CHAPTER 4 



The Binomial and Normal 

 Frequency Distributions 



The discussions and illustrations of Chapter 2 involved situations 

 in which groups of measurements (usually numerous) had been 

 taken under specified conditions, and we had in mind only an efficient 

 summarization of the data. The ACE scores in Table 2.01 were 

 cited as an example. In a sense, we simply took what we got and 

 thereafter applied statistical methods to reduce a bulk of data to a 

 more comprehensible form without losing any essential information. 

 More generally, however, populations of numerical measurements 

 must be studied by means of samples because so many measurements 

 are involved that it is not feasible, efficient, or even possible to obtain 

 and to analyze the whole of the population. 



Two different types of populations will be considered. In one type, 

 the chance variable will be a qualitative one such as male or female, 

 dead or alive, own an automobile or do not own an automobile. The 

 population will consist of individual members, each falling into one 

 of just two classes according to the qualitative designation adopted. 

 The other type of population to be considered will be based upon 

 a variable which is measured along a continuous scale, such as the 

 weight of an individual, the volume of a gas, or the bushel yield of 

 a variety of wheat. 



As regards populations in which a qualitative variable is used, 

 attention herein will be confined to what is called a binomial popula- 

 tion because each member of the population falls into one of only two 

 classes. The proportion of a binomial population which belongs to 

 one of the two classes will be measured by the fraction p, leaving the 

 fraction falling into the other class to be 1 — p = q. For example, 

 if all the babies born in New York City during a given year were 

 to be classified as male or female, p might be the fraction who were 

 males. If p = .51, then q = 1 — .51 = .49. The sex would be the 

 qualitative variable mentioned above, and has but two "values": 



76 



