68 BELL SYSTEM TECHNICAL JOURNAL 



which means that the sample numbers must be small, and yet we see 

 from the solution derived from Pearson the significance of the sizes of 

 both the original and the second sample. Thus, he '" shows that the 

 standard deviation a is given by the equation 



a— *=/>f/(i+f-;). (9) 



MiiUimodal Distributions. These occur frotiuently in engineering 

 work and particularly in connection with the inspection of large 

 quantities of apparatus. One such instance has already been referred 

 to in the discussion of the data given in Table II, and another is 

 illustrated by the data given in Fig. 1. Prof. Pearson '' has developed 

 a method for determining analytically whether or not the observed 

 distribution is such as may be expected to have arisen from the com- 

 bination of two normal components, the mean values of which are 

 different. The method involves the solution of a ninth degree equa- 

 tion. As a result, the arithmetic work is in many cases prohibitive. 

 This method cannot be applied to the data given in Fig. 1 primarily 

 because the number of observations is not sufficiently great. 



Pearson's Closed Type Cunes.^- One of the best known statistical 

 methods for graduating data is that developed by Prof. Pearson. His 

 system of closed type curves arises from the solution of the differ- 

 ential equation derived upon the assumption that the distribution 

 is uni-modal and touches the axis when 3' = 0. In the hands of Pearson 

 and his school great success has been attained in graduating data 

 collected from widely different fields, although primarily from these 

 of biolog>', psychology', and economics. The choice of curve to 

 represent a given distribution rests primarily upon a consideration 

 of a criterion involving two constants, /3i = V ;fe and ^«, both of which 

 have been defined previously in footnote 21. 



In the early study of the distributions of efficiencies of product 

 transmitters an attempt was made to apply this system of curves. 

 For example, the Pearson types are indicated in Table II. In no 

 instance, however, was it possible to obtain a very satisfactory fit 

 between the observed and the theoretical distributions. Further- 

 more, the arithmetical work required to calculate a theoretical dis- 

 tribution in this way is excessive. W'e must also consider what 

 physical significance can be attached to the different types of curves. 

 The answer is not definite. Under certain conditions the generalized 



"Pearson, K— Philosophical ^faga:illc—\907, pp. 365-378. 

 "I'carson, K.— Philosophical Mag(i:iiic — Vol. I, 1901, pp. 115-119. 

 " Eldcrton — Frequency Curves and Correlation. 





