THE STATISTICAL STUDY OF VARIATION 



45 



The coefficients that appear are what they are because precisely those 

 combinations are possible. There is but one combination in which there 

 are no heads, there are four combinations consisting of 1 head and 3 

 tails, there are six combinations possible of 2 heads and 2 tails, there 

 are four combinations of 3 heads and 1 tail, and again but 1 with no tails. 

 But this is simply the expansion of the binomial (1 + I) 4 . The prob- 

 ability that when n coins are tossed exactly ra of them will be heads 

 and the rest tails, therefore, is given by the m + 1st term of the binomial, 

 expansion (1 + l) n - When n is small a symmetrical frequency 

 polygon is obtained somewhat similar to that given by plotting the yields 

 of individual oat plants. When n is very large more points are obtained 



-3(T 



-2U -IT Q, M Q 3 (T 



2(T 



3IT 



Fio. 19. A normal curve or curve of error showing the relationship between the quar- 

 tile, i.e., the probable error of a single variate, and the standard deviation. Q = .6745<r. 

 In this curve the mode, median and mean are identical. The quartile equals the probable 

 error of a single variate because by definition one- half of the variates lie within its limits; 

 therefore the chances are even that any variate lies within or without it. The proportions 

 of variates within certain areas of the curve are as follows: 



within M Q, 50 % of the variates, within M <r, 68.3 % of the variates, 

 within M 2Q, 82.3 % of the variates, within M 2<r, 95.5 % of the variates, 

 within M 3Q, 95.7 % of the variates, within M 3<r, 99.7 % of the variates. 



and the polygon becomes a regular curve, the normal probability curve 

 or curve of error. It is called the "curve of error" because if a refined 

 set of direct measurements are made and plotted as abscissas, the corre- 

 sponding ordinates represent the frequencies or probabilities that each 

 will occur. The mean is the most probable value and is assumed to be 

 the true value and the deviations from the mean are errors. Positive 

 errors lie to the right and negative errors lie to the left of the mean. 

 Positive and negative errors are equally likely to occur if they are gov- 

 erned by chance only and as the errors increase in magnitude the 

 frequency with which they occur becomes less and less. 



Let us assume that we have a perfectly normal frequency curve such 

 as that represented in Fig. 19, and we shall be able to demonstrate the 

 meaning of some of the constants that we have learned to calculate for it. 

 This curve represents observations on a large number of individuals and 



