UNKNOWN MEAN OF SAMPLED UNIVERSE 639 



certainty in the investigator's mind that the true value of ni lies 

 closer and closer to the assumed most probable figure, M. 



With these two assumptions incorporated in equation (10) we may 

 now write 



P(m)dm =f{t)dt = A"'(l + t^y^^i^^'^dt, (10') 



in which 





1/ — c 



The formula (10') is a "Student" ^ frequency form with the argu- 

 ments n and replaced hy n -\- 2 -\- c -\- N and 



respectively. 



Fig. 2 shows curves plotted for ranges of t such that 



A'" 1 (1 + t^y^^'^^'^dt = .50, .80, .90, and .9973,» 



and the errors in the mean corresponding to any of these probabilities, 

 after determining t, may be found by evaluating z — m in equation 

 (14). 



IV. Solution of a Typical Example 



Five samples of retardation coils rated at 47 ohms are taken from a 

 large lot, and careful measurements show them to have resistances of 

 46.30, 44.40, 47.72, 50.50, and 45.58 ohms respectively. We are 

 asked to determine the probable and 99.73 per cent errors of the 

 average of these resistances, assuming that the samples have been 

 drawn from a normal universe. 



The average of these five values is x = 46.90 ohms and their standard 

 deviation about this average is found to be ^ = 2.097. 



From the preceding discussion it is evident that as many answers 

 to this problem may be obtained as there are assumptions made 

 regarding, in general, the a priori distributions of the mean and 



8 Student: "The Probable Error of a Mean," Biometrika, Vol. VI, No. 1, March 

 1908. 



3 Student: "New Tables for Testing the Significance of Observations," Metron, 

 Vol. V, No. 3, I-XII-1925. Tables I and II, pages 114-118, for values of n' = 2 

 to 21. 



