The Frequency Distribution of the Unknown Mean 

 of a Sampled Universe 



By E. C. MOLINA and R. I. WILKINSON 



In drawing conclusions as to the reliability of the mean of a sample it 

 ia important that all relevant information be taken into consideration. 

 The mathematical analysis in this paper is based on the Laplacian Bayes 

 Theorem which implicitly comprehends the results of a sample together 

 with the a priori knowledge available concerning the parameters of the 

 universe. 



The discussion is limited to a universe assumed to be normal but whose 

 mean and precision constant are unknown. Several simplifying, yet quite 

 reasonable, assumptions regarding the forms and independence of the 

 a priori frequency distribution of the true mean and standard deviation are 

 incorporated in the analysis so that numerical answers may more easily 

 be deduced. 



Conclusions, properly drawn, are usually quite definitely dependent 

 upon the a priori assumptions made, and especially so in the case of small 

 samples. A considerable space is, therefore, devoted to the solution of a 

 problem in which the sample is only five, taking up a wide variety of these 

 a priori assumptions. They give, in consequence, a wide range of numerical 

 results, appearing in the form of probable errors in the mean of the sample. 

 Each set of assumptions is briefly discussed indicating how the sampling 

 technician may be able to make a selection consistent with his a priori 

 knowledge of a particular problem. 



EVERY observation or series of observations upon the items 

 composing a "universe" or "population" may be regarded as 

 constituting a sample. We may divide sampling into two broad 

 natural classes, (1) Sampling of Attributes, and (2) Sampling of 

 Variables. The theory of the first class concerns itself with some 

 particular characteristic, such as the color red, which each item of 

 the universe definitely does or does not possess, and endeavors to 

 assign, ultimately, a numerical value to the probability that the 

 number or proportion of the items in the universe having this character- 

 istic lies within any given range. The second division comprehends 

 that wide variety of problems in which each item of the universe 

 displays to a greater or less degree the same particular quality, such 

 as length, weight, or resistance. After having drawn a random 

 sample of items, probability theory is called upon to assert with what 

 likelihood certain important descriptive constants or "parameters" 

 of the universe lie within any given ranges. 



In either class the problem is legitimately attacked by means of a 

 posteriori probability theory. This theory makes use of the two 

 important distinct kinds of knowledge which, in varying amounts, 

 are always at hand, namely, (1) a priori or preexisting information 

 regarding the universe and the possible values which the unknown 



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