δ
375:η
371:σ
360:ε
356:σ
347:δ
300:η
291:∗
248:ε
239:∗
2932:32810265
2887:15049234
2682:(1943).
2405:See also
2372:vertices
2368:triangle
2207:centroid
1013:Solution
70:, is an
2923:7498316
2879:2456092
2845:8448852
2809:2635646
2744:3620485
2637:2973072
2607:2635758
2565:, 2004.
2137:is the
2930:
2920:
2885:
2877:
2843:
2806:
2750:
2742:
2709:
2690:
2672:262186
2670:
2635:
2604:
2117:where
909:
903:
84:errors
2883:S2CID
2804:JSTOR
2764:(PDF)
2748:S2CID
2740:JSTOR
2633:JSTOR
2602:JSTOR
2427:Notes
2928:PMID
2841:PMID
2707:ISBN
2688:ISBN
2668:PMID
1005:See
482:and
422:and
332:and
50:and
38:and
2918:PMC
2908:doi
2867:doi
2863:115
2831:doi
2794:doi
2732:doi
2658:doi
2625:doi
2592:doi
2545:doi
2541:128
2296:If
2267:If
1025:):
913:min
58:In
2945::
2926:.
2916:.
2904:48
2902:.
2898:.
2881:.
2875:MR
2873:.
2861:.
2839:.
2827:39
2825:.
2821:.
2802:.
2788:.
2784:.
2746:.
2738:.
2728:85
2726:.
2666:.
2654:25
2652:.
2648:.
2631:.
2621:20
2619:.
2600:.
2586:.
2582:.
2539:.
2390:.
2350:A
189:,
174:,
62:,
2934:.
2910::
2889:.
2869::
2847:.
2833::
2812:.
2796::
2790:6
2754:.
2734::
2715:.
2696:.
2674:.
2660::
2639:.
2627::
2610:.
2594::
2588:5
2551:.
2547::
2525:.
2501:.
2477:.
2453:.
2441:.
2346:.
2332:S
2310:0
2304:S
2281:0
2278:=
2275:S
2249:j
2245:z
2235:n
2232:1
2226:=
2218:z
2190:2
2186:)
2177:z
2167:j
2163:z
2159:(
2152:=
2149:S
2125:i
2105:)
2100:j
2096:y
2092:,
2087:j
2083:x
2079:(
2057:j
2053:y
2049:i
2046:+
2041:j
2037:x
2033:=
2028:j
2024:z
1995:1
1992:=
1957:.
1954:)
1949:i
1945:x
1939:1
1917:0
1895:i
1891:y
1887:(
1878:+
1873:2
1868:1
1848:1
1829:+
1824:i
1820:x
1816:=
1806:i
1796:x
1784:,
1776:x
1769:1
1744:y
1739:=
1734:0
1712:,
1704:y
1701:x
1697:s
1693:2
1684:2
1679:y
1676:x
1672:s
1665:4
1662:+
1657:2
1653:)
1647:x
1644:x
1640:s
1628:y
1625:y
1621:s
1617:(
1612:+
1607:x
1604:x
1600:s
1588:y
1585:y
1581:s
1574:=
1569:1
1519:.
1514:2
1505:y
1489:2
1485:y
1479:=
1469:2
1465:)
1456:y
1446:i
1442:y
1438:(
1429:n
1426:1
1420:=
1411:y
1408:y
1404:s
1396:,
1388:y
1377:x
1363:y
1360:x
1354:=
1346:)
1338:y
1328:i
1324:y
1320:(
1317:)
1309:x
1299:i
1295:x
1291:(
1282:n
1279:1
1273:=
1264:y
1261:x
1257:s
1249:,
1244:2
1235:x
1219:2
1215:x
1209:=
1199:2
1195:)
1186:x
1176:i
1172:x
1168:(
1159:n
1156:1
1150:=
1141:x
1138:x
1134:s
1126:,
1121:i
1117:y
1107:n
1104:1
1098:=
1086:y
1077:i
1073:x
1063:n
1060:1
1054:=
1042:x
1023:n
1019:i
990:R
987:S
984:S
972:n
968:x
964:,
958:,
948:1
944:x
940:,
935:1
927:,
922:0
898:)
891:2
887:)
876:i
872:x
863:i
859:x
855:(
849:+
844:2
840:)
829:i
825:x
819:1
806:0
793:i
789:y
785:(
780:(
773:n
768:1
765:=
762:i
750:2
737:1
732:=
727:)
718:2
703:2
698:i
688:+
681:2
666:2
661:i
649:(
642:n
637:1
634:=
631:i
623:=
620:R
617:S
614:S
588:,
579:x
573:1
565:+
560:0
552:=
543:y
516:1
513:=
490:y
470:x
430:y
410:x
387:.
380:2
365:2
350:=
334:η
330:ε
309:,
304:i
296:+
286:i
282:x
278:=
269:i
265:x
257:,
252:i
244:+
234:i
230:y
226:=
217:i
213:y
195:*
193:i
191:x
187:*
185:i
183:y
178:i
176:x
171:i
169:y
141:δ
133:δ
115:δ
92:y
88:x
52:y
48:x
44:y
40:y
36:x
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
