Continuous Variation

If we examine a number of individuals of the same species, we find that no two of them are exactly alike in all particulars. If, however, we arrange them according to some one character, for example, according to the height, we find that there is a gradation more or less perfect from one end of the series to the other. Thus, if we were to take at random a hundred men, and stand them in line arranged according to their height, the tops of their heads, if joined, would form a nearly continuous line; the line will, of course, incline downward from the tallest to the shortest man. This illustrates individual variation. An arrangement of this kind fails to bring out one of the most important facts connected with individual differences. If the line is more carefully examined, it will be found that somewhere near the middle the men are much more nearly of the same height, or rather there are more men having about the same height than there are near the ends of the line. Another arrangement will bring this out better. If we stand in a line all the men from 60 to 61.9 inches, and in another parallel line all those between 62 and 63.9, then those between 64 and 65.9, then between 66 and 67.9 inches in height, etc., it will be found that there are more men in some of these lines than in others. The longest line will be that containing the men of about 65 inches; the two lines formed out of men on each side of this one will contain somewhat fewer men, and the next ones fewer still, and so on. If we looked at our new group of men from above, we should have a figure triangular in outline, the so-called frequency polygon, Figure [3 B]. With a larger amount of data of this sort it is possible to construct a curve, the curve of frequency, Figure [3 A]. In order to obtain this curve of frequency, it is of course not necessary to actually put the individuals in line, but the curve can be drawn on paper from the measurements. We sort out the measurements into classes as in the case given above. The classes are laid off at regular intervals along a base-line by placing points at definite intervals. Perpendiculars are then erected at each point, the height of each being proportional to the frequency with which each class occurs. If now we join the tops of these perpendiculars, the curve of frequency is the result.

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]]

[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.

If, instead of examining the variations of the individuals of the race, we study the variations in the different organs of the same individual, we find in many cases that certain organs vary together. Thus the right and the left leg nearly always vary in the same direction, also the first joints of the index and middle fingers, and the stature and the forearm. On the other hand, the length of the clavicle and that of the humerus do not vary together to the same extent; and the breadth and height of the skull even less so.

No. of Veins10111213141516171819202122
First Tree—————147941——
Second Tree———34982—————

We may also study those cases in which a particular organ is repeated a number of times in the same individual, as are the leaves of a tree. If the leaves of the same tree are examined in respect, for example, to the number of veins that each contains, we find that the number varies, and that the results give a variation polygon exactly like that when different individuals are compared with one another. Let us take the illustration given by Pearson. He counted the veins on each side of the midrib of the leaves of the beech. If a number of leaves be collected from one tree, and the same number from another, and if all those having fifteen veins are put in one vertical column, and all those with sixteen in another, as shown in the following table, it will be found that each tree has a mode of its own. Thus in the first tree the mode is represented by nine individuals having eighteen veins, and in the second by nine individuals having fifteen veins. So far as this character is concerned we might have interchanged certain of the individual leaves, but we could not have interchanged the two series. They are individual to the two trees. Now in what does this individuality consist? Clearly there are most leaves in one tree with eighteen ribs, and most in the other with fifteen ribs.

If we contrast these results with those obtained by picking at random a large number of leaves from different beech trees, we have no longer types of individuals, but racial characters. Pearson has given the following table to illustrate these points: