Sec. 4.1 



THE BINOMIAL FREQUENCY DISTRIBUTION 



83 



TABLE 4.12 



Comparison of Observed and Expected Frequencies of Heads (H) and 

 Tails (T) When 5 Pennies (Assumed Unbiased) Are Flipped 100, 500, 



1000, and 2000 Times 



* Actually each of these numbers is 62.5 but was rounded off this way to 

 keep the sum of the observed and expected frequencies equal. 



After the 2000 trials involving 10,000 tosses the ratio of heads to 

 tails is 0.94 to 1. Hence there apparently is a weak but definite 

 tendency for tails to appear more frequently than heads; that is, p 

 is not exactly equal to 1/2. Methods will be described in Chapter 5 

 for deciding when a coin, say, is biased, and for estimating the de- 

 gree of bias. 



If the observed frequencies in any column of Table 4.12 are taken 

 as the / and the r is listed merely as 5, 4, 3, 2, 1, and 0, we have an 

 observed frequency distribution, as in Chapter 2. If the expected 

 frequencies (which follow a mathematical law) are used as the / 

 column and r again is listed as 5, 4, 3, 2, 1, and 0, we have a the- 

 oretical frequency distribution of the sort being discussed in this 

 chapter. 



In view of the existence of a general mathematical expression for 

 the binomial frequency distribution (as in formula 3.32), we might 

 be curious to know if such statistical measures as the arithmetic 

 mean and the standard deviation can be determined just from the 

 n and p which determine the distribution. This is, in fact, true, as 

 will be shown partially below. 



The discussion of mathematical expectation given in Chapter 3 

 included the information that the arithmetic mean of the number 

 of occurrences of an event E over many trials coincides with the ex- 



