318 



BELL SYSTEM TECHNICAL JOURNAL 



Solution : 



Customary practice would answer: the average of the five values 

 plus or minus 3 times their standard deviation. The better answer is: 



3 



the average plus or minus 



.774G 



times the standard de\iation. 



This follows from Professor Pearson's work previously quoted. He 

 has shown that the most probable observed standard deviation 5 of 



C£ 



: ±|± ?::::;:::::: :::::::::::::r ::::::::::::::::::::::::::: 



OF S 



AMPLE — T1 



Fig. 7 — Curves giving the most proljable value of the true standard deviation a 



I When the average 5 of standard deviations of many samples is known. ra='s 



II When the standard deviations of one sample is known, ra =5 



a sample of n from a normal distribution with standard deviation a 



_ In — 2 



is s = \l (X. Substituting the value n = 5 in this equation we get 



\ n 



s . 



A curve of the values of — vs. n is presented in Fig. 7 for reference 



(T 



in solving problems of this character. 



Problem Type 4, Determination of Most Probable Root ]\Ii:an 



Square Deviation of the Universe When Several 



Samples of n Pieces Each Have Been Examined 



Example 1: 



One thousand transmitters, known to have different efficiencies, 

 have been tested five times each for efficiency. Find the standard 

 deviation of the machine method of measurement. 



