join the tops of these perpendiculars, the curve of frequency is the result. [Illustration: Fig. 3.—Curves of frequency, etc. A, normal curve. B, showing the method of arranging individuals in lines containing similar kinds of individuals. C, curve that is skew to the right. D, polygon of frequencies of horns of rhinoceros beetles. (After Davenport.)] “In arranging the individuals it will be found, as has been said, that certain groups contain more individuals. They will form the longest line. This value that occurs with the greatest frequency is called the mode. The position of this modal class in the polygon is one of the points of importance, and the spread of the polygon at its base is another. A polygon with a low mode and a broad range means great variability. The range may, however, be much affected by a single individual standing far removed from the rest, so that a polygon containing such an individual might appear to show greater variation than really exists. Therefore 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 squared departures. This measure is called the standard deviation. The standard deviation is of great importance, because it is the index of variability.”[21] Footnote 21: Davenport, C. B., “The Statistical Study of Biological Problems,” _Popular Science Monthly_, September, 1900. Of the different kinds of polygons there are two main sorts, the simple and the complex. The former have only a single mode, the latter have more than one mode. Some simple polygons lie symmetrically on each side of the mode, Figure 3 A; others are unsymmetrical or skew, Figure 3 B. The skew polygon generally extends out on one side farther than on the other. It has been suggested that when a polygon is symmetrical the species is not changing, and when skew that the species is evolving in the direction of the longer base. This assumes that the sort of variation measured by these curves is of the kind of which evolution is made up, but this is a question that we must further consider. How far the change indicated by the skew curve may be carried is also another point for further examination. A complex polygon of variation, Figure 3 D, has been sometimes interpreted to mean that two subgroups exist in a species, as is well shown in the case of the rhinoceros beetle described by Bateson. Two kinds of male individuals exist, some with long horns, others with short horns; each with a mode of its own, the two polygons overlapping. Other complex polygons may be due to changes occurring at different times in the life of the individual, as old age, for example.