82 BINOMIAL AND NORMAL DISTRIBUTIONS Ch. 4 



particular situation which produced the data summarized in a given 

 table. 



We might wish to know what the median r is for a binomial dis- 

 tribution. By Table 4.11, 37.7 per cent of the numbers are seen to 

 be O's, l's, 2's, 3's, and 4's. If 5's are included, the percentage runs 

 past 50 (needed for the median) to 62.3; therefore the median r 

 must be 5. It is not some decimal fraction between 4 and 5 because 

 no such numbers even exist on the scale of measurement of r. 



-i * L 



8 



10 



1.00 

 .90 | 



- .80 § 

 .70 £ 

 .60 | 

 .50 J2 

 .40 | 

 .30 £ 



H.20| 

 .10 cr 

 .00 



12 3 4 5 6 7 



Number of occurrences (r) 



Figure 4.13. The r.c.j. distribution for the binomial distribution with p = 1/2 



and n = 10. 



The median of the binomial distribution just considered also can 

 be obtained from the r.c.j. distribution of Figure 4.13 by reading 

 horizontally from the point where r.c.j. = .50 until we come to the 

 first ordinate on the left, which is high enough to be intersected by 

 the horizontal line from r.c.j. — .50. 



It is interesting to compare a frequency distribution which was 

 obtained by actual trials with that which would be expected mathe- 

 matically under the specified conditions. This is done approximately 

 in Table 4.12 for a situation in which 5 pennies were flipped 2000 

 times. It was assumed that the pennies were unbiased, although 

 this is known not to be strictly true for any actual coin. It should 

 be apparent from previous discussions that the mathematically ex- 

 pected proportions of the 6 possible combinations of heads and tails 

 listed in column 1 of Table 4.12 are 1:5:10:10:5:1. The resulting 

 expected numbers of occurrences of each of the possibilities are given 

 to the nearest whole number under the heading "Exp." in columns 

 3, 5, 7, and 9. 



