Ch. 4 BINOMIAL AND NORMAL DISTRIBUTIONS 77 



male and female. If a baby were to be chosen at random from 

 among those born in the specified year, its classification as male or 

 female would be a member of this binomial population. Under the 

 above assumptions, the probability that such a selection will turn 

 out to be male is p = .51. 



If n repeated observations, or trials, are made on a binomial pop- 

 ulation in which the proportion p is staying fixed, and if attention is 

 fixed upon the number of individuals in each of the two classes, these 

 numbers are variable from one set of n trials to another. For exam- 

 ple, it was noted in Chapter 3 that the probability that r males, say, 

 and n — r females would be observed is given by C„, r (p) r (l — p) n ~ r . 

 In other words, r is a chance variable. The relative frequencies with 

 which r will have the values 0, 1, 2, ... , and n after a great many sets 

 of n random trials from a binomial population constitutes a binomial 

 frequency distribution. This distribution will be of more direct in- 

 terest to us than the binomial population in itself because the binomial 

 frequency distribution describes results which are obtained in the 

 process of sampling a binomial population. 



There are many types of populations for which the random vari- 

 able is of the second type discussed at the beginning of this chapter, 

 namely, a measurement referred to a continuous scale, such as weight. 

 Probably, the most important populations of this sort are those called 

 normal populations. It will be convenient to describe this type of 

 population by means of a mathematical formula for its frequency 

 distribution. This will be done in a later section. 



It seems obvious that we cannot possibly learn much by sampling 

 a population which cannot be clearly and concisely described ; hence 

 there is need for a mathematical description, or classification, of 

 populations. We choose to study types of populations by means of 

 their frequency distributions because that — or something equivalent 

 — constitutes the fullest description we can obtain for a particular 

 population. As noted above, the discussion in this chapter will be 

 devoted to two of the most important types of frequency distribu- 

 tions: one, the binomial, is appropriate to qualitative measurements 

 of a certain kind ; the other, the normal, typifies continuous numerical 

 measurements of types quite frequently met in practice. Between 

 these two theoretical distributions, a great many of the uses of 

 statistical analysis will be introduced. 



