34 GENETICS IN RELATION TO AGRICULTURE 



principle has been expressed by King in terms, which fit well the imaginary 

 case under discussion, as follows: "A moderately large number of items 

 chosen at random from among a very large group are almost sure, on 

 the average, to have the characteristics of the large group." It must not 

 be inferred that any partial group of individuals no matter how large, 

 will give exactly the same results as would be obtained by the use of the 

 entire mass. But the averages will be close and the probability of in- 

 accuracy due to accidental error diminishes as the numbers increase 

 because individual errors tend in the long run to counteract each other. 



Law of Deviations from the Average. If, now, one lot of 500 beans 

 be measured to the nearest millimeter and then arranged in columns from 

 left to right according to width beginning with the narrowest beans, the 

 result will be very similar to Fig. 14. It will be noticed first that the 

 middle classes contain the most beans while the classes on the extreme 

 left and right are very small. The black vertical line M indicates the 

 average width or mean of all the beans and the column with the most 

 beans in it represents the most frequent width of beans and is called 

 the mode. The columns nearest the average value on either side contain 

 the most beans and the further the column is from the average the fewer 

 the beans in it. Thus we see that the .majority of the beans show 

 only slight deviations from the average while a few exhibit wide deviations 

 therefrom. Statistical study has proved that it is a general rule with 

 fluctuations that individuals showing extreme deviations in either 

 direction for a given character are comparatively rare, while individuals 

 exhibiting smaller deviations, and hence occupying a position inter- 

 mediate between the two extremes are especially frequent. In other 

 words, continuous variations usually appear in frequencies such that, 

 if we represent these frequencies graphically, we obtain a polygon which 

 resembles more or less the normal variability curve. Such a polygon 

 is produced by connecting the ends of the columns in Fig. 14. 



The Normal Curve and its Significance. The normal variability 

 curve is a theoretical curve which pictures the result of expanding the 

 binomial (a + 6) n when a = b 1 and n is assumed to be indefinitely 

 great. By the binomial theorem 



(a + 6) 1 =1 + 1 



(a + 6) 2 = 1 + 2]+ 1 



(a + 6) 3 =1+3 + 3 + 1 



(a + 6)4 =1+4 + 6 + 4 + 1 



(a + 6) 5 =1+5 + 10 + 10 + 5 + 1 



(a + 6) 6 =1 + 6 + 15 + 20 + 15 + 6 + 1 



(a + by =1 + 7 + 21+35+35 + 21+7 + 1 



(a + 6) 8 =1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 



(a + 6) 9 = 1+9 + 36 + 84 + 126 + 126 + 84 + 36 + 9 + 1 



(a + 6) 10 = 1 + 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1. 



