TABLE 16.— METHODS OF AVERAGING DATA (concluded) 39 



The probable errors of the a and the b of equation (8) are given by 



0.675 



VI T x*y 2 — 2* y xy + x 2 y 2 , 1 

 T=T2 L ?rr>- "- fl J 



fr = 0.675 ^^ [£-£_*] 



(9) 



For unequally weighted measurements of which the errors of measurement are associ- 

 ated with the determinations of the y's, 



_ Huux^Zunyi — Zu'tx^ziuXiyi 

 °~ •LivZzViXi 2 — (I,Zi'iX i y 



, 2w'Liu i x i y i — ZiViXiZiviVi ..„. 



°~ 2to2«W-(2w/i*i) a ( ' 



Where the errois of measurement are associated with the ^-determination only, the cor- 

 responding coefficients of an equation of the type x = a' + b'y can be obtained by merely 

 interchanging x and 31 in equation (8). 



Where the errors of measurement are associated with both the x- and the y- determina- 

 tions, the expressions are complicated. 13 



13 Worthing, A. C, and Geffner, J., Treatment of experimental data, p. 259, John Wiley and Sons, 

 New York, 1943. Used by permission. 



Part 3. — Least-squares equation of the type y = a -\- bx 4. ex 2 4. dx 3 to represent a 

 series of observed (x, y) values 



For the general case involving irregularly spaced x-values, the formulae for a, b, c, etc., 

 are very complex." However, for the case of equally weighted observations with errors 

 of measurement associated entirely with the y-values in which succeeding .r-values are 

 equally spaced, the mechanics of the computations for least-squares constants are very 

 greatly simplified, thanks to tables computed by Baily 16 and by Cox and Matuschak. 1 ' The 

 procedure requires a change of the jr-variable to yield a new X-variable with a zero-value 

 at the midpoint of the series. In case of an even number of terms, the shift is given by 



Xe= X ~ X (11) 



Ax 



of which Ax is the even spacing between successive jr-values ; and, if the number of terms 

 is odd, the shift is given by 



X. = i=f (12) 



The further procedure consists in determining the appropriate summations indicated in 

 Table 17, the appropriate fc-terms giv^n as a function of the number of terms n in Tables 19 

 and 20, combining the appropriate summations and £-terms, to give parameters for the 

 equation y = f(X) , and finally transferring the function to the original coordinate system 

 to yield y = ji(x). 



How to apply the simplified procedure to determine the coefficient of x 2 in the least- 

 squares equation y = a + bx + ex 2 to represent the xy values of the first two columns of 

 the following tabulations is shown in the remainder of the tabulation. 



c' = k&X 2 y — k&y 

 n — 6 



£ 5 = 16,741,071 X lO" 10 

 k<= 19,531,250 X lO" 9 

 k^X 2 y = 6.2005 cm 

 k,2y = 5.6523 cm 

 c' = 0.5482 cm 

 Ax = 3 sec 

 c = 4c'/ ' ( Ax ) 2 = 0.244 cm/sec 2 



"Birge, R. T., and Shea, J. D., Univ. California Publ. Math., vol. 2, p. 67, 192/; Worthing, A. G., 

 nd Geffner, J., Treatment of experimental data, p. 250, John Wiley and Sons, New York, 1943. 

 u Baily, J. L., Ann. Math. Statistics, vol. 2, p. 355, 1931. 

 16 Cox, G. C, and Matuschak, Margaret, Journ. Phys. Chem., vol. 45, p. 362, 1941. 



SMITHSONIAN PHYSICAL TABLES 



