37 

 TABLES 16-25.— TREATMENT OF EXPERIMENTAL DATA* 



TABLE 16.— METHODS OF AVERAGING DATA 



When a number of measurements are made of any quantity variations will be found. 

 The question is: What is the best representative value for the quantity thus measured: 

 and how shall the precision of the measurements be stated? The arithmetic mean of all 

 the readings is generally taken as the best value. To tell something about the precision 

 of the final result any one of five measures of variation which arc discussed in books dealing 

 with this subject may be given. The.-e measures of deviation are : 



p = probable error 



a = the average deviation (from the arithmetic mean) 



a = the standard deviation 



l//i =. the reciprocal of the modulus of precision 



k/w= the reciprocal of the "precision constant" 



Of these precision indexes the standard deviation, a. is most easily computed. For the 

 set of observed values .r,, .r 2 . ..r„ of equal weight, the a for a single observation is given by 



and for the mean bv 



V^ef-V 



i ( .r — X 



a _ I S(.r — -f)- _ / 2L 

 V^ \ «(« — 1) = \ 



The ratios of these precision indexes to one another for a normal (or Gaussian) 



distribution are: _ 



/> : a : a : 1 h : k «• : : 0.476936 : 1 \ w : \ ( 1 2 ) : 1.000 : \ it 

 or roughly as p : a : a : 1/h : k .«■ : : 7 : 8 : 10 : 14 : 25 



Most experimental data can be represented by an equation of some form. One of the 

 recommended methods for determining the coefficients of such equations is the use of a 

 least-squares solution. This means that an attempt is made to find values for the coefficients 

 such that the sum of the squares of the deviations of the experimental points from the 

 resulting curve has the least possible value. Certain tables are of help in making such 

 solutions (Tables 16-26), and reference should be made to books or papers on this subject 

 for their use. 



An example of one method of finding the coefficients of such selected equations (based 

 on "Treatment of Experimental Data," bv Worthing and Geffner. published bv Wiley. 

 1943) follows. 



Part 1. — Least squares adjustment of measurements of linearly related quantities 



Let Qu Qi...Qk be the k adjusted, but initially unknown, values of the linearly related 

 quantities. Let A\, A' 2 ....V„ be »(> '0 measured values of Q's or of linear combinations 

 of two or more, Q's. 



Let Ai, A 2 . . .An be the adjustments or corrections that must be applied to the measured 

 A"s to yield consistent least-squares values for the Q's. See below for a simple illustration. 



As observation equations we have 



aiQi + biQt+...ktQ* — X t = \, 



a-Qi + IhQi +... k t Q k - X, = A. ( 1) 



«„(?, + /-„(?. + . . .knQ k - X„ = X, 



of which fli, /', . . .k, are constants, whose values are frequently + 1, — 1, or 0. 



From the observation equations k normal equations are formed. For equally weighted 



observed values of A', they are 



[a ( a l ]Q i + [a,b,]Q*+ r«iifi.]()» + ...l«iifri1l)*— latX',1 =0 

 [Mi](?i+[M»i](?*+ !' 1 iri](?a + ...l' , iitiic ) *— r/'.A'i] = (2) 



[/;,<7,]0> + \k,h,~\0: + \k,c,1Q*+...\kik,}Q k — lk,X,] =0 



* Prepared by the late A. G. Worthing, of the I'niversity of Pittsburgh. 



(continued} 



SMITHSONIAN PHYSICAL TABLES 



