CITY SURVEYING 135 



PRECISION 



If a quantity, as a distance or an angle, is measured very 

 accurately several times by the same method, it is usually 

 found that the results vary slightly from one another. The 

 true measure of the quantity is taken to be the mean of the 

 different results obtained that is, the sum of these results 

 divided by their number. This mean is called the mean value, 

 or most probable value. 



By the law of probabilities it may be determined that the 

 error made in using the mean value does not exceed a certain 

 quantity, called its probable error. This quantity may be posi- 

 tive or negative, that is, the exact value may be greater or 

 smaller than the mean value. It serves as a measure of the 

 accuracy obtained by the use of the mean value. 



Let the probable error be denoted by p] the sum of the 

 squares of the differences between the actual measurements 

 and the mean value, the latter being called residuals, by 2r 2 ; 

 and the number of measurements made by m. Then, 



.6745 



m-l) 



EXAMPLE. A distance was measured four times, the results 

 of the measurements being, respectively, 501.07, 501.06, 

 501.05, and 501.08 ft. Determine: (a) the mean value M 

 of the distance; (b) the probable error p. 



SOLUTION. (a) Since 501 is common to all the measurements, 



.07 +.06 +.05 + .08 

 M = 501+ 501.065 



(&) To apply the formula for p, m = 4, m 1 = 3, and 

 ^ = 501.065-501.07= -.005 

 vz = 501. 065 -501. 06= .005 

 i* = 501 .065 -501. 05= .015 

 V4 = 501. 065 -501. 08- -.015 

 2z>2= (- .005)2+ (.005)2+ (.015)2+ (- .015)2 = .0005 



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 \4X3 



Therefore, p = .6745 \ C2=I = =fc .0044. 



4X3 



Weighted Measurements. If the measurements are not 

 made under the same conditions, so that there are reasons to 



