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 + 1)*.. The prob- 
ability that when n coins are tossed exactly m of them will be heads 
and the rest tails, therefore, is given by the m+ Ist term of the binomial 
expansion (1+ 1)". When » 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 
/1\ 
7 =—CO2(i | OM CSCOTSC*«s 
Fic. 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 = .67450. 
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 %ofthe variates, within M+ o, 68.3 % of the variates, 
within M + 2Q, 82.3 % of the variates, within M + 2c, 95.5 % of the variates, 
within M + 3Q, 95.7 % of the variates, within M + 3c, 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 
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