THE STATISTICAL STUDY OF EVOLUTION. 451 



'ribs' (Fig. 3). In any hundred individuals from one locality the num- 

 ber of ribs may vary from 15 to 21. If we put in one pile the shells 

 having the same number of ribs and arrange the piles in order, from 

 lr5 to 20, upon a level base we shall get a figure which is the frequency 

 polygon for the ribs of Pecten shells from the given locality (Fig. 4). 



Frequency polygons may be obtained in the same way from meas- 

 urements or countings made on almost any organ of any plant or 

 animal. The shapes of the polygons are probably never exactly alike in 

 different organs; consequently, there is a field for the comparing of 

 pol5^gons and for drawing interpretations from differences in their form. 



In comparing frequency polygons attention should be directed, first 

 of all, to two characters ; namely, the position and relative proportion of 

 individuals included in the modal class and the spread of the polygon 

 at the base. This spread is known technically as the range. While 



Fig. 3. Scallop Shell with 15 and 20 Ribs 

 Respectively. 



Fig. 4. Self-foemed Frequency Poly'gon 

 OF Pecten Ribs. 



some frequency polygons have a high mode and narrow range others 

 have a low mode and a broad range. The importance of this fact is 

 tliat a narrow range implies relatively small variability, since relatively 

 few individuals depart far from the modal condition. On the other 

 hand, wide range implies great variability. Range, however, is not an 

 accurate measure of variability because it is too easily affected by the 

 accidental occurrence of even one aberrant individual. We need a 

 measure of variability that shall take into account the departures of all 

 the individuals from the mode. One such measure is the arithmetical 

 average of all the departures from the mean in both directions; and 

 this measure has been widely employed. At present another method is 

 preferred; namely, the square root of the average of the squared de- 

 partures. This measure is called the standard deviation. The standard 

 deviation is of great importance, because it is the index of variability. 

 This index in the case of measured organs is, like the range, a con- 

 crete number; consequently indices are not always comparable, being 

 expressed, e. ^r., in feet, millimeters, degrees or pounds. So it has been 

 proposed to reduce all indices (except those based on countings) to 



