38 TABLE 16.— METHODS OF AVERAGING DATA (continued) 



of which, as representative bracketed L 1 coefficients, we have 

 [diGu] = aid -f- a? a i 4 a3<i3 4- • .Oiifln 



[a<bi] = aibi -\- a-ibi 4 a 3 b 3 4 . . .a n b n (3) 



[a 4 X<] = oj, + a 2 X 2 4 a 3 X 3 +. . .a n X n 



[&{Oi] = kxdx + kidi -f" k 3 a 3 -\- . . .kndn 



Solutions of equation (2) yield the least-squares adjusted values of Qx, Qi. . .Qy. 



For unequally weighted values of X, that is Wi, iv 2 ,...w n for XiX 2 . . .X n , the normal 

 equdtions become 



[wiOiaJQ, -f [widib^Qi + [zvtdiC^Qz +. . . [wid l k l ]Q k — [uuatXt] =0 



[w i b t a i ]Q 1 + [iv<b i b i ~\Q i + [w t bid1Q 3 + • ■• [Wib t k't]Q h — [wibiXil = (4) 



IwikiOtlQi 4 [wik t bt1Qi+ IwtktCtlQ, + . : . IvnkikAQu— [a; t AJi] =0 



of which 



[WiOiOi] = IVxdxdx -\- IViChd? 4" lV 3 d 3 d 3 4 • • .Wndndn 



[iVidibi] = lihdibi 4 IVidibi 4 WaOsta 4 • • -ZVndnbn (5) 



[Wt&iOU] = Wxkxdx 4 Wlkia2 4 Wxkzd* 4 • • -Wnkndn 



The weights Wi, Wi. . .w n associated with the Xi, X 2 . . .X„ and with the successive ob- 

 servation equations are taken as inversely proportional to the squares of the probable 

 errors (or of the standard deviations) of the corresponding X's. It is customary to take 

 simple rounded numbers for the proportional values. A precise set of 28, 50, 41, and 78 

 may be rounded to 3, 5, 4, and 8. 



As a simple application, consider the elevations of stations B, C, and D above A. Let 

 those elevations in order be Qx, Q 2 , and Q 3 . Let the quantities measured and the observed 

 elevations be such as to yield the following observation equations : 



Gi —10 ft = Ai 



Q 2 — 18 ft = A, 



Qx— 4ft = A, (6) 



-Qx + Q* - 9 ft = A, 



Q 2 -03-12ft = A 5 



Qx — Q 3 — 5ft = A. 



The coefficients a it 6i, and ex are seen to be 1,0, and 0. The values of the other coefficients 

 are obvious. Substitution in equation (2) yields for the normal equations 



30 2 - Q 2 — Qx— 6ft = 



-Gi4 3Q 2 - 03 - 39 ft = (7) 



- Qx - Q* 4 3Q 3 4 13 ft = 



Solutions of equation (7) yield % ft, Yl\ ft, and 4f ft for the elevations of B, C, and D 

 above A. 



Part 2. — Least-squares equations of the type y = a 4 bx, to represent a series of 



observed (x,y) values 



For equally weighted pairs of (x,y) of which the errors of measurement are associated 

 with the determinations of the y's 



_ 2x 2 2y — l.x'Lxy _ x 2 y — x x y 

 °~ nZ.r 2 — (2x) 2 ~ ~x~ 2 — x 2 



n'Zxy — Z-rZy _ xy — xy 

 m2x 2 — (2x) 2 



of which 



nx = 2x, n.r 2 =z 2x 2 , xy = n'Lxy 

 irx i = CZx) 2 , etc. 



(continued) 

 SMITHSONIAN PHYSICAL TABLES 



, nzxy — zxzy _ xy — xy ,n\ 



