SOCIETY OF THE UNIVERSITY OF ABERDEEN. 



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In the binomial curve the ordinates are proportional to the 

 coefficients of a binomial expansion. Let us suppose that we toss 

 6 coins 2" or 64 times, then since the probabilities of a head and of a 

 tail turning up are each equal to ^, the frequency or number of times 

 which 0, 1, 2, 3, 4, 5 and 6 heads will turn up is represented by the 

 following binomial expansion : 



64 (>- + i)" = 1 + 6 + 15 + 20 + 15 + 6 + 1. 



The result may be stated as follows : 



No. of heads 0, 1, 2, 3, 4, 5, 6. 

 Frequencies 1, 6, 15, 20, 15, 6, 1. 



According to this if we toss 6 coins 64 times, 3 heads (the mean 

 number) ought to turn up 20 times ; and 6 heads once ; 1 and 5 

 heads 6 times ; 2 and 4 heads 15 times. The results of tossing will 

 not correspond exactly with this theory unless the number of coins 

 is very large. If a curve be drawn using the frequency at each unit 

 on the base line as an ordinate we get a binomial curve. (See 

 sketch. ) 



01 23456 



If we measure, say, the head breadths of a large number of 

 people and classify our measurements so as to find the frequency of 

 each dimension, and then draw a frequency curve, we shall find that 

 it is approximately of the same form as the binomial curve. The 

 binomial curve which we have discussed is symmetrical about the 

 mean, that is the mean, dimension corresponds with the dimension 

 which is most frequent. This dimension which occurs most fre- 

 quently is called the Mode. If the pennies with which we toss are 

 loaded so that the chances of a head or a tail turning up are not 



equal, then our curve would not be symmetrical about the mean 



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