Sec. 10.13] GEIGER-MULLER COUNTERS 319 



radiation detected is then purely random. In a time interval /, which is 

 small compared to the half-life of the substance, the probability P of observ- 

 ing n particles is given [19] by Poisson's distribution curve for random 

 events. 



X" 

 P( n ) = — e~» 



nl 



where N is the true average number of particles detected per unit time. For 

 large numbers of counts the difference between the true average and the com- 

 puted average (the most probable value) is always small, and no significant 

 error is introduced by using the latter value. 



Random errors accompany all measurements; they are indeterminate and 

 presumably arise from a multitude of unknown factors which influence 

 slightly the measured values. They can, however, be treated by statistical 

 methods which provide a measure of the possible error introduced by random 

 fluctuations. The true value of a quantity, in this instance the counting 

 rate, is seldom known, but in its place the most probable or average value 

 of the set of measurements must be used. Residuals may now be defined 

 as the difference between each measurement %i and the average value n; 

 Xi = n% — n. If the residuals are truly random, there will be about the 

 same number of negative as positive values, and small values will occur with 

 greater frequency than large values. It can be shown that under this con- 

 dition the distribution of the residuals is nearly symmetrical about the average 

 value and follows approximately the Gauss error curve (normal distribution 

 curve) 



e -xV2<r S h 



P (x) - — =__<> 



7T V 7r 



-hr-x 



where a — standard deviation (see below) 



h = 3^c 2 = index of precision 

 If h is large, a greater proportion of the errors are small and hence grouped 

 close to n; if h is small, the spread in data is greater. 



Aside from the intrinsic form of the distribution of data and errors to be 

 expected, the important facts concerning a set of measurements are estimates 

 of reliability in terms of the variability of the data. Three measures of 

 variability are commonly used for this purpose: average deviation from the 

 mean, standard deviation from the mean, and the probable error. Of these, 

 the probable error is most often used for estimating the statistical error in 

 counting measurements. 



10.13. Average Deviation. If n is the average number of counts per unit 

 time for M measurements of the same sample under identical physical 

 conditions, the average deviation of the residuals from the mean value of the 

 set of measurements is (in counts per minute) 



