Table 30 



LEAST SQUARES (FORMULAE) 



59 



Observation equations : 



ajZi + bxZo + . . . Iiz q = Mi, weight pj 

 a2Zi + b2Z2 4- . . . l 2 z q = M 2 . weight p 2 



a n Zi + b„z 2 + . . . ] n z q = M n , weight p n . 



Auxiliary equations : 



[paa] =piaf +p 2 a| + . . . p n a^. 

 [pab] = piaibi + p 2 a 2 b 2 + . . . p n a n bn. 



[paM] = piaiM x + p 2 a 2 M 2 + . . . p n a n M n . 



Normal equations : 



[paa]zj + [pab]z 2 + . . . [pal]z q = [paM] 

 [pabjz x + [pbb]z 2 + . . . [pbl]z q = [pbM] 



[plajzi + [plb]z 2 +'...' [pll]'z q = [plM]. 



Solution of normal equations in the form, 



zi = AjI paM] + Bj[pbM] + . • . L lL plM] 

 z 2 = A 2 [paM] + B 2 LpbM] + • . . L 2 [plM] 



Zq = A n [paM] + B„[pbMJ + .'. . L„[plM], 



gives : 



wherein 



weight of zj = pzi = (Ax) -1 ; probable error of z\ = — — 



VPzi 



r 

 weight of z 2 = pz 2 = (B2)- 1 ; probable error of z 2 = — 



VPza 



r 

 weight of z q = p z = (Ln)- 1 ; probable error of z q = — — 



VPz„ 



r = probable error of observation of weight unity 

 = 0.6745-1/ — — ■ (q unknowns.) 



-q 



Arithmetical mean, n observations: 



/S v* 0.84532 V 



= 0.6745 ■* I = — ^ (approx.) =probable error of ob- 



» n — 1 \/n(n — 1)' servation of weight unity. 



/ 2 v* _ 0.8453 2 v x , , , 



o = 0.6745-*/ — — (approx ) = probable error 



Vn(n-i) n \/n-i of mean. 



Weighted mean, n observations: 



, /Zpv* r / Spy' 



r = 0.6745 V 7-T, = ro = ^^ =a6 745 V(^T^p 



Probable error (R) of a function (Z) of several observed quantities zi, z 2 , . . . whose 

 probable errors are respectively, r lf r<>, . . . . 



Z = f (z t , Zo, . . .) 



Examples : z = z x ± z 2 + • • • R 2 = A 4- 4 + . . . 



Z = Azj ± BZ3 ± . . . R2 = A? r, 4- B*rJ+ . . . 



Z = zj z 2 . R 2 = z 2 rj 4- z 3 2 r 2 . 

 See Birge, Calculation of errors by the method of least squares, Phys. Rev., 40, 207, 

 1932. 



Smithsonian Tables. 



