136 CITY SURVEYING 



believe that some of them are more accurate than others, the 

 results must be weighted. That measurement r r hose accuracy 

 is supposed to be the least usually receives a weight of 1; a 

 measurement whose accuracy appears to be twice as great 

 receives a weight of 2; etc. After the measurements have 

 been weighted, each measurement is multiplied by the number 

 representing its weight, the products are added, and the sum 

 is divided by the sum of the weight numbers. This result 

 is the mean value, or most probable value, of the quantity. 

 Thus, in the preceding example, if the first measurement is 

 of the least weight, while the second is twice as great as the 

 first, and the third and fourth are each two and one-half times 

 as great as the first, the weights of the four measurements are 

 respectively, 1, 2, 2.5, and 2.5, and the mean value M is 

 .07X1 + .06X2 + .05X2.5 + .08X2.5 



1 + 2 + 2.5 + 2.5 



If the weight of any measurement is denoted by h, then the 

 probable error 



in which 2 (hv*) is the sum of the products of the squares of the 

 residuals by their corresponding weights, and "2,h is the sum of 

 all the weights. 



EXAMPLE. Determine the probable error p in the preceding 

 example, the weights of the four measurements being, respec- 

 tively, 1, 2, 2.5, and 2.5. 



SOLUTION. The mean value M has been found to be 501.064. 

 The values of the residuals are as follows: 



1/1-501.064-501.07= -.006 

 V2 = 501. 064 -501. 06= +.004 

 1/3 = 501.064-501.05= +.014 

 V4 = 601. 064 -501. 08= -.016 

 Then, 



2(/;2) = lXt-.006)*+2X(+.004)2+2.5X(+.014) 

 +2.5X(-.016)=.001198, and (2/t-l)2A = 7X8 

 Substituting in the formula, 



8X7 



