# Popular Science Monthly/Volume 59/September 1901/The Statistical Study of Evolution

(1901)
The Statistical Study of Evolution by Charles Benedict Davenport

 THE STATISTICAL STUDY OF EVOLUTION.

By Professor C. B. DAVENPORT,

UNIVERSITY OF CHICAGO.

AS I was going through the chemical section of the John Crerar Library the other day I stopped to examine two books. The first was one of those dawning-chemical works—Glauber's—which enjoyed a wide reputation in the sixteenth and seventeenth centuries. It contained an abundance of speculation and of recipes; but the recipes were of a purely qualitative sort—mix this and this, the amount is immaterial. The second book was the lectures of Van't Hoff, marking the most recent development of the science of chemistry. This book is full of formulæ and tables; numbers, signs and quantities fill every page; they are symbols not only of quantitative relations but also of the direction and cause of advance of chemistry since the days of the alchemists. The history of chemistry is the history of the other quantitative natural sciences, of astronomy and physics. As science advances its methods become more and more quantitative.

Biology is often contrasted with physics and chemistry as a qualitative science. But there is nothing so fundamentally dissimilar in the phenomena of chemistry and biology that they must necessarily be studied so differently. Both treat of matter, not only statically but also dynamically. But the phenomena of biology are more complex than those of chemistry, the things to be described and compared are more numerous; and so the science, which is hardly more than a century old, is still in the descriptive and comparative stage. But the history of science in general justifies the prediction that biology, too, will in time use chiefly quantitative methods in studying processes.

Evolution is an organic process. It has been studied in various ways. Many have sought by ingenious logic to discover its workings. Others have reasoned by analogy from the effects of artificial breeding. Others still, having observed a probable factor of evolution in one case, have argued for its universality. But all of these methods have been qualitative. More recently, on the other hand, there has been undertaken an exact, quantitative study of the condition of species in nature under different environments, to determine exactly the effects that the different environments have produced. This new method, which now demands our attention, is nothing less than the quantitative study of evolution.

We all know that a lot of people, taken at random, show individual physical differences. Even when the people form a homogeneous lot and are of one sex, are all adults, and are of one race, and when such individuals as are very pathological or freaks are excluded, we still find that they differ in stature, weight, proportions of trunk and appendages, color of hair and eyes, proportions of facial features and various other characters. These differences are so slight and multitudinous that the language of adjectives fails to indicate them; measurements must he made in order that they may be expressed numerically.

But how can these numbers tell us anything about the evolution of the human species? The general principle is this: The differences between the races of man are of the same kind as, and differ only in degree from the differences between the individuals of a race. Consequently, the laws of individual variation in a race may be relied on to illuminate the method of origin of the race or species under consideration. Just how, will be clearer after we have considered the sorts of individual variation.

The laws of nature are got at only with the key of the proper method. And it is only within recent years that an adequate quantitative method has been developed in biology. It will now be necessary to consider this method.

Imagine a file of 40 men arranged in order of stature—the tallest at the head of the column. The crown of their heads will form a flowing curve, nearly level in the middle of its course and becoming more oblique at the ends. The reason for this is that the middle statures are much m.ore common than the extreme ones. Half the people have a stature within two inches of the average (Fig. 1). The general features of such a curve are common to all classes of men, but the details differ in different classes of men. The characteristic differences are measured by what are called the constants of the curve.

The stature of the middle man in the file will give us very nearly the first constant, the mean or average. The average is obtained exactly by adding all the individual statures together and dividing by the number of men (e. g., by 40). The average is used constantly and as a matter of course by nearly every one who wishes to compare two comparable series of numbers. It is awkward to compare all the separate data so we let the average stand for the lot. The average is, however, an entirely ideal quantity which need not agree with the measurement of any individual; and it is a little curious that it is so universally used in statistics. Of a series of measurements made on one and the same dimension, the average is demonstrably nearest the true value, and consequently engineers, physicists, astronomers and others who aim at the greatest possible precision in the measurement of the individual phenomenon must make use of it. But in measurements made on a series of distinct individuals the average does not signify the value nearest the truth, and we cannot infer that it is the most significant single representative of the series. The average has this disadvantage, moreover, that the introduction of a few very extraordinary individuals has undue influence on the result. Thus in calculating the average income of American colleges, one institution with an income of \$1,200,000 increases the average by an amount (\$2,500) equal to the total income of about 5 per cent, of the 'colleges and universities.' The average has indeed been over-rated and over-used, as though it were always the best single representative of a series; whereas there are other representatives which are sometimes superior. Among these is the middle value, which is usually got without much calculation. It

Fig. 1. File of Forty University of Chicago Students arranged (approximately) in order of Height.

is the value above and below which fifty per cent, of the cases lie. In the case of income of American colleges the middle value is not far from \$15,000, while the average is \$43,000; the former amount unfortunately gives the truer idea of the usual American college; for about 80 per cent, of the colleges have an income of less than \$43,000 Still another representative value is the geometric mean which is especially important in many biologic and economic statistics. The geometric mean is the number corresponding to the average of the logarithms of the individual quantities.

Finally, if there is one representative of a biological series that is more apt to be significant than any other, it is the value that occurs with the greatest frequency or, in other words, the commonest value. Since this value may be said figuratively to be the most fashionable one, it has been called the mode. The peculiar value of the mode lies in this, that it is not the result of calculation and is not an ideal value merely, but is the prevailing or typical actual condition. In biological statistics the mode should always be considered.

No single value can, however, adequately take the place of all the values obtained. Nevertheless, it is necessary to combine these data in some unit for purposes of comparison. .The best unit is the so-called frequency polygon.' The frequency polygon is got first by sorting out the data into a number of equally extensive 'classes'; then by laying off these classes as a series of points at equal intervals of space along a horizontal base-line; by erecting perpendiculars proportional to the frequency of each class, and by joining with a line the tops of all these perpendiculars. Or, if the tops be united by a flowing line, the frequency polygon is replaced by the frequency curve.

Fig. 2. Bird’s-eye View of 40 University of Chicago Students arranged in Files by Classes of Stature.

Such frequency polygons may also be obtained, without drawing on paper, by putting the individuals belonging to the same class in the same vertical column and arranging the columns in order along a common base-line. For example, we may separate our university students into stature classes as follows: 56 to 57.9 inches, 58 to 59.9, 60 to 61.9; 62 to 63.9; 64 to 65.9; 66 to 67.9; 68 to 69.9; 70 to 71.9; place those falling into the same stature class in a file; and place the files next each other in order, all starting from a common base-line. Then if we take a bird's-eye view' of this body of students, we get the frequency polygon of their statures (Fig. 2). The construction of frequency polygons may be illustrated by another example. The common scallop shells of the Atlantic coast have a variable number of 'ribs' (Fig. 3). In any hundred individuals from one locality the number 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 15 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 measurements 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 polygons 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-formed Frequency Polygon 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 that 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 departures. 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 concrete number; consequently indices are not always comparable, being expressed, e. g., in feet, millimeters, degrees or pounds. So it has been proposed to reduce all indices (except those based on countings) to percentages of the average value. This proportional index is called the coefficient of variability. So much then for the principal operations: counting or measuring; seriation of the numbers; determination of the mode, average, standard deviation, and coefficient of variability.

The results of comparative variation studies, so far, fall into two

Fig. 5. Frequency Polygon of Statures of 25,820 American Males; Lower Polygon, of 100 University Students.

general categories. First, we have got some notion of the classes of variation polygons that may occur among organisms; secondly, we have some evidence as to the significance of these different forms. As the science is only about five years old it is but to be expected that a satisfactory solution of even the principal problems concerning polygon interpretation is still to be worked out. The beginnings are, however, instructive.

Variation polygons are of two main sorts—simple and complex. Simple variation polygons possess only a single mode, whilst complex

Fig. 6. Types of Frequency Curves. From K. Pearson.

polygons usually show a trace of more than one mode. Simple variation polygons are of several types. Some are symmetrical about the mode as in Fig. 5. Others are more or less unsynmetrical or skew, as

Fig. 7. Curves of unselected and selected lots of white daisies.
 ———— curves of frequencies of various members of ray flowers in wild daisies. — - — ⁠"⁠"⁠"⁠"⁠"⁠"⁠in descendants of 12 or 13-rayed wild daisies. ---- ⁠"⁠"⁠"⁠"⁠"⁠"⁠of ray flowers in descendants of 21 rayed wild daisies.

they are technically called (Fig. 6). Skew polygons usually tail out at the base further from the mode on one side than on the other. The polygon is said to be skew in the direction of this longer partial base. Why some distributions are symmetrical and others skew is not fully understood, but at present it seems probable that we get symmetrical polygons when the organ measured is not undergoing evolution and, on the other hand, that unsymmetrical polygons indicate evolutionary progress. Also the direction of skewness is probably determined by the direction of evolution. At present, however, we can only say that in many cases the skew polygon tails off (or is skew) in the direction from which the race is evolving. This conclusion, which I believe to be new, is based upon certain results of experiments as well as upon data gathered from material which had developed under natural conditions. Of this material the most important for our purpose is that in which two polygons have apparently arisen by a splitting off from the original polygon of the two extremes which now form two distinct and widely separated types. The first case (Fig. 7) is derived from the common

 Fig. 8. Polygon of frequencies of lengths in millimeters of cephalic bones of 343 rhinoceros beetles. From data of Bateson. Fig. 9. Polygon of frequency of lengths of wing, in hundredths of a millimeter, for a lot of chinch bugs.

white daisy. In the figure the full-line polygon gives the frequency distribution of the ray-flowers in a collection of wild daisies. This polygon has a ${\displaystyle +}$ skewness of 1.18. The left-hand, dot-and-dash, polygon gives the ray frequences in the descendants of 12-or 13-rayed wild plants. The positive skewness is increased as a result of this selection to ${\displaystyle +}$1.92. The right-hand polygon gives the ray frequencies in the descendants of the 21-rayed plants, a single highly aberrant case of 32 rays being omitted. The skewness is —0.13. In this case we have experimental evidence that curves are skew towards the original, ancestral, condition. The cases in which the frequency curve is bimodal frequently signify that two races are arising out of a former ancestral intermediate condition; i. e., they correspond to the right and left-hand polygons of Fig. 7. Consequently we may expect them to he skew in opposite directions and so we find them to he. For example, Bateson has measured the horns on the heads of 343 rhinoceros beetles; the frequency curve is shown in Fig. 8. The left-hand polygon has a skewness of ${\displaystyle +}$ 0.48; the right-hand polygon of -0.03. One might infer that the right-hand form, the long-horned beetles, had diverged less from the ancestral condition than the short-horned beetles. Again, my pupil, Mr. Garber, has obtained a bimodal distribution polygon in the length of the chinch bug's wing (Fig. 9). The

 Fig. 10. Frequency Polygon of the Number of Petals in Buttercups. Fig. 11. Polygon of frequency of lengths in millimeters, of Pecten shells gathered at random from a fisherman's shell heap.

short-winged form has a skewness of +.44; the long winged form of -.43. In this case also the ancestral form lies between the present modes. It is obvious that we may get cases in which two modes, representing conditions in different places, have moved, to different extents, in the same direction. Thus the index (breadth ÷ length) of the shell of Littorina, a marine snail, as measured by Bumpus, has at Newport a mode of 90; at Casco Bay of 93. The skewness is positive in both •places and greater (+.24) at the more southern point than at Casco Bay (+.13). This result indicates that the Littorina came from a more northern home, for which we have confirmatory historical evidence, and that these ancestral races were rounder, having an index possibly not far from 96. Likewise the Littorinas from South Kincardineshire, Scotland, have a modal index of 88 and a positive skewness of 0.065; while those of the Humber, having a mode of 91, have a skewness of ${\displaystyle +}$ 0.048. These figures suggest an ancestral index of about 97, or about the same as before. The form of the frequency polygon may thus enable us to infer the ancestral condition of a race or species and may consequently help us to get at the history of the race.

Skewness may, as we have seen, depart to any extent from a nearly symmetrical condition. The extreme case occurs when the mode lies at one end of the range. This case is sometimes found among plants. It indicates that the group has in respect to the character in question reached an extreme condition (Fig. 10).

Complex frequency polygons have various interpretations. We have already seen that one of these interpretations at least is a splitting of one race into two. Another kind of complex polygon is due to

Fig. 12. Complex curve of length in cm. of fasciated stem of 146 individuals of Crepis biennis derived from fasciated ancestors. One individual of 19 centimeters omitted. From de Vries, ’95, Bulletin Scientifique, Tome 27, p. 397. Ordinates, number of individuals; Abscissæ, length of fasciated stem.

differences of age. Suppose an animal that breeds at one restricted season of the year and that annually nearly doubles in size. If we make a collection of a lot of these animals from a place at one time, we shall include individuals of different ages such as, for instance, six months, one year and six months, two years and six months, and so on. If now the length of all these individuals be measured we shall obtain a series of modes of which each corresponds to one of the broods (Fig. 11). Still again, two modes may appear when the material is not perfectly homogeneous although the age be constant. For instance the material may contain both normal and abnormal individuals. An example of this sort of polygon is given in Fig. 13. A very complex curve is afforded by the number of the ray flowers of composite plants. If the lappets of a thousand white daisies be counted it will be found that there is not a single mode only but a series of them. These modes increase in height from one extremity of the range, reach a great mode at one point and then diminish again (Fig. 13). It appears also that the modes do not occur at haphazard.

Fig. 13. Polygon of frequency of numbers of ray flowers of the white daisy gathered at random from various german localities. From data of Ludwig.

but chiefly in the series of numbers: 1, 2, 3, 5, 8, 13, 21, 44, and 65. This is a mathematical series in which each term is the sum of the two preceding. Also the ratios of these numbers, namely, 12, 13, 25, 38, and so on, have long been known to represent the arrangement of leaves on a stem; and this seems to be why the numbers of this series are so prominent in the rays of the flower head.

The comparative study of frequency polygons, such as we have been making, enables us, it will be seen, to distinguish different kinds of variation and to make that philosophical classification which is the first step in advancing knowledge. Although the causes of variation are not at once revealed, we are directed to working hypotheses that can be tested by experiment. It is not too much to say that the frequency polygon is the key to the first door that has barred true progress in the difficult subject of the origin of organic diversity.

In what has gone before we have considered variation of single organs or qualities of a species. Yet, although we have to study the variation of organs taken one at a time, in nature no organ undergoes variation by itself alone. For the parts of the body are so knit together, their morphological kinship or their physiological interdependence is such, that when one organ deviates from the mode many others deviate also. This fact has long been known as correlation of variation. A recognition of the law by Cuvier was the justification, slight though it was, of his premature attempts to reconstruct an extinct form from one of its bones. Now, correlation is of great importance in the origin of species; it makes it easier to understand how evolution can take place. For example, when it was objected that natural selection by acting on one part at a time could hardly build up so complex a structure as the eye with so many mutually dependent parts, Darwin was able to rejoin that the principle of correlation comes in to ensure that when any one part is improved all other parts shall vary to meet the new conditions. And in general, a knowledge of correlation is necessary in order to complement the study of individual variation and to perfect our investigations upon the origin of species. And correlation must be studied quantitatively. A proper method has been afforded by Galton and Pearson. That method may be briefly stated. Let us suppose that we desire to find the degree of correlation in variation, or deviation from the mean, between an organ A, called subject, and a second organ B, called relative. We first take all the individuals of one (subject) class; that is, individuals whose subject organ deviates from the mean by a constant quantity, p. We next find for those individuals the average deviation-from-the-mean of the organ B, and call it q. We then find the ration q/p; this is the partial index of correlation. We find this ratio for every subject class. The average of the ratios is the index of correlation sought. The ratio will not exceed unity; because q is bound in the long run not to exceed p. When q ${\displaystyle =}$ p, correlation is perfect and is equal to 1. When the index of correlation is zero, correlation is absent; when the index is negative, correlation is inverse and a large organ is associated with a small one. A good example of organs with strong positive correlation is the right and left arm. Inverse correlation is rarer; an example is stature and cephalic index. The results of studying correlation quantitatively are interesting, as showing how intimately bound together the most remote parts of the body are. Take for example the following table of correlation of parts of the human skeleton, from Pearson's 'Grammar of Science.'

 Femur and tibia 81 to 0.89 Femur and humerus 84 to 0.87 Humerus and radius 74 to 0.84 Humerus and ulna 75 to 0.86 Clavicle and humerus 44 to 0.63 Clavicle and scapula 12 to 0.16 Stature and femur 80 to 0.81 Stature and humerus 77 to 0.81 Stature and fore-arm 0.37 Stature and cephalic index — 0.8 Length and breadth of skull 29 to 0.49 Breadth and height of skull 10 to 0.34 Length and capacity of skull 50 to 0.89 Length x breadth x height and capacity of skull 70 to 0.8 Weight and length (babies) 62 to 0.64 Weight and stature (adolescents) 50 to 0.72 Right and left femur 0.96 Right and left first joint of ring finger 0.93 First joints of right hand, index and middle fingers 0.9 First joints of right hand, index and little fingers 0.82 Metacarpal phalanges, right hand, index and middle fingers 0.94 Metacarpal phalanges, right hand index and little fingers 0.89 Strength of pull and stature 22 to 0.3 Strength of pull and weight 34 to 0.54

A study of this table shows us how justified was Darwin's contention that the evolution of one organ necessarily means the evolution of many parts of the body.

The modern methods of studying evolution have still another application. It is sometimes said that variation and heredity are the two factors of evolution. Heredity is, however, only a special case of correlated variation; a correlation between parents and offspring or between any two blood relatives. So evolution is reduced |to |a single factor—variation, simple and correlated.

|As a criticism of the new methods of studying variation it has been urged that, after all, they deal not so much with the causes of evolution as with the mere results. |to |this criticism it may be rejoined that the first step toward the determination of the causes of a phenomenon is a precise knowledge of the limitations and conditions of the phenomenon itself; and this is what the quantitative study of variation gives. Science has been more retarded by wasted efforts |to |explain erroneous data than by conscientious attempts |to |discover the precise facts. For when the facts are correctly known the true explanation often follows at once. Even if the explanation does not follow at once the proper direction of experimentation |to |discover causes is indicated. Statistics tell us not only the exact static condition of species to-day under the varying circumstances of environment; but they will enable us |to |measure precisely the results of any change in environment, artificially or naturally brought about. We shall thus be able not only to tell what are the factors of phylogenetic change, but also the rate of such change. We shall get possession of the laws of evolution so that we can not only reconstruct the past, but also predict the future development of a race.

The importance of knowing the methods of evolution is partly theoretical, like the importance of astronomical investigation, and partly practical. For, on the one hand, a rapid and thoroughgoing improvement of the human race can probably be effected only by understand and applying these methods; and on the other hand, the improvement of live-stock and of food plants must depend on a knowledge of the laws of phylogenesis. How appalling is our ignorance, for example, concerning the effect of a mixing of races as contrasted with pure breeding; a matter of infinite importance in a country like ours containing numerous races and subspecies of men. How little do we know of the direct effect of climate on 'blood'; a matter of concern in a land with such diversified geography. In our fast-filling earth all problems will some day be secondary to that of raising more grain or beef to the acre; then at least the biologic-evolutionary problems will be recognized as paramount. It is for us to anticipate in part the future demands on biology. The State Experiment Stations of our day are doing something in this direction, but for the most part in too narrow a fashion. For the future, broad, far-reaching experiments in evolution are required, with a quantitative study of causes and results.