CORRECTION OF DATA FOR ERRORS 315 



Curves showing the most frequently used errors of averages meas- 

 ured in terms of z (i.e. in terms of the ratio of the error to the observed 

 standard de\'iation) are given in Fig. 6. The error curves for n less 

 than 30 have been obtained with the aid of "Student's" original 

 tables, those for n between 30 and 100 have been obtained from the 

 normal law integral tables using the standard deviation of s; i.e. 



/ as given by "Student." For n greater than 100, customary 



■yn — S 



error theory has been used.^ 



Typical Practical Applications 



But few, if any, recent developments of statistical theory are of 

 more general application in most fields of scientific research and en- 

 gineering than the one herein described.^" This follows because the 

 theory herein discussed must be used in calculating the required prob- 

 able error (or other measure of dispersion) of the averages obtained 

 from small numbers of observations. The number of applications 

 of this character is legion. 



Problem Type 1, Determination of Error of Average 



Example 1 : 



Five samples of granular carbon taken from a crucible show 

 resistances of 47.5, 49.4, 43.2, 48.0 and 46.2 ohms respectively. What 

 are the probable and 99.73% errors of the average of these resistances? 



Solution: 



The observed values of average resistance X, and standard devia- 



tion 5 = \j— are 46.9 ohms and 2.097 ohms respectively. Hence 



from Fig. 6 we see that the probable and 99.73% errors are respect- 

 ively .3725 = .780 ohms and 3. 335 = .699 ohms respectively whereas 

 from customary theory they would be .3025 = .633 ohms and 1.345 = 270 



' For the curves in this figure as in the preceding one, I have assumed the customary 

 theory for the case where the true vahie of X is known so that the root mean square 



. ^ 



error of the average X of sample of size n is the ratio -7=. Of course, as we know 



from customary error theory, if we assume no knowledge of the true value of X, we 



should use , 



\w— 1 



'"Since this paper was written, a very interesting article, " Statistics in Adminis- 

 tration," has appeared in Nature (V. 117, pp. 37-38, Jan. 9, 1926), calling attention 

 to the importance of the theory of small samjjles. 



