[{"data":1,"prerenderedAt":4533},["ShallowReactive",2],{"blog-\u002Fblog\u002Fweighted-observations-mpv":3,"blog-surround-\u002Fblog\u002Fweighted-observations-mpv":4522},{"id":4,"title":5,"author":6,"body":7,"category":4490,"description":4491,"extension":4492,"image":4493,"meta":4494,"navigation":4514,"path":4515,"publishedAt":4516,"readTime":4517,"seo":4518,"sitemap":4519,"stem":4520,"__hash__":4521},"blog\u002Fblog\u002F16.weighted-observations-mpv.md","Weighted Observations: Using Statistical Reliability to Find the MPV","Surveyiiing Expert",{"type":8,"value":9,"toc":4462},"minimark",[10,15,33,37,222,226,330,335,942,946,950,1151,1320,1764,1768,1992,1996,2005,2667,4168,4172,4226,4230,4352,4356,4444,4448,4451,4455],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,19,23,24,27,28,32],"p",{},"In high-precision surveying, not all measurements are created equal. Some observations are taken by more experienced observers, some use superior instrumentation, and some are averages of larger data sets. To combine these varied measurements into a single result, surveyors use ",[20,21,22],"strong",{},"Weighted Observations"," to find the ",[20,25,26],{},"Most Probable Value (MPV)"," ",[29,30,31],"span",{},"1, 2",".",[11,34,36],{"id":35},"why-this-matters","Why This Matters",[16,38,39,40,145,146,218,219,32],{},"If you treat a measurement with a standard error of ",[29,41,44,82],{"className":42},[43],"katex",[29,45,48],{"className":46},[47],"katex-mathml",[49,50,52],"math",{"xmlns":51},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[53,54,55,77],"semantics",{},[56,57,58,62],"mrow",{},[59,60,61],"mo",{},"±",[63,64,65,69],"msup",{},[66,67,68],"mn",{},"5",[56,70,71,75],{},[59,72,74],{"mathvariant":73},"normal","′",[59,76,74],{"mathvariant":73},[78,79,81],"annotation",{"encoding":80},"application\u002Fx-tex","\\pm 5''",[29,83,87],{"className":84,"ariaHidden":86},[85],"katex-html","true",[29,88,91,96,100],{"className":89},[90],"base",[29,92],{"className":93,"style":95},[94],"strut","height:0.8352em;vertical-align:-0.0833em;",[29,97,61],{"className":98},[99],"mord",[29,101,103,106],{"className":102},[99],[29,104,68],{"className":105},[99],[29,107,110],{"className":108},[109],"msupsub",[29,111,114],{"className":112},[113],"vlist-t",[29,115,118],{"className":116},[117],"vlist-r",[29,119,123],{"className":120,"style":122},[121],"vlist","height:0.7519em;",[29,124,126,131],{"style":125},"top:-3.063em;margin-right:0.05em;",[29,127],{"className":128,"style":130},[129],"pstrut","height:2.7em;",[29,132,138],{"className":133},[134,135,136,137],"sizing","reset-size6","size3","mtight",[29,139,141],{"className":140},[99,137],[29,142,144],{"className":143},[99,137],"′′"," the same as one with ",[29,147,149,174],{"className":148},[43],[29,150,152],{"className":151},[47],[49,153,154],{"xmlns":51},[53,155,156,171],{},[56,157,158,160],{},[59,159,61],{},[63,161,162,165],{},[66,163,164],{},"1",[56,166,167,169],{},[59,168,74],{"mathvariant":73},[59,170,74],{"mathvariant":73},[78,172,173],{"encoding":80},"\\pm 1''",[29,175,177],{"className":176,"ariaHidden":86},[85],[29,178,180,183,186],{"className":179},[90],[29,181],{"className":182,"style":95},[94],[29,184,61],{"className":185},[99],[29,187,189,192],{"className":188},[99],[29,190,164],{"className":191},[99],[29,193,195],{"className":194},[109],[29,196,198],{"className":197},[113],[29,199,201],{"className":200},[117],[29,202,204],{"className":203,"style":122},[121],[29,205,206,209],{"style":125},[29,207],{"className":208,"style":130},[129],[29,210,212],{"className":211},[134,135,136,137],[29,213,215],{"className":214},[99,137],[29,216,144],{"className":217},[99,137],", you degrade the quality of your final result. Weighting allows the surveyor to mathematically favor more reliable data, ensuring that the most precise observations have the greatest influence on the final coordinates or elevations ",[29,220,221],{},"2, 3",[11,223,225],{"id":224},"theory","Theory",[16,227,228,229,262,263,326,327,32],{},"The weight of an observation (",[29,230,232,247],{"className":231},[43],[29,233,235],{"className":234},[47],[49,236,237],{"xmlns":51},[53,238,239,245],{},[56,240,241],{},[242,243,244],"mi",{},"W",[78,246,244],{"encoding":80},[29,248,250],{"className":249,"ariaHidden":86},[85],[29,251,253,257],{"className":252},[90],[29,254],{"className":255,"style":256},[94],"height:0.6833em;",[29,258,244],{"className":259,"style":261},[99,260],"mathnormal","margin-right:0.1389em;",") is a measure of its relative reliability compared to other observations in the same set. By definition, weight is inversely proportional to the variance (",[29,264,266,286],{"className":265},[43],[29,267,269],{"className":268},[47],[49,270,271],{"xmlns":51},[53,272,273,283],{},[56,274,275],{},[63,276,277,280],{},[242,278,279],{},"σ",[66,281,282],{},"2",[78,284,285],{"encoding":80},"\\sigma^2",[29,287,289],{"className":288,"ariaHidden":86},[85],[29,290,292,296],{"className":291},[90],[29,293],{"className":294,"style":295},[94],"height:0.8141em;",[29,297,299,303],{"className":298},[99],[29,300,279],{"className":301,"style":302},[99,260],"margin-right:0.0359em;",[29,304,306],{"className":305},[109],[29,307,309],{"className":308},[113],[29,310,312],{"className":311},[117],[29,313,315],{"className":314,"style":295},[121],[29,316,317,320],{"style":125},[29,318],{"className":319,"style":130},[129],[29,321,323],{"className":322},[134,135,136,137],[29,324,282],{"className":325},[99,137],") of the measurement ",[29,328,329],{},"3, 4",[331,332,334],"h3",{"id":333},"the-most-probable-value-mpv","The Most Probable Value (MPV)",[16,336,337,338,549,550,745,746,27,749,751,752],{},"The MPV is the closest approximation to the true value that can be achieved from a set of data. For a series of weighted observations ",[29,339,341,384],{"className":340},[43],[29,342,344],{"className":343},[47],[49,345,346],{"xmlns":51},[53,347,348,381],{},[56,349,350,358,361,367,369,372,374],{},[351,352,353,356],"msub",{},[242,354,355],{},"L",[66,357,164],{},[59,359,360],{"separator":86},",",[351,362,363,365],{},[242,364,355],{},[66,366,282],{},[59,368,360],{"separator":86},[59,370,371],{},"…",[59,373,360],{"separator":86},[351,375,376,378],{},[242,377,355],{},[242,379,380],{},"n",[78,382,383],{"encoding":80},"L_1, L_2, \\dots, L_n",[29,385,387],{"className":386,"ariaHidden":86},[85],[29,388,390,394,440,444,449,489,492,495,499,502,505,508],{"className":389},[90],[29,391],{"className":392,"style":393},[94],"height:0.8778em;vertical-align:-0.1944em;",[29,395,397,400],{"className":396},[99],[29,398,355],{"className":399},[99,260],[29,401,403],{"className":402},[109],[29,404,407,431],{"className":405},[113,406],"vlist-t2",[29,408,410,426],{"className":409},[117],[29,411,414],{"className":412,"style":413},[121],"height:0.3011em;",[29,415,417,420],{"style":416},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[29,418],{"className":419,"style":130},[129],[29,421,423],{"className":422},[134,135,136,137],[29,424,164],{"className":425},[99,137],[29,427,430],{"className":428},[429],"vlist-s","​",[29,432,434],{"className":433},[117],[29,435,438],{"className":436,"style":437},[121],"height:0.15em;",[29,439],{},[29,441,360],{"className":442},[443],"mpunct",[29,445],{"className":446,"style":448},[447],"mspace","margin-right:0.1667em;",[29,450,452,455],{"className":451},[99],[29,453,355],{"className":454},[99,260],[29,456,458],{"className":457},[109],[29,459,461,481],{"className":460},[113,406],[29,462,464,478],{"className":463},[117],[29,465,467],{"className":466,"style":413},[121],[29,468,469,472],{"style":416},[29,470],{"className":471,"style":130},[129],[29,473,475],{"className":474},[134,135,136,137],[29,476,282],{"className":477},[99,137],[29,479,430],{"className":480},[429],[29,482,484],{"className":483},[117],[29,485,487],{"className":486,"style":437},[121],[29,488],{},[29,490,360],{"className":491},[443],[29,493],{"className":494,"style":448},[447],[29,496,371],{"className":497},[498],"minner",[29,500],{"className":501,"style":448},[447],[29,503,360],{"className":504},[443],[29,506],{"className":507,"style":448},[447],[29,509,511,514],{"className":510},[99],[29,512,355],{"className":513},[99,260],[29,515,517],{"className":516},[109],[29,518,520,541],{"className":519},[113,406],[29,521,523,538],{"className":522},[117],[29,524,527],{"className":525,"style":526},[121],"height:0.1514em;",[29,528,529,532],{"style":416},[29,530],{"className":531,"style":130},[129],[29,533,535],{"className":534},[134,135,136,137],[29,536,380],{"className":537},[99,260,137],[29,539,430],{"className":540},[429],[29,542,544],{"className":543},[117],[29,545,547],{"className":546,"style":437},[121],[29,548],{}," with corresponding weights ",[29,551,553,591],{"className":552},[43],[29,554,556],{"className":555},[47],[49,557,558],{"xmlns":51},[53,559,560,588],{},[56,561,562,568,570,576,578,580,582],{},[351,563,564,566],{},[242,565,244],{},[66,567,164],{},[59,569,360],{"separator":86},[351,571,572,574],{},[242,573,244],{},[66,575,282],{},[59,577,360],{"separator":86},[59,579,371],{},[59,581,360],{"separator":86},[351,583,584,586],{},[242,585,244],{},[242,587,380],{},[78,589,590],{"encoding":80},"W_1, W_2, \\dots, W_n",[29,592,594],{"className":593,"ariaHidden":86},[85],[29,595,597,600,641,644,647,687,690,693,696,699,702,705],{"className":596},[90],[29,598],{"className":599,"style":393},[94],[29,601,603,606],{"className":602},[99],[29,604,244],{"className":605,"style":261},[99,260],[29,607,609],{"className":608},[109],[29,610,612,633],{"className":611},[113,406],[29,613,615,630],{"className":614},[117],[29,616,618],{"className":617,"style":413},[121],[29,619,621,624],{"style":620},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[29,622],{"className":623,"style":130},[129],[29,625,627],{"className":626},[134,135,136,137],[29,628,164],{"className":629},[99,137],[29,631,430],{"className":632},[429],[29,634,636],{"className":635},[117],[29,637,639],{"className":638,"style":437},[121],[29,640],{},[29,642,360],{"className":643},[443],[29,645],{"className":646,"style":448},[447],[29,648,650,653],{"className":649},[99],[29,651,244],{"className":652,"style":261},[99,260],[29,654,656],{"className":655},[109],[29,657,659,679],{"className":658},[113,406],[29,660,662,676],{"className":661},[117],[29,663,665],{"className":664,"style":413},[121],[29,666,667,670],{"style":620},[29,668],{"className":669,"style":130},[129],[29,671,673],{"className":672},[134,135,136,137],[29,674,282],{"className":675},[99,137],[29,677,430],{"className":678},[429],[29,680,682],{"className":681},[117],[29,683,685],{"className":684,"style":437},[121],[29,686],{},[29,688,360],{"className":689},[443],[29,691],{"className":692,"style":448},[447],[29,694,371],{"className":695},[498],[29,697],{"className":698,"style":448},[447],[29,700,360],{"className":701},[443],[29,703],{"className":704,"style":448},[447],[29,706,708,711],{"className":707},[99],[29,709,244],{"className":710,"style":261},[99,260],[29,712,714],{"className":713},[109],[29,715,717,737],{"className":716},[113,406],[29,718,720,734],{"className":719},[117],[29,721,723],{"className":722,"style":526},[121],[29,724,725,728],{"style":620},[29,726],{"className":727,"style":130},[129],[29,729,731],{"className":730},[134,135,136,137],[29,732,380],{"className":733},[99,260,137],[29,735,430],{"className":736},[429],[29,738,740],{"className":739},[117],[29,741,743],{"className":742,"style":437},[121],[29,744],{},", the MPV is the ",[20,747,748],{},"Weighted Mean",[29,750,31],{},":\n",[29,753,755,802],{"className":754},[43],[29,756,758],{"className":757},[47],[49,759,760],{"xmlns":51},[53,761,762,799],{},[56,763,764,768,771],{},[765,766,767],"mtext",{},"MPV",[59,769,770],{},"=",[772,773,774,793],"mfrac",{},[56,775,776,779,783,785,788,790],{},[59,777,778],{},"∑",[59,780,782],{"stretchy":781},"false","(",[242,784,355],{},[59,786,787],{},"×",[242,789,244],{},[59,791,792],{"stretchy":781},")",[56,794,795,797],{},[59,796,778],{},[242,798,244],{},[78,800,801],{"encoding":80},"\\text{MPV} = \\frac{\\sum (L \\times W)}{\\sum 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Principles",[331,947,949],{"id":948},"_1-relationship-between-weight-and-error","1. Relationship Between Weight and Error",[16,951,952,953,983,984,751,987,1147,1148,32],{},"The fundamental rule for assigning weights based on standard error (",[29,954,956,970],{"className":955},[43],[29,957,959],{"className":958},[47],[49,960,961],{"xmlns":51},[53,962,963,967],{},[56,964,965],{},[242,966,279],{},[78,968,969],{"encoding":80},"\\sigma",[29,971,973],{"className":972,"ariaHidden":86},[85],[29,974,976,980],{"className":975},[90],[29,977],{"className":978,"style":979},[94],"height:0.4306em;",[29,981,279],{"className":982,"style":302},[99,260],") is ",[29,985,986],{},"4",[29,988,990,1017],{"className":989},[43],[29,991,993],{"className":992},[47],[49,994,995],{"xmlns":51},[53,996,997,1014],{},[56,998,999,1001,1004],{},[242,1000,244],{},[59,1002,1003],{},"∝",[772,1005,1006,1008],{},[66,1007,164],{},[63,1009,1010,1012],{},[242,1011,279],{},[66,1013,282],{},[78,1015,1016],{"encoding":80},"W \\propto \\frac{1}{\\sigma^2}",[29,1018,1020,1038],{"className":1019,"ariaHidden":86},[85],[29,1021,1023,1026,1029,1032,1035],{"className":1022},[90],[29,1024],{"className":1025,"style":256},[94],[29,1027,244],{"className":1028,"style":261},[99,260],[29,1030],{"className":1031,"style":821},[447],[29,1033,1003],{"className":1034},[825],[29,1036],{"className":1037,"style":821},[447],[29,1039,1041,1045],{"className":1040},[90],[29,1042],{"className":1043,"style":1044},[94],"height:1.1901em;vertical-align:-0.345em;",[29,1046,1048,1051,1144],{"className":1047},[99],[29,1049],{"className":1050},[842,843],[29,1052,1054],{"className":1053},[772],[29,1055,1057,1135],{"className":1056},[113,406],[29,1058,1060,1132],{"className":1059},[117],[29,1061,1064,1109,1117],{"className":1062,"style":1063},[121],"height:0.8451em;",[29,1065,1066,1069],{"style":859},[29,1067],{"className":1068,"style":863},[129],[29,1070,1072],{"className":1071},[134,135,136,137],[29,1073,1075],{"className":1074},[99,137],[29,1076,1078,1081],{"className":1077},[99,137],[29,1079,279],{"className":1080,"style":302},[99,260,137],[29,1082,1084],{"className":1083},[109],[29,1085,1087],{"className":1086},[113],[29,1088,1090],{"className":1089},[117],[29,1091,1094],{"className":1092,"style":1093},[121],"height:0.7463em;",[29,1095,1097,1101],{"style":1096},"top:-2.786em;margin-right:0.0714em;",[29,1098],{"className":1099,"style":1100},[129],"height:2.5em;",[29,1102,1106],{"className":1103},[134,1104,1105,137],"reset-size3","size1",[29,1107,282],{"className":1108},[99,137],[29,1110,1111,1114],{"style":886},[29,1112],{"className":1113,"style":863},[129],[29,1115],{"className":1116,"style":894},[893],[29,1118,1120,1123],{"style":1119},"top:-3.394em;",[29,1121],{"className":1122,"style":863},[129],[29,1124,1126],{"className":1125},[134,135,136,137],[29,1127,1129],{"className":1128},[99,137],[29,1130,164],{"className":1131},[99,137],[29,1133,430],{"className":1134},[429],[29,1136,1138],{"className":1137},[117],[29,1139,1142],{"className":1140,"style":1141},[121],"height:0.345em;",[29,1143],{},[29,1145],{"className":1146},[926,843],"\nThis means that if measurement B is twice as precise as measurement A (half the error), it receives four times the weight ",[29,1149,1150],{},"3",[331,1152,1154,1155,792],{"id":1153},"_2-standard-error-of-the-weighted-mean-sxˉws_barx_wsxˉw","2. Standard Error of the Weighted Mean (",[29,1156,1158,1189],{"className":1157},[43],[29,1159,1161],{"className":1160},[47],[49,1162,1163],{"xmlns":51},[53,1164,1165,1186],{},[56,1166,1167],{},[351,1168,1169,1172],{},[242,1170,1171],{},"S",[351,1173,1174,1183],{},[1175,1176,1177,1180],"mover",{"accent":86},[242,1178,1179],{},"x",[59,1181,1182],{},"ˉ",[242,1184,1185],{},"w",[78,1187,1188],{"encoding":80},"S_{\\bar{x}_w}",[29,1190,1192],{"className":1191,"ariaHidden":86},[85],[29,1193,1195,1199],{"className":1194},[90],[29,1196],{"className":1197,"style":1198},[94],"height:0.9334em;vertical-align:-0.2501em;",[29,1200,1202,1206],{"className":1201},[99],[29,1203,1171],{"className":1204,"style":1205},[99,260],"margin-right:0.0576em;",[29,1207,1209],{"className":1208},[109],[29,1210,1212,1311],{"className":1211},[113,406],[29,1213,1215,1308],{"className":1214},[117],[29,1216,1219],{"className":1217,"style":1218},[121],"height:0.2474em;",[29,1220,1222,1225],{"style":1221},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[29,1223],{"className":1224,"style":130},[129],[29,1226,1228],{"className":1227},[134,135,136,137],[29,1229,1231],{"className":1230},[99,137],[29,1232,1234,1270],{"className":1233},[99,137],[29,1235,1238],{"className":1236},[99,1237,137],"accent",[29,1239,1241],{"className":1240},[113],[29,1242,1244],{"className":1243},[117],[29,1245,1248,1257],{"className":1246,"style":1247},[121],"height:0.5678em;",[29,1249,1251,1254],{"style":1250},"top:-2.7em;",[29,1252],{"className":1253,"style":130},[129],[29,1255,1179],{"className":1256},[99,260,137],[29,1258,1259,1262],{"style":1250},[29,1260],{"className":1261,"style":130},[129],[29,1263,1267],{"className":1264,"style":1266},[1265],"accent-body","left:-0.2222em;",[29,1268,1182],{"className":1269},[99,137],[29,1271,1273],{"className":1272},[109],[29,1274,1276,1299],{"className":1275},[113,406],[29,1277,1279,1296],{"className":1278},[117],[29,1280,1283],{"className":1281,"style":1282},[121],"height:0.1645em;",[29,1284,1286,1289],{"style":1285},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[29,1287],{"className":1288,"style":1100},[129],[29,1290,1292],{"className":1291},[134,1104,1105,137],[29,1293,1185],{"className":1294,"style":1295},[99,260,137],"margin-right:0.0269em;",[29,1297,430],{"className":1298},[429],[29,1300,1302],{"className":1301},[117],[29,1303,1306],{"className":1304,"style":1305},[121],"height:0.143em;",[29,1307],{},[29,1309,430],{"className":1310},[429],[29,1312,1314],{"className":1313},[117],[29,1315,1318],{"className":1316,"style":1317},[121],"height:0.2501em;",[29,1319],{},[16,1321,1322,1323,751,1325,1731],{},"The precision of the calculated weighted mean is given by ",[29,1324,986],{},[29,1326,1328,1399],{"className":1327},[43],[29,1329,1331],{"className":1330},[47],[49,1332,1333],{"xmlns":51},[53,1334,1335,1396],{},[56,1336,1337,1351,1353,1355],{},[351,1338,1339,1341],{},[242,1340,1171],{},[351,1342,1343,1349],{},[1175,1344,1345,1347],{"accent":86},[242,1346,1179],{},[59,1348,1182],{},[242,1350,1185],{},[59,1352,770],{},[59,1354,61],{},[1356,1357,1358],"msqrt",{},[772,1359,1360,1379],{},[56,1361,1362,1364,1366,1368,1370,1377],{},[59,1363,778],{},[59,1365,782],{"stretchy":781},[242,1367,244],{},[59,1369,787],{},[63,1371,1372,1375],{},[242,1373,1374],{},"r",[66,1376,282],{},[59,1378,792],{"stretchy":781},[56,1380,1381,1383,1385,1388,1390,1392,1394],{},[59,1382,782],{"stretchy":781},[242,1384,380],{},[59,1386,1387],{},"−",[66,1389,164],{},[59,1391,792],{"stretchy":781},[59,1393,778],{},[242,1395,244],{},[78,1397,1398],{"encoding":80},"S_{\\bar{x}_w} = \\pm \\sqrt{\\frac{\\sum (W \\times r^2)}{(n-1) \\sum W}}",[29,1400,1402,1525],{"className":1401,"ariaHidden":86},[85],[29,1403,1405,1408,1516,1519,1522],{"className":1404},[90],[29,1406],{"className":1407,"style":1198},[94],[29,1409,1411,1414],{"className":1410},[99],[29,1412,1171],{"className":1413,"style":1205},[99,260],[29,1415,1417],{"className":1416},[109],[29,1418,1420,1508],{"className":1419},[113,406],[29,1421,1423,1505],{"className":1422},[117],[29,1424,1426],{"className":1425,"style":1218},[121],[29,1427,1428,1431],{"style":1221},[29,1429],{"className":1430,"style":130},[129],[29,1432,1434],{"className":1433},[134,135,136,137],[29,1435,1437],{"className":1436},[99,137],[29,1438,1440,1471],{"className":1439},[99,137],[29,1441,1443],{"className":1442},[99,1237,137],[29,1444,1446],{"className":1445},[113],[29,1447,1449],{"className":1448},[117],[29,1450,1452,1460],{"className":1451,"style":1247},[121],[29,1453,1454,1457],{"style":1250},[29,1455],{"className":1456,"style":130},[129],[29,1458,1179],{"className":1459},[99,260,137],[29,1461,1462,1465],{"style":1250},[29,1463],{"className":1464,"style":130},[129],[29,1466,1468],{"className":1467,"style":1266},[1265],[29,1469,1182],{"className":1470},[99,137],[29,1472,1474],{"className":1473},[109],[29,1475,1477,1497],{"className":1476},[113,406],[29,1478,1480,1494],{"className":1479},[117],[29,1481,1483],{"className":1482,"style":1282},[121],[29,1484,1485,1488],{"style":1285},[29,1486],{"className":1487,"style":1100},[129],[29,1489,1491],{"className":1490},[134,1104,1105,137],[29,1492,1185],{"className":1493,"style":1295},[99,260,137],[29,1495,430],{"className":1496},[429],[29,1498,1500],{"className":1499},[117],[29,1501,1503],{"className":1502,"style":1305},[121],[29,1504],{},[29,1506,430],{"className":1507},[429],[29,1509,1511],{"className":1510},[117],[29,1512,1514],{"className":1513,"style":1317},[121],[29,1515],{},[29,1517],{"className":1518,"style":821},[447],[29,1520,770],{"className":1521},[825],[29,1523],{"className":1524,"style":821},[447],[29,1526,1528,1532,1535],{"className":1527},[90],[29,1529],{"className":1530,"style":1531},[94],"height:1.84em;vertical-align:-0.61em;",[29,1533,61],{"className":1534},[99],[29,1536,1539],{"className":1537},[99,1538],"sqrt",[29,1540,1542,1722],{"className":1541},[113,406],[29,1543,1545,1719],{"className":1544},[117],[29,1546,1549,1696],{"className":1547,"style":1548},[121],"height:1.23em;",[29,1550,1554,1558],{"className":1551,"style":1553},[1552],"svg-align","top:-3.8em;",[29,1555],{"className":1556,"style":1557},[129],"height:3.8em;",[29,1559,1562],{"className":1560,"style":1561},[99],"padding-left:1em;",[29,1563,1565,1568,1693],{"className":1564},[99],[29,1566],{"className":1567},[842,843],[29,1569,1571],{"className":1570},[772],[29,1572,1574,1685],{"className":1573},[113,406],[29,1575,1577,1682],{"className":1576},[117],[29,1578,1580,1618,1626],{"className":1579,"style":856},[121],[29,1581,1582,1585],{"style":859},[29,1583],{"className":1584,"style":863},[129],[29,1586,1588],{"className":1587},[134,135,136,137],[29,1589,1591,1594,1597,1600,1603,1606,1609,1612,1615],{"className":1590},[99,137],[29,1592,782],{"className":1593},[842,137],[29,1595,380],{"className":1596},[99,260,137],[29,1598,1387],{"className":1599},[919,137],[29,1601,164],{"className":1602},[99,137],[29,1604,792],{"className":1605},[926,137],[29,1607],{"className":1608,"style":880},[447,137],[29,1610,778],{"className":1611,"style":876},[873,874,875,137],[29,1613],{"className":1614,"style":880},[447,137],[29,1616,244],{"className":1617,"style":261},[99,260,137],[29,1619,1620,1623],{"style":886},[29,1621],{"className":1622,"style":863},[129],[29,1624],{"className":1625,"style":894},[893],[29,1627,1628,1631],{"style":897},[29,1629],{"className":1630,"style":863},[129],[29,1632,1634],{"className":1633},[134,135,136,137],[29,1635,1637,1640,1643,1646,1649,1679],{"className":1636},[99,137],[29,1638,778],{"className":1639,"style":876},[873,874,875,137],[29,1641,782],{"className":1642},[842,137],[29,1644,244],{"className":1645,"style":261},[99,260,137],[29,1647,787],{"className":1648},[919,137],[29,1650,1652,1656],{"className":1651},[99,137],[29,1653,1374],{"className":1654,"style":1655},[99,260,137],"margin-right:0.0278em;",[29,1657,1659],{"className":1658},[109],[29,1660,1662],{"className":1661},[113],[29,1663,1665],{"className":1664},[117],[29,1666,1668],{"className":1667,"style":1093},[121],[29,1669,1670,1673],{"style":1096},[29,1671],{"className":1672,"style":1100},[129],[29,1674,1676],{"className":1675},[134,1104,1105,137],[29,1677,282],{"className":1678},[99,137],[29,1680,792],{"className":1681},[926,137],[29,1683,430],{"className":1684},[429],[29,1686,1688],{"className":1687},[117],[29,1689,1691],{"className":1690,"style":936},[121],[29,1692],{},[29,1694],{"className":1695},[926,843],[29,1697,1699,1702],{"style":1698},"top:-3.19em;",[29,1700],{"className":1701,"style":1557},[129],[29,1703,1707],{"className":1704,"style":1706},[1705],"hide-tail","min-width:1.02em;height:1.88em;",[1708,1709,1715],"svg",{"xmlns":1710,"width":1711,"height":1712,"viewBox":1713,"preserveAspectRatio":1714},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.88em","0 0 400000 1944","xMinYMin slice",[1716,1717],"path",{"d":1718},"M983 90\nl0 -0\nc4,-6.7,10,-10,18,-10 H400000v40\nH1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7\ns-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744\nc-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30\nc26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722\nc56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5\nc53.7,-170.3,84.5,-266.8,92.5,-289.5z\nM1001 80h400000v40h-400000z",[29,1720,430],{"className":1721},[429],[29,1723,1725],{"className":1724},[117],[29,1726,1729],{"className":1727,"style":1728},[121],"height:0.61em;",[29,1730],{},[1732,1733,1734,1735,1763],"em",{},"(Where ",[29,1736,1738,1751],{"className":1737},[43],[29,1739,1741],{"className":1740},[47],[49,1742,1743],{"xmlns":51},[53,1744,1745,1749],{},[56,1746,1747],{},[242,1748,1374],{},[78,1750,1374],{"encoding":80},[29,1752,1754],{"className":1753,"ariaHidden":86},[85],[29,1755,1757,1760],{"className":1756},[90],[29,1758],{"className":1759,"style":979},[94],[29,1761,1374],{"className":1762,"style":1655},[99,260]," is the residual of each observation from the weighted mean).",[11,1765,1767],{"id":1766},"field-workflow","Field Workflow",[1769,1770,1771,1775,1778,1782,1894,1898,1974,1978,1983,1987],"steps",{},[331,1772,1774],{"id":1773},"data-collection","Data Collection",[16,1776,1777],{},"Collect all relevant measurements for the target quantity (e.g., an angle measured by different teams).",[331,1779,1781],{"id":1780},"calculate-individual-precision","Calculate Individual Precision",[16,1783,1784,1785,1891,1892,32],{},"Determine the standard error (",[29,1786,1788,1810],{"className":1787},[43],[29,1789,1791],{"className":1790},[47],[49,1792,1793],{"xmlns":51},[53,1794,1795,1807],{},[56,1796,1797],{},[351,1798,1799,1801],{},[242,1800,1171],{},[1175,1802,1803,1805],{"accent":86},[242,1804,1179],{},[59,1806,1182],{},[78,1808,1809],{"encoding":80},"S_{\\bar{x}}",[29,1811,1813],{"className":1812,"ariaHidden":86},[85],[29,1814,1816,1820],{"className":1815},[90],[29,1817],{"className":1818,"style":1819},[94],"height:0.8333em;vertical-align:-0.15em;",[29,1821,1823,1826],{"className":1822},[99],[29,1824,1171],{"className":1825,"style":1205},[99,260],[29,1827,1829],{"className":1828},[109],[29,1830,1832,1883],{"className":1831},[113,406],[29,1833,1835,1880],{"className":1834},[117],[29,1836,1838],{"className":1837,"style":1218},[121],[29,1839,1840,1843],{"style":1221},[29,1841],{"className":1842,"style":130},[129],[29,1844,1846],{"className":1845},[134,135,136,137],[29,1847,1849],{"className":1848},[99,137],[29,1850,1852],{"className":1851},[99,1237,137],[29,1853,1855],{"className":1854},[113],[29,1856,1858],{"className":1857},[117],[29,1859,1861,1869],{"className":1860,"style":1247},[121],[29,1862,1863,1866],{"style":1250},[29,1864],{"className":1865,"style":130},[129],[29,1867,1179],{"className":1868},[99,260,137],[29,1870,1871,1874],{"style":1250},[29,1872],{"className":1873,"style":130},[129],[29,1875,1877],{"className":1876,"style":1266},[1265],[29,1878,1182],{"className":1879},[99,137],[29,1881,430],{"className":1882},[429],[29,1884,1886],{"className":1885},[117],[29,1887,1889],{"className":1888,"style":437},[121],[29,1890],{},") for each set of observations to assess its internal consistency ",[29,1893,329],{},[331,1895,1897],{"id":1896},"assign-weight-ratios","Assign Weight Ratios",[16,1899,1900,1901,1970,1971,32],{},"Calculate the weight for each set using ",[29,1902,1904,1927],{"className":1903},[43],[29,1905,1907],{"className":1906},[47],[49,1908,1909],{"xmlns":51},[53,1910,1911,1924],{},[56,1912,1913,1915,1918],{},[66,1914,164],{},[242,1916,1917],{"mathvariant":73},"\u002F",[63,1919,1920,1922],{},[242,1921,1171],{},[66,1923,282],{},[78,1925,1926],{"encoding":80},"1\u002FS^2",[29,1928,1930],{"className":1929,"ariaHidden":86},[85],[29,1931,1933,1937,1941],{"className":1932},[90],[29,1934],{"className":1935,"style":1936},[94],"height:1.0641em;vertical-align:-0.25em;",[29,1938,1940],{"className":1939},[99],"1\u002F",[29,1942,1944,1947],{"className":1943},[99],[29,1945,1171],{"className":1946,"style":1205},[99,260],[29,1948,1950],{"className":1949},[109],[29,1951,1953],{"className":1952},[113],[29,1954,1956],{"className":1955},[117],[29,1957,1959],{"className":1958,"style":295},[121],[29,1960,1961,1964],{"style":125},[29,1962],{"className":1963,"style":130},[129],[29,1965,1967],{"className":1966},[134,135,136,137],[29,1968,282],{"className":1969},[99,137],". Simplify these into a usable weight ratio ",[29,1972,1973],{},"2, 4",[331,1975,1977],{"id":1976},"compute-weighted-products","Compute Weighted Products",[16,1979,1980,1981,32],{},"Multiply each measured value (or its fractional part for easier math) by its assigned weight ",[29,1982,282],{},[331,1984,1986],{"id":1985},"final-reduction","Final Reduction",[16,1988,1989,1990,32],{},"Divide the sum of the weighted products by the total sum of the weights to find the MPV ",[29,1991,282],{},[11,1993,1995],{"id":1994},"step-by-step-example","Step-by-Step Example",[16,1997,1998,2001,2002,2004],{},[20,1999,2000],{},"Problem:"," An angle is measured by three observers. Determine the MPV based on their recorded means and standard errors ",[29,2003,282],{},":",[2006,2007,2008,2235,2451],"ul",{},[2009,2010,2011,2012],"li",{},"Observer A: ",[29,2013,2015,2067],{"className":2014},[43],[29,2016,2018],{"className":2017},[47],[49,2019,2020],{"xmlns":51},[53,2021,2022,2064],{},[56,2023,2024,2032,2040,2051,2053],{},[63,2025,2026,2029],{},[66,2027,2028],{},"89",[59,2030,2031],{},"∘",[63,2033,2034,2037],{},[66,2035,2036],{},"54",[59,2038,74],{"mathvariant":73,"lspace":2039,"rspace":2039},"0em",[63,2041,2042,2045],{},[66,2043,2044],{},"36",[56,2046,2047,2049],{},[59,2048,74],{"mathvariant":73},[59,2050,74],{"mathvariant":73},[59,2052,61],{},[63,2054,2055,2058],{},[66,2056,2057],{},"0.7",[56,2059,2060,2062],{},[59,2061,74],{"mathvariant":73},[59,2063,74],{"mathvariant":73},[78,2065,2066],{"encoding":80},"89^\\circ 54' 36'' \\pm 0.7''",[29,2068,2070,2192],{"className":2069,"ariaHidden":86},[85],[29,2071,2073,2076,2080,2111,2114,2146,2149,2182,2186,2189],{"className":2072},[90],[29,2074],{"className":2075,"style":95},[94],[29,2077,2079],{"className":2078},[99],"8",[29,2081,2083,2087],{"className":2082},[99],[29,2084,2086],{"className":2085},[99],"9",[29,2088,2090],{"className":2089},[109],[29,2091,2093],{"className":2092},[113],[29,2094,2096],{"className":2095},[117],[29,2097,2100],{"className":2098,"style":2099},[121],"height:0.6741em;",[29,2101,2102,2105],{"style":125},[29,2103],{"className":2104,"style":130},[129],[29,2106,2108],{"className":2107},[134,135,136,137],[29,2109,2031],{"className":2110},[919,137],[29,2112,68],{"className":2113},[99],[29,2115,2117,2120],{"className":2116},[99],[29,2118,986],{"className":2119},[99],[29,2121,2123],{"className":2122},[109],[29,2124,2126],{"className":2125},[113],[29,2127,2129],{"className":2128},[117],[29,2130,2132],{"className":2131,"style":122},[121],[29,2133,2134,2137],{"style":125},[29,2135],{"className":2136,"style":130},[129],[29,2138,2140],{"className":2139},[134,135,136,137],[29,2141,2143],{"className":2142},[99,137],[29,2144,74],{"className":2145},[99,137],[29,2147,1150],{"className":2148},[99],[29,2150,2152,2156],{"className":2151},[99],[29,2153,2155],{"className":2154},[99],"6",[29,2157,2159],{"className":2158},[109],[29,2160,2162],{"className":2161},[113],[29,2163,2165],{"className":2164},[117],[29,2166,2168],{"className":2167,"style":122},[121],[29,2169,2170,2173],{"style":125},[29,2171],{"className":2172,"style":130},[129],[29,2174,2176],{"className":2175},[134,135,136,137],[29,2177,2179],{"className":2178},[99,137],[29,2180,144],{"className":2181},[99,137],[29,2183],{"className":2184,"style":2185},[447],"margin-right:0.2222em;",[29,2187,61],{"className":2188},[919],[29,2190],{"className":2191,"style":2185},[447],[29,2193,2195,2198,2202],{"className":2194},[90],[29,2196],{"className":2197,"style":122},[94],[29,2199,2201],{"className":2200},[99],"0.",[29,2203,2205,2209],{"className":2204},[99],[29,2206,2208],{"className":2207},[99],"7",[29,2210,2212],{"className":2211},[109],[29,2213,2215],{"className":2214},[113],[29,2216,2218],{"className":2217},[117],[29,2219,2221],{"className":2220,"style":122},[121],[29,2222,2223,2226],{"style":125},[29,2224],{"className":2225,"style":130},[129],[29,2227,2229],{"className":2228},[134,135,136,137],[29,2230,2232],{"className":2231},[99,137],[29,2233,144],{"className":2234},[99,137],[2009,2236,2237,2238],{},"Observer B: ",[29,2239,2241,2289],{"className":2240},[43],[29,2242,2244],{"className":2243},[47],[49,2245,2246],{"xmlns":51},[53,2247,2248,2286],{},[56,2249,2250,2256,2262,2273,2275],{},[63,2251,2252,2254],{},[66,2253,2028],{},[59,2255,2031],{},[63,2257,2258,2260],{},[66,2259,2036],{},[59,2261,74],{"mathvariant":73,"lspace":2039,"rspace":2039},[63,2263,2264,2267],{},[66,2265,2266],{},"42",[56,2268,2269,2271],{},[59,2270,74],{"mathvariant":73},[59,2272,74],{"mathvariant":73},[59,2274,61],{},[63,2276,2277,2280],{},[66,2278,2279],{},"1.2",[56,2281,2282,2284],{},[59,2283,74],{"mathvariant":73},[59,2285,74],{"mathvariant":73},[78,2287,2288],{"encoding":80},"89^\\circ 54' 42'' \\pm 1.2''",[29,2290,2292,2409],{"className":2291,"ariaHidden":86},[85],[29,2293,2295,2298,2301,2330,2333,2365,2368,2400,2403,2406],{"className":2294},[90],[29,2296],{"className":2297,"style":95},[94],[29,2299,2079],{"className":2300},[99],[29,2302,2304,2307],{"className":2303},[99],[29,2305,2086],{"className":2306},[99],[29,2308,2310],{"className":2309},[109],[29,2311,2313],{"className":2312},[113],[29,2314,2316],{"className":2315},[117],[29,2317,2319],{"className":2318,"style":2099},[121],[29,2320,2321,2324],{"style":125},[29,2322],{"className":2323,"style":130},[129],[29,2325,2327],{"className":2326},[134,135,136,137],[29,2328,2031],{"className":2329},[919,137],[29,2331,68],{"className":2332},[99],[29,2334,2336,2339],{"className":2335},[99],[29,2337,986],{"className":2338},[99],[29,2340,2342],{"className":2341},[109],[29,2343,2345],{"className":2344},[113],[29,2346,2348],{"className":2347},[117],[29,2349,2351],{"className":2350,"style":122},[121],[29,2352,2353,2356],{"style":125},[29,2354],{"className":2355,"style":130},[129],[29,2357,2359],{"className":2358},[134,135,136,137],[29,2360,2362],{"className":2361},[99,137],[29,2363,74],{"className":2364},[99,137],[29,2366,986],{"className":2367},[99],[29,2369,2371,2374],{"className":2370},[99],[29,2372,282],{"className":2373},[99],[29,2375,2377],{"className":2376},[109],[29,2378,2380],{"className":2379},[113],[29,2381,2383],{"className":2382},[117],[29,2384,2386],{"className":2385,"style":122},[121],[29,2387,2388,2391],{"style":125},[29,2389],{"className":2390,"style":130},[129],[29,2392,2394],{"className":2393},[134,135,136,137],[29,2395,2397],{"className":2396},[99,137],[29,2398,144],{"className":2399},[99,137],[29,2401],{"className":2402,"style":2185},[447],[29,2404,61],{"className":2405},[919],[29,2407],{"className":2408,"style":2185},[447],[29,2410,2412,2415,2419],{"className":2411},[90],[29,2413],{"className":2414,"style":122},[94],[29,2416,2418],{"className":2417},[99],"1.",[29,2420,2422,2425],{"className":2421},[99],[29,2423,282],{"className":2424},[99],[29,2426,2428],{"className":2427},[109],[29,2429,2431],{"className":2430},[113],[29,2432,2434],{"className":2433},[117],[29,2435,2437],{"className":2436,"style":122},[121],[29,2438,2439,2442],{"style":125},[29,2440],{"className":2441,"style":130},[129],[29,2443,2445],{"className":2444},[134,135,136,137],[29,2446,2448],{"className":2447},[99,137],[29,2449,144],{"className":2450},[99,137],[2009,2452,2453,2454],{},"Observer C: ",[29,2455,2457,2505],{"className":2456},[43],[29,2458,2460],{"className":2459},[47],[49,2461,2462],{"xmlns":51},[53,2463,2464,2502],{},[56,2465,2466,2472,2478,2489,2491],{},[63,2467,2468,2470],{},[66,2469,2028],{},[59,2471,2031],{},[63,2473,2474,2476],{},[66,2475,2036],{},[59,2477,74],{"mathvariant":73,"lspace":2039,"rspace":2039},[63,2479,2480,2483],{},[66,2481,2482],{},"33",[56,2484,2485,2487],{},[59,2486,74],{"mathvariant":73},[59,2488,74],{"mathvariant":73},[59,2490,61],{},[63,2492,2493,2496],{},[66,2494,2495],{},"1.0",[56,2497,2498,2500],{},[59,2499,74],{"mathvariant":73},[59,2501,74],{"mathvariant":73},[78,2503,2504],{"encoding":80},"89^\\circ 54' 33'' \\pm 1.0''",[29,2506,2508,2625],{"className":2507,"ariaHidden":86},[85],[29,2509,2511,2514,2517,2546,2549,2581,2584,2616,2619,2622],{"className":2510},[90],[29,2512],{"className":2513,"style":95},[94],[29,2515,2079],{"className":2516},[99],[29,2518,2520,2523],{"className":2519},[99],[29,2521,2086],{"className":2522},[99],[29,2524,2526],{"className":2525},[109],[29,2527,2529],{"className":2528},[113],[29,2530,2532],{"className":2531},[117],[29,2533,2535],{"className":2534,"style":2099},[121],[29,2536,2537,2540],{"style":125},[29,2538],{"className":2539,"style":130},[129],[29,2541,2543],{"className":2542},[134,135,136,137],[29,2544,2031],{"className":2545},[919,137],[29,2547,68],{"className":2548},[99],[29,2550,2552,2555],{"className":2551},[99],[29,2553,986],{"className":2554},[99],[29,2556,2558],{"className":2557},[109],[29,2559,2561],{"className":2560},[113],[29,2562,2564],{"className":2563},[117],[29,2565,2567],{"className":2566,"style":122},[121],[29,2568,2569,2572],{"style":125},[29,2570],{"className":2571,"style":130},[129],[29,2573,2575],{"className":2574},[134,135,136,137],[29,2576,2578],{"className":2577},[99,137],[29,2579,74],{"className":2580},[99,137],[29,2582,1150],{"className":2583},[99],[29,2585,2587,2590],{"className":2586},[99],[29,2588,1150],{"className":2589},[99],[29,2591,2593],{"className":2592},[109],[29,2594,2596],{"className":2595},[113],[29,2597,2599],{"className":2598},[117],[29,2600,2602],{"className":2601,"style":122},[121],[29,2603,2604,2607],{"style":125},[29,2605],{"className":2606,"style":130},[129],[29,2608,2610],{"className":2609},[134,135,136,137],[29,2611,2613],{"className":2612},[99,137],[29,2614,144],{"className":2615},[99,137],[29,2617],{"className":2618,"style":2185},[447],[29,2620,61],{"className":2621},[919],[29,2623],{"className":2624,"style":2185},[447],[29,2626,2628,2631,2634],{"className":2627},[90],[29,2629],{"className":2630,"style":122},[94],[29,2632,2418],{"className":2633},[99],[29,2635,2637,2641],{"className":2636},[99],[29,2638,2640],{"className":2639},[99],"0",[29,2642,2644],{"className":2643},[109],[29,2645,2647],{"className":2646},[113],[29,2648,2650],{"className":2649},[117],[29,2651,2653],{"className":2652,"style":122},[121],[29,2654,2655,2658],{"style":125},[29,2656],{"className":2657,"style":130},[129],[29,2659,2661],{"className":2660},[134,135,136,137],[29,2662,2664],{"className":2663},[99,137],[29,2665,144],{"className":2666},[99,137],[2668,2669,2670,3214,3285,3763,4012],"ol",{},[2009,2671,2672,2741],{},[20,2673,2674,2675,2740],{},"Calculate Weights (",[29,2676,2678,2699],{"className":2677},[43],[29,2679,2681],{"className":2680},[47],[49,2682,2683],{"xmlns":51},[53,2684,2685,2697],{},[56,2686,2687,2689,2691],{},[66,2688,164],{},[242,2690,1917],{"mathvariant":73},[63,2692,2693,2695],{},[242,2694,1171],{},[66,2696,282],{},[78,2698,1926],{"encoding":80},[29,2700,2702],{"className":2701,"ariaHidden":86},[85],[29,2703,2705,2708,2711],{"className":2704},[90],[29,2706],{"className":2707,"style":1936},[94],[29,2709,1940],{"className":2710},[99],[29,2712,2714,2717],{"className":2713},[99],[29,2715,1171],{"className":2716,"style":1205},[99,260],[29,2718,2720],{"className":2719},[109],[29,2721,2723],{"className":2722},[113],[29,2724,2726],{"className":2725},[117],[29,2727,2729],{"className":2728,"style":295},[121],[29,2730,2731,2734],{"style":125},[29,2732],{"className":2733,"style":130},[129],[29,2735,2737],{"className":2736},[134,135,136,137],[29,2738,282],{"className":2739},[99,137],"):",[2006,2742,2743,2901,3058],{},[2009,2744,2745],{},[29,2746,2748,2784],{"className":2747},[43],[29,2749,2751],{"className":2750},[47],[49,2752,2753],{"xmlns":51},[53,2754,2755,2781],{},[56,2756,2757,2764,2766,2768,2770,2776,2778],{},[351,2758,2759,2761],{},[242,2760,244],{},[242,2762,2763],{},"A",[59,2765,770],{},[66,2767,164],{},[242,2769,1917],{"mathvariant":73},[63,2771,2772,2774],{},[66,2773,2057],{},[66,2775,282],{},[59,2777,770],{},[66,2779,2780],{},"2.04",[78,2782,2783],{"encoding":80},"W_A = 1\u002F0.7^2 = 2.04",[29,2785,2787,2843,2891],{"className":2786,"ariaHidden":86},[85],[29,2788,2790,2793,2834,2837,2840],{"className":2789},[90],[29,2791],{"className":2792,"style":1819},[94],[29,2794,2796,2799],{"className":2795},[99],[29,2797,244],{"className":2798,"style":261},[99,260],[29,2800,2802],{"className":2801},[109],[29,2803,2805,2826],{"className":2804},[113,406],[29,2806,2808,2823],{"className":2807},[117],[29,2809,2812],{"className":2810,"style":2811},[121],"height:0.3283em;",[29,2813,2814,2817],{"style":620},[29,2815],{"className":2816,"style":130},[129],[29,2818,2820],{"className":2819},[134,135,136,137],[29,2821,2763],{"className":2822},[99,260,137],[29,2824,430],{"className":2825},[429],[29,2827,2829],{"className":2828},[117],[29,2830,2832],{"className":2831,"style":437},[121],[29,2833],{},[29,2835],{"className":2836,"style":821},[447],[29,2838,770],{"className":2839},[825],[29,2841],{"className":2842,"style":821},[447],[29,2844,2846,2849,2853,2882,2885,2888],{"className":2845},[90],[29,2847],{"className":2848,"style":1936},[94],[29,2850,2852],{"className":2851},[99],"1\u002F0.",[29,2854,2856,2859],{"className":2855},[99],[29,2857,2208],{"className":2858},[99],[29,2860,2862],{"className":2861},[109],[29,2863,2865],{"className":2864},[113],[29,2866,2868],{"className":2867},[117],[29,2869,2871],{"className":2870,"style":295},[121],[29,2872,2873,2876],{"style":125},[29,2874],{"className":2875,"style":130},[129],[29,2877,2879],{"className":2878},[134,135,136,137],[29,2880,282],{"className":2881},[99,137],[29,2883],{"className":2884,"style":821},[447],[29,2886,770],{"className":2887},[825],[29,2889],{"className":2890,"style":821},[447],[29,2892,2894,2898],{"className":2893},[90],[29,2895],{"className":2896,"style":2897},[94],"height:0.6444em;",[29,2899,2780],{"className":2900},[99],[2009,2902,2903],{},[29,2904,2906,2942],{"className":2905},[43],[29,2907,2909],{"className":2908},[47],[49,2910,2911],{"xmlns":51},[53,2912,2913,2939],{},[56,2914,2915,2922,2924,2926,2928,2934,2936],{},[351,2916,2917,2919],{},[242,2918,244],{},[242,2920,2921],{},"B",[59,2923,770],{},[66,2925,164],{},[242,2927,1917],{"mathvariant":73},[63,2929,2930,2932],{},[66,2931,2279],{},[66,2933,282],{},[59,2935,770],{},[66,2937,2938],{},"0.69",[78,2940,2941],{"encoding":80},"W_B = 1\u002F1.2^2 = 0.69",[29,2943,2945,3001,3049],{"className":2944,"ariaHidden":86},[85],[29,2946,2948,2951,2992,2995,2998],{"className":2947},[90],[29,2949],{"className":2950,"style":1819},[94],[29,2952,2954,2957],{"className":2953},[99],[29,2955,244],{"className":2956,"style":261},[99,260],[29,2958,2960],{"className":2959},[109],[29,2961,2963,2984],{"className":2962},[113,406],[29,2964,2966,2981],{"className":2965},[117],[29,2967,2969],{"className":2968,"style":2811},[121],[29,2970,2971,2974],{"style":620},[29,2972],{"className":2973,"style":130},[129],[29,2975,2977],{"className":2976},[134,135,136,137],[29,2978,2921],{"className":2979,"style":2980},[99,260,137],"margin-right:0.0502em;",[29,2982,430],{"className":2983},[429],[29,2985,2987],{"className":2986},[117],[29,2988,2990],{"className":2989,"style":437},[121],[29,2991],{},[29,2993],{"className":2994,"style":821},[447],[29,2996,770],{"className":2997},[825],[29,2999],{"className":3000,"style":821},[447],[29,3002,3004,3007,3011,3040,3043,3046],{"className":3003},[90],[29,3005],{"className":3006,"style":1936},[94],[29,3008,3010],{"className":3009},[99],"1\u002F1.",[29,3012,3014,3017],{"className":3013},[99],[29,3015,282],{"className":3016},[99],[29,3018,3020],{"className":3019},[109],[29,3021,3023],{"className":3022},[113],[29,3024,3026],{"className":3025},[117],[29,3027,3029],{"className":3028,"style":295},[121],[29,3030,3031,3034],{"style":125},[29,3032],{"className":3033,"style":130},[129],[29,3035,3037],{"className":3036},[134,135,136,137],[29,3038,282],{"className":3039},[99,137],[29,3041],{"className":3042,"style":821},[447],[29,3044,770],{"className":3045},[825],[29,3047],{"className":3048,"style":821},[447],[29,3050,3052,3055],{"className":3051},[90],[29,3053],{"className":3054,"style":2897},[94],[29,3056,2938],{"className":3057},[99],[2009,3059,3060],{},[29,3061,3063,3099],{"className":3062},[43],[29,3064,3066],{"className":3065},[47],[49,3067,3068],{"xmlns":51},[53,3069,3070,3096],{},[56,3071,3072,3079,3081,3083,3085,3091,3093],{},[351,3073,3074,3076],{},[242,3075,244],{},[242,3077,3078],{},"C",[59,3080,770],{},[66,3082,164],{},[242,3084,1917],{"mathvariant":73},[63,3086,3087,3089],{},[66,3088,2495],{},[66,3090,282],{},[59,3092,770],{},[66,3094,3095],{},"1.00",[78,3097,3098],{"encoding":80},"W_C = 1\u002F1.0^2 = 1.00",[29,3100,3102,3158,3205],{"className":3101,"ariaHidden":86},[85],[29,3103,3105,3108,3149,3152,3155],{"className":3104},[90],[29,3106],{"className":3107,"style":1819},[94],[29,3109,3111,3114],{"className":3110},[99],[29,3112,244],{"className":3113,"style":261},[99,260],[29,3115,3117],{"className":3116},[109],[29,3118,3120,3141],{"className":3119},[113,406],[29,3121,3123,3138],{"className":3122},[117],[29,3124,3126],{"className":3125,"style":2811},[121],[29,3127,3128,3131],{"style":620},[29,3129],{"className":3130,"style":130},[129],[29,3132,3134],{"className":3133},[134,135,136,137],[29,3135,3078],{"className":3136,"style":3137},[99,260,137],"margin-right:0.0715em;",[29,3139,430],{"className":3140},[429],[29,3142,3144],{"className":3143},[117],[29,3145,3147],{"className":3146,"style":437},[121],[29,3148],{},[29,3150],{"className":3151,"style":821},[447],[29,3153,770],{"className":3154},[825],[29,3156],{"className":3157,"style":821},[447],[29,3159,3161,3164,3167,3196,3199,3202],{"className":3160},[90],[29,3162],{"className":3163,"style":1936},[94],[29,3165,3010],{"className":3166},[99],[29,3168,3170,3173],{"className":3169},[99],[29,3171,2640],{"className":3172},[99],[29,3174,3176],{"className":3175},[109],[29,3177,3179],{"className":3178},[113],[29,3180,3182],{"className":3181},[117],[29,3183,3185],{"className":3184,"style":295},[121],[29,3186,3187,3190],{"style":125},[29,3188],{"className":3189,"style":130},[129],[29,3191,3193],{"className":3192},[134,135,136,137],[29,3194,282],{"className":3195},[99,137],[29,3197],{"className":3198,"style":821},[447],[29,3200,770],{"className":3201},[825],[29,3203],{"className":3204,"style":821},[447],[29,3206,3208,3211],{"className":3207},[90],[29,3209],{"className":3210,"style":2897},[94],[29,3212,3095],{"className":3213},[99],[2009,3215,3216,3219],{},[20,3217,3218],{},"Simplify Ratios (optional, but good for checks):",[2006,3220,3221],{},[2009,3222,3223,3224],{},"Total weight ",[29,3225,3227,3248],{"className":3226},[43],[29,3228,3230],{"className":3229},[47],[49,3231,3232],{"xmlns":51},[53,3233,3234,3245],{},[56,3235,3236,3238,3240,3242],{},[59,3237,778],{},[242,3239,244],{},[59,3241,770],{},[66,3243,3244],{},"3.73",[78,3246,3247],{"encoding":80},"\\sum W = 3.73",[29,3249,3251,3276],{"className":3250,"ariaHidden":86},[85],[29,3252,3254,3258,3261,3264,3267,3270,3273],{"className":3253},[90],[29,3255],{"className":3256,"style":3257},[94],"height:1em;vertical-align:-0.25em;",[29,3259,778],{"className":3260,"style":876},[873,874,875],[29,3262],{"className":3263,"style":448},[447],[29,3265,244],{"className":3266,"style":261},[99,260],[29,3268],{"className":3269,"style":821},[447],[29,3271,770],{"className":3272},[825],[29,3274],{"className":3275,"style":821},[447],[29,3277,3279,3282],{"className":3278},[90],[29,3280],{"className":3281,"style":2897},[94],[29,3283,3244],{"className":3284},[99],[2009,3286,3287,3290],{},[20,3288,3289],{},"Multiply Seconds by Weights:",[2006,3291,3292,3449,3606],{},[2009,3293,3294],{},[29,3295,3297,3336],{"className":3296},[43],[29,3298,3300],{"className":3299},[47],[49,3301,3302],{"xmlns":51},[53,3303,3304,3333],{},[56,3305,3306,3316,3318,3320,3322],{},[63,3307,3308,3310],{},[66,3309,2044],{},[56,3311,3312,3314],{},[59,3313,74],{"mathvariant":73},[59,3315,74],{"mathvariant":73},[59,3317,787],{},[66,3319,2780],{},[59,3321,770],{},[63,3323,3324,3327],{},[66,3325,3326],{},"73.44",[56,3328,3329,3331],{},[59,3330,74],{"mathvariant":73},[59,3332,74],{"mathvariant":73},[78,3334,3335],{"encoding":80},"36'' \\times 2.04 = 73.44''",[29,3337,3339,3389,3407],{"className":3338,"ariaHidden":86},[85],[29,3340,3342,3345,3348,3380,3383,3386],{"className":3341},[90],[29,3343],{"className":3344,"style":95},[94],[29,3346,1150],{"className":3347},[99],[29,3349,3351,3354],{"className":3350},[99],[29,3352,2155],{"className":3353},[99],[29,3355,3357],{"className":3356},[109],[29,3358,3360],{"className":3359},[113],[29,3361,3363],{"className":3362},[117],[29,3364,3366],{"className":3365,"style":122},[121],[29,3367,3368,3371],{"style":125},[29,3369],{"className":3370,"style":130},[129],[29,3372,3374],{"className":3373},[134,135,136,137],[29,3375,3377],{"className":3376},[99,137],[29,3378,144],{"className":3379},[99,137],[29,3381],{"className":3382,"style":2185},[447],[29,3384,787],{"className":3385},[919],[29,3387],{"className":3388,"style":2185},[447],[29,3390,3392,3395,3398,3401,3404],{"className":3391},[90],[29,3393],{"className":3394,"style":2897},[94],[29,3396,2780],{"className":3397},[99],[29,3399],{"className":3400,"style":821},[447],[29,3402,770],{"className":3403},[825],[29,3405],{"className":3406,"style":821},[447],[29,3408,3410,3413,3417],{"className":3409},[90],[29,3411],{"className":3412,"style":122},[94],[29,3414,3416],{"className":3415},[99],"73.4",[29,3418,3420,3423],{"className":3419},[99],[29,3421,986],{"className":3422},[99],[29,3424,3426],{"className":3425},[109],[29,3427,3429],{"className":3428},[113],[29,3430,3432],{"className":3431},[117],[29,3433,3435],{"className":3434,"style":122},[121],[29,3436,3437,3440],{"style":125},[29,3438],{"className":3439,"style":130},[129],[29,3441,3443],{"className":3442},[134,135,136,137],[29,3444,3446],{"className":3445},[99,137],[29,3447,144],{"className":3448},[99,137],[2009,3450,3451],{},[29,3452,3454,3493],{"className":3453},[43],[29,3455,3457],{"className":3456},[47],[49,3458,3459],{"xmlns":51},[53,3460,3461,3490],{},[56,3462,3463,3473,3475,3477,3479],{},[63,3464,3465,3467],{},[66,3466,2266],{},[56,3468,3469,3471],{},[59,3470,74],{"mathvariant":73},[59,3472,74],{"mathvariant":73},[59,3474,787],{},[66,3476,2938],{},[59,3478,770],{},[63,3480,3481,3484],{},[66,3482,3483],{},"28.98",[56,3485,3486,3488],{},[59,3487,74],{"mathvariant":73},[59,3489,74],{"mathvariant":73},[78,3491,3492],{"encoding":80},"42'' \\times 0.69 = 28.98''",[29,3494,3496,3546,3564],{"className":3495,"ariaHidden":86},[85],[29,3497,3499,3502,3505,3537,3540,3543],{"className":3498},[90],[29,3500],{"className":3501,"style":95},[94],[29,3503,986],{"className":3504},[99],[29,3506,3508,3511],{"className":3507},[99],[29,3509,282],{"className":3510},[99],[29,3512,3514],{"className":3513},[109],[29,3515,3517],{"className":3516},[113],[29,3518,3520],{"className":3519},[117],[29,3521,3523],{"className":3522,"style":122},[121],[29,3524,3525,3528],{"style":125},[29,3526],{"className":3527,"style":130},[129],[29,3529,3531],{"className":3530},[134,135,136,137],[29,3532,3534],{"className":3533},[99,137],[29,3535,144],{"className":3536},[99,137],[29,3538],{"className":3539,"style":2185},[447],[29,3541,787],{"className":3542},[919],[29,3544],{"className":3545,"style":2185},[447],[29,3547,3549,3552,3555,3558,3561],{"className":3548},[90],[29,3550],{"className":3551,"style":2897},[94],[29,3553,2938],{"className":3554},[99],[29,3556],{"className":3557,"style":821},[447],[29,3559,770],{"className":3560},[825],[29,3562],{"className":3563,"style":821},[447],[29,3565,3567,3570,3574],{"className":3566},[90],[29,3568],{"className":3569,"style":122},[94],[29,3571,3573],{"className":3572},[99],"28.9",[29,3575,3577,3580],{"className":3576},[99],[29,3578,2079],{"className":3579},[99],[29,3581,3583],{"className":3582},[109],[29,3584,3586],{"className":3585},[113],[29,3587,3589],{"className":3588},[117],[29,3590,3592],{"className":3591,"style":122},[121],[29,3593,3594,3597],{"style":125},[29,3595],{"className":3596,"style":130},[129],[29,3598,3600],{"className":3599},[134,135,136,137],[29,3601,3603],{"className":3602},[99,137],[29,3604,144],{"className":3605},[99,137],[2009,3607,3608],{},[29,3609,3611,3650],{"className":3610},[43],[29,3612,3614],{"className":3613},[47],[49,3615,3616],{"xmlns":51},[53,3617,3618,3647],{},[56,3619,3620,3630,3632,3634,3636],{},[63,3621,3622,3624],{},[66,3623,2482],{},[56,3625,3626,3628],{},[59,3627,74],{"mathvariant":73},[59,3629,74],{"mathvariant":73},[59,3631,787],{},[66,3633,3095],{},[59,3635,770],{},[63,3637,3638,3641],{},[66,3639,3640],{},"33.00",[56,3642,3643,3645],{},[59,3644,74],{"mathvariant":73},[59,3646,74],{"mathvariant":73},[78,3648,3649],{"encoding":80},"33'' \\times 1.00 = 33.00''",[29,3651,3653,3703,3721],{"className":3652,"ariaHidden":86},[85],[29,3654,3656,3659,3662,3694,3697,3700],{"className":3655},[90],[29,3657],{"className":3658,"style":95},[94],[29,3660,1150],{"className":3661},[99],[29,3663,3665,3668],{"className":3664},[99],[29,3666,1150],{"className":3667},[99],[29,3669,3671],{"className":3670},[109],[29,3672,3674],{"className":3673},[113],[29,3675,3677],{"className":3676},[117],[29,3678,3680],{"className":3679,"style":122},[121],[29,3681,3682,3685],{"style":125},[29,3683],{"className":3684,"style":130},[129],[29,3686,3688],{"className":3687},[134,135,136,137],[29,3689,3691],{"className":3690},[99,137],[29,3692,144],{"className":3693},[99,137],[29,3695],{"className":3696,"style":2185},[447],[29,3698,787],{"className":3699},[919],[29,3701],{"className":3702,"style":2185},[447],[29,3704,3706,3709,3712,3715,3718],{"className":3705},[90],[29,3707],{"className":3708,"style":2897},[94],[29,3710,3095],{"className":3711},[99],[29,3713],{"className":3714,"style":821},[447],[29,3716,770],{"className":3717},[825],[29,3719],{"className":3720,"style":821},[447],[29,3722,3724,3727,3731],{"className":3723},[90],[29,3725],{"className":3726,"style":122},[94],[29,3728,3730],{"className":3729},[99],"33.0",[29,3732,3734,3737],{"className":3733},[99],[29,3735,2640],{"className":3736},[99],[29,3738,3740],{"className":3739},[109],[29,3741,3743],{"className":3742},[113],[29,3744,3746],{"className":3745},[117],[29,3747,3749],{"className":3748,"style":122},[121],[29,3750,3751,3754],{"style":125},[29,3752],{"className":3753,"style":130},[129],[29,3755,3757],{"className":3756},[134,135,136,137],[29,3758,3760],{"className":3759},[99,137],[29,3761,144],{"className":3762},[99,137],[2009,3764,3765,3768],{},[20,3766,3767],{},"Sum and Divide:",[2006,3769,3770,3885],{},[2009,3771,3772],{},[29,3773,3775,3810],{"className":3774},[43],[29,3776,3778],{"className":3777},[47],[49,3779,3780],{"xmlns":51},[53,3781,3782,3807],{},[56,3783,3784,3786,3788,3790,3792,3794,3796],{},[59,3785,778],{},[59,3787,782],{"stretchy":781},[242,3789,355],{},[242,3791,244],{},[59,3793,792],{"stretchy":781},[59,3795,770],{},[63,3797,3798,3801],{},[66,3799,3800],{},"135.42",[56,3802,3803,3805],{},[59,3804,74],{"mathvariant":73},[59,3806,74],{"mathvariant":73},[78,3808,3809],{"encoding":80},"\\sum (LW) = 135.42''",[29,3811,3813,3843],{"className":3812,"ariaHidden":86},[85],[29,3814,3816,3819,3822,3825,3828,3831,3834,3837,3840],{"className":3815},[90],[29,3817],{"className":3818,"style":3257},[94],[29,3820,778],{"className":3821,"style":876},[873,874,875],[29,3823,782],{"className":3824},[842],[29,3826,355],{"className":3827},[99,260],[29,3829,244],{"className":3830,"style":261},[99,260],[29,3832,792],{"className":3833},[926],[29,3835],{"className":3836,"style":821},[447],[29,3838,770],{"className":3839},[825],[29,3841],{"className":3842,"style":821},[447],[29,3844,3846,3849,3853],{"className":3845},[90],[29,3847],{"className":3848,"style":122},[94],[29,3850,3852],{"className":3851},[99],"135.4",[29,3854,3856,3859],{"className":3855},[99],[29,3857,282],{"className":3858},[99],[29,3860,3862],{"className":3861},[109],[29,3863,3865],{"className":3864},[113],[29,3866,3868],{"className":3867},[117],[29,3869,3871],{"className":3870,"style":122},[121],[29,3872,3873,3876],{"style":125},[29,3874],{"className":3875,"style":130},[129],[29,3877,3879],{"className":3878},[134,135,136,137],[29,3880,3882],{"className":3881},[99,137],[29,3883,144],{"className":3884},[99,137],[2009,3886,3887],{},[29,3888,3890,3926],{"className":3889},[43],[29,3891,3893],{"className":3892},[47],[49,3894,3895],{"xmlns":51},[53,3896,3897,3923],{},[56,3898,3899,3902,3904,3906,3908,3910,3912],{},[765,3900,3901],{},"MPV Seconds",[59,3903,770],{},[66,3905,3800],{},[242,3907,1917],{"mathvariant":73},[66,3909,3244],{},[59,3911,770],{},[63,3913,3914,3917],{},[66,3915,3916],{},"36.3",[56,3918,3919,3921],{},[59,3920,74],{"mathvariant":73},[59,3922,74],{"mathvariant":73},[78,3924,3925],{"encoding":80},"\\text{MPV Seconds} = 135.42 \u002F 3.73 = 36.3''",[29,3927,3929,3951,3970],{"className":3928,"ariaHidden":86},[85],[29,3930,3932,3936,3942,3945,3948],{"className":3931},[90],[29,3933],{"className":3934,"style":3935},[94],"height:0.6944em;",[29,3937,3939],{"className":3938},[99,814],[29,3940,3901],{"className":3941},[99],[29,3943],{"className":3944,"style":821},[447],[29,3946,770],{"className":3947},[825],[29,3949],{"className":3950,"style":821},[447],[29,3952,3954,3957,3961,3964,3967],{"className":3953},[90],[29,3955],{"className":3956,"style":3257},[94],[29,3958,3960],{"className":3959},[99],"135.42\u002F3.73",[29,3962],{"className":3963,"style":821},[447],[29,3965,770],{"className":3966},[825],[29,3968],{"className":3969,"style":821},[447],[29,3971,3973,3976,3980],{"className":3972},[90],[29,3974],{"className":3975,"style":122},[94],[29,3977,3979],{"className":3978},[99],"36.",[29,3981,3983,3986],{"className":3982},[99],[29,3984,1150],{"className":3985},[99],[29,3987,3989],{"className":3988},[109],[29,3990,3992],{"className":3991},[113],[29,3993,3995],{"className":3994},[117],[29,3996,3998],{"className":3997,"style":122},[121],[29,3999,4000,4003],{"style":125},[29,4001],{"className":4002,"style":130},[129],[29,4004,4006],{"className":4005},[134,135,136,137],[29,4007,4009],{"className":4008},[99,137],[29,4010,144],{"className":4011},[99,137],[2009,4013,4014,4017,4018,27,4166,32],{},[20,4015,4016],{},"Result:"," MPV = ",[29,4019,4021,4055],{"className":4020},[43],[29,4022,4024],{"className":4023},[47],[49,4025,4026],{"xmlns":51},[53,4027,4028,4052],{},[56,4029,4030,4036,4042],{},[63,4031,4032,4034],{},[66,4033,2028],{},[59,4035,2031],{},[63,4037,4038,4040],{},[66,4039,2036],{},[59,4041,74],{"mathvariant":73,"lspace":2039,"rspace":2039},[63,4043,4044,4046],{},[66,4045,3916],{},[56,4047,4048,4050],{},[59,4049,74],{"mathvariant":73},[59,4051,74],{"mathvariant":73},[78,4053,4054],{"encoding":80},"89^\\circ 54' 36.3''",[29,4056,4058],{"className":4057,"ariaHidden":86},[85],[29,4059,4061,4064,4067,4096,4099,4131,4134],{"className":4060},[90],[29,4062],{"className":4063,"style":122},[94],[29,4065,2079],{"className":4066},[99],[29,4068,4070,4073],{"className":4069},[99],[29,4071,2086],{"className":4072},[99],[29,4074,4076],{"className":4075},[109],[29,4077,4079],{"className":4078},[113],[29,4080,4082],{"className":4081},[117],[29,4083,4085],{"className":4084,"style":2099},[121],[29,4086,4087,4090],{"style":125},[29,4088],{"className":4089,"style":130},[129],[29,4091,4093],{"className":4092},[134,135,136,137],[29,4094,2031],{"className":4095},[919,137],[29,4097,68],{"className":4098},[99],[29,4100,4102,4105],{"className":4101},[99],[29,4103,986],{"className":4104},[99],[29,4106,4108],{"className":4107},[109],[29,4109,4111],{"className":4110},[113],[29,4112,4114],{"className":4113},[117],[29,4115,4117],{"className":4116,"style":122},[121],[29,4118,4119,4122],{"style":125},[29,4120],{"className":4121,"style":130},[129],[29,4123,4125],{"className":4124},[134,135,136,137],[29,4126,4128],{"className":4127},[99,137],[29,4129,74],{"className":4130},[99,137],[29,4132,3979],{"className":4133},[99],[29,4135,4137,4140],{"className":4136},[99],[29,4138,1150],{"className":4139},[99],[29,4141,4143],{"className":4142},[109],[29,4144,4146],{"className":4145},[113],[29,4147,4149],{"className":4148},[117],[29,4150,4152],{"className":4151,"style":122},[121],[29,4153,4154,4157],{"style":125},[29,4155],{"className":4156,"style":130},[129],[29,4158,4160],{"className":4159},[134,135,136,137],[29,4161,4163],{"className":4162},[99,137],[29,4164,144],{"className":4165},[99,137],[29,4167,282],{},[11,4169,4171],{"id":4170},"practical-tips","Practical Tips",[2006,4173,4174,4182],{},[2009,4175,4176,4179,4180,32],{},[20,4177,4178],{},"The \"Fractional\" Shortcut:"," When calculating the MPV of an angle or large number, only apply the weighting to the differing seconds or decimals to keep the numbers manageable ",[29,4181,282],{},[2009,4183,4184,4187,4188,4222,4223,32],{},[20,4185,4186],{},"Rejection of Outliers:"," Before weighting, any observation whose residual is greater than a specified rejection criterion (e.g., ",[29,4189,4191,4207],{"className":4190},[43],[29,4192,4194],{"className":4193},[47],[49,4195,4196],{"xmlns":51},[53,4197,4198,4204],{},[56,4199,4200,4202],{},[66,4201,1150],{},[242,4203,1171],{},[78,4205,4206],{"encoding":80},"3S",[29,4208,4210],{"className":4209,"ariaHidden":86},[85],[29,4211,4213,4216,4219],{"className":4212},[90],[29,4214],{"className":4215,"style":256},[94],[29,4217,1150],{"className":4218},[99],[29,4220,1171],{"className":4221,"style":1205},[99,260]," or Chauvenet’s criterion) should be discarded ",[29,4224,4225],{},"3, 5",[11,4227,4229],{"id":4228},"common-mistakes","Common Mistakes",[2006,4231,4232,4346],{},[2009,4233,4234,4273,4274,4277,4278,4343,4344,32],{},[20,4235,4236,4237,2740],{},"Weighting by Error (",[29,4238,4240,4258],{"className":4239},[43],[29,4241,4243],{"className":4242},[47],[49,4244,4245],{"xmlns":51},[53,4246,4247,4255],{},[56,4248,4249,4251,4253],{},[66,4250,164],{},[242,4252,1917],{"mathvariant":73},[242,4254,1171],{},[78,4256,4257],{"encoding":80},"1\u002FS",[29,4259,4261],{"className":4260,"ariaHidden":86},[85],[29,4262,4264,4267,4270],{"className":4263},[90],[29,4265],{"className":4266,"style":3257},[94],[29,4268,1940],{"className":4269},[99],[29,4271,1171],{"className":4272,"style":1205},[99,260]," A common beginner error is weighting by the error itself rather than the ",[20,4275,4276],{},"square"," of the error (",[29,4279,4281,4302],{"className":4280},[43],[29,4282,4284],{"className":4283},[47],[49,4285,4286],{"xmlns":51},[53,4287,4288,4300],{},[56,4289,4290,4292,4294],{},[66,4291,164],{},[242,4293,1917],{"mathvariant":73},[63,4295,4296,4298],{},[242,4297,1171],{},[66,4299,282],{},[78,4301,1926],{"encoding":80},[29,4303,4305],{"className":4304,"ariaHidden":86},[85],[29,4306,4308,4311,4314],{"className":4307},[90],[29,4309],{"className":4310,"style":1936},[94],[29,4312,1940],{"className":4313},[99],[29,4315,4317,4320],{"className":4316},[99],[29,4318,1171],{"className":4319,"style":1205},[99,260],[29,4321,4323],{"className":4322},[109],[29,4324,4326],{"className":4325},[113],[29,4327,4329],{"className":4328},[117],[29,4330,4332],{"className":4331,"style":295},[121],[29,4333,4334,4337],{"style":125},[29,4335],{"className":4336,"style":130},[129],[29,4338,4340],{"className":4339},[134,135,136,137],[29,4341,282],{"className":4342},[99,137],"). Weighting must be based on variance ",[29,4345,329],{},[2009,4347,4348,4351],{},[20,4349,4350],{},"Assuming Equal Weight:"," Never assume observations are equal if the conditions (equipment, weather, distances) changed significantly between sets.",[11,4353,4355],{"id":4354},"faq","FAQ",[4357,4358,4359,4368,4376],"accordion",{},[4360,4361,4363],"accordion-item",{"label":4362},"What is a 'Unit Weight'?",[16,4364,4365,4366,32],{},"It is an observation assigned a weight of 1.0. All other weights in the system are then expressed as multiples of this unit weight ",[29,4367,1973],{},[4360,4369,4371],{"label":4370},"Can weights be assigned subjectively?",[16,4372,4373,4374,32],{},"While statistical weighting is preferred, engineers sometimes assign weights based on 'field judgement' (e.g., 1 for poor conditions, 3 for excellent), though this lacks mathematical rigor ",[29,4375,164],{},[4360,4377,4379],{"label":4378},"Does the MPV equal the True Value?",[16,4380,4381,4382,4412,4413,4441,4442,32],{},"Mathematically, the MPV is only equal to the true value (",[29,4383,4385,4399],{"className":4384},[43],[29,4386,4388],{"className":4387},[47],[49,4389,4390],{"xmlns":51},[53,4391,4392,4397],{},[56,4393,4394],{},[242,4395,4396],{},"X",[78,4398,4396],{"encoding":80},[29,4400,4402],{"className":4401,"ariaHidden":86},[85],[29,4403,4405,4408],{"className":4404},[90],[29,4406],{"className":4407,"style":256},[94],[29,4409,4396],{"className":4410,"style":4411},[99,260],"margin-right:0.0785em;",") when the number of observations (",[29,4414,4416,4429],{"className":4415},[43],[29,4417,4419],{"className":4418},[47],[49,4420,4421],{"xmlns":51},[53,4422,4423,4427],{},[56,4424,4425],{},[242,4426,380],{},[78,4428,380],{"encoding":80},[29,4430,4432],{"className":4431,"ariaHidden":86},[85],[29,4433,4435,4438],{"className":4434},[90],[29,4436],{"className":4437,"style":979},[94],[29,4439,380],{"className":4440},[99,260],") reaches infinity ",[29,4443,164],{},[11,4445,4447],{"id":4446},"conclusion","Conclusion",[16,4449,4450],{},"Weighting is the tool that transforms a collection of measurements into a unified, defensible result. By linking weight to the inverse of variance, the surveyor ensures that the most precise field work dictates the quality of the final engineering design.",[11,4452,4454],{"id":4453},"references","References",[16,4456,4457,4458,4461],{},"Schofield, W. (2001). ",[1732,4459,4460],{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann.",{"title":4463,"searchDepth":4464,"depth":4464,"links":4465},"",2,[4466,4467,4468,4472,4477,4484,4485,4486,4487,4488,4489],{"id":13,"depth":4464,"text":14},{"id":35,"depth":4464,"text":36},{"id":224,"depth":4464,"text":225,"children":4469},[4470],{"id":333,"depth":4471,"text":334},3,{"id":944,"depth":4464,"text":945,"children":4473},[4474,4475],{"id":948,"depth":4471,"text":949},{"id":1153,"depth":4471,"text":4476},"2. Standard Error of the Weighted Mean (SxˉwS_{\\bar{x}_w}Sxˉw​​)",{"id":1766,"depth":4464,"text":1767,"children":4478},[4479,4480,4481,4482,4483],{"id":1773,"depth":4471,"text":1774},{"id":1780,"depth":4471,"text":1781},{"id":1896,"depth":4471,"text":1897},{"id":1976,"depth":4471,"text":1977},{"id":1985,"depth":4471,"text":1986},{"id":1994,"depth":4464,"text":1995},{"id":4170,"depth":4464,"text":4171},{"id":4228,"depth":4464,"text":4229},{"id":4354,"depth":4464,"text":4355},{"id":4446,"depth":4464,"text":4447},{"id":4453,"depth":4464,"text":4454},"Surveying Math","A technical guide to applying weights to surveying measurements to determine the Most Probable Value (MPV) and assess indices of precision.","md","\u002Fimages\u002Fweighted-observations.jpg",{"authorCredentials":4495,"authorBio":4496,"commentCount":4497,"tags":4498,"relatedTools":4503,"relatedArticles":4506,"sources":4509},"LS, P.Eng","Expert in geomatics and statistical analysis for engineering control networks.",0,[4499,4500,748,4501,4502],"Statistics","Error Analysis","Precision","Most Probable Value",[4504,4505],"Scientific Calculator","Least Squares Software",[4507,4508],"combination-of-errors-propagation","bowditch-adjustment-math",[4510],{"title":4511,"type":4512,"note":4513},"Engineering Surveying (Fifth Edition)","Book","Primary source for indices of precision and weighting theory.",true,"\u002Fblog\u002Fweighted-observations-mpv","2024-04-04",18,{"title":5,"description":4491},{"loc":4515},"blog\u002F16.weighted-observations-mpv","J_SB6j5wnxGpag1W8bvlVTAMdnNdAxsAJlHgwZB4iwk",[4523,4528],{"title":4524,"path":4525,"stem":4526,"description":4527,"children":-1},"Inverted Sights: Levelling Techniques for Overheads and Soffits","\u002Fblog\u002Finverted-sights-underground-and-overhead-levelling","blog\u002F15.inverted-sights-underground-and-overhead-levelling","A technical guide to performing precise levelling on the underside of structures, tunnels, and bridges using inverted staff techniques.",{"title":4529,"path":4530,"stem":4531,"description":4532,"children":-1},"Prismoidal vs. End-Area: Selecting the Right Volume Formula","\u002Fblog\u002Fprismoidal-vs-end-area-volumes","blog\u002F17.prismoidal-vs-end-area-volumes","A technical comparison of the Trapezoidal (End-Area) and Prismoidal rules for calculating earthwork volumes in civil engineering.",1783851950355]