of age (Biometrika, vol. i. part 1, p. 41), the mean stature is
66-91 in., the modal 66-96 in., the median lying between the
two. In other cases the difference between the three may be
considerable.
As an example of extreme asymmetry we may take
de Vries's record of the frequency with which given numbers of
petals occur in a certain race of buttercups. Pearson has shown
(Phil. Trans., A., 1893) that this frequency may be closely represented
by the curve whose equation is
y = 0.211225x-0.332(7.3253 - x)3.142.
The curve, and the observations it represents, are drawn in fig. 2. The two are compared numerically in Table II. Here the mode is at 4.5 petals, the mean at 5.6 petals, the median lying of course between the two.
Fig. 1.
Table II.
| Numbers of petals | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| Frequency observed | 133 | 55 | 23 | 7 | 2 | 2 | 0 |
| Frequency given by Pearson's curve | 136.9 | 48.5 | 22.6 | 9.6 | 3.4 | 0.8 | 0.2 |
|
| Fig. 2. |
The distributions represented in figs. 1 and 2 may be taken as examples of three common forms of series into which the individuals of a race may be arranged with respect to a single character; a comparison of them will show how little can be learnt from a mere statement of racial type, without some knowledge of the way in which deviations from the type are distributed.
The variability of structures which are repeated in the body of the same individual (serial homologues) has been studied by Pearson and his pupils with important results. The simplest of such repeated elements are the cells of the tissues, more complex are cell-aggregates, from hairs, scales, teeth and the like, up to limbs or metameres in animals, or the leaves and their homologues in plants. Serially homologous structures, borne on the same body, are commonly differentiated into sets, the mean character of a set produced in one part of the body, or during one period of life, differing from the mean character of a set produced in a different region or at a different time. Such differentiation may be measured by determining the correlation between the position or the time of production and the character of the organs produced, the methods by which the correlation is measured being those described in the article Error, Law of. An excellent example of structures differentiated according to position is given by the appendages borne on the stem of an ordinary flowering plant—the one or two seed leaves; the stem leaves, which may or may not be differentiated into secondary sets; and the various floral organs borne at the apex of the stem or its lateral branches. The change which often occurs in the mean character and variability of the flowers produced at different periods of the flowering season by the same plant is an example of differentiation associated with time of production; as this kind of differentiation is less familiar than differentiation according to the region of production, it may be well to give an example. In a group of plants of Aster prenanthoides, examined by G. H. Shull (American Naturalist, xxxvi., 1902), the mean number of bracts, ray-florets and disc florets, and the standard deviation of each, was determined on four different days, with the following result:—
Table III.
| Sept. 27. | Sept. 30. | Oct. 4. | Oct. 8. | |
| Mean No. of bracts | 47.41 | 44.34 | 43.83 | 41.92 |
| Standard deviation | 5.52 | 5.15 | 5.28 | 4.89 |
| Mean No. of ray-florets | 30.77 | 28.71 | 28.25 | 26.34 |
| Standard deviation | 3.99 | 3.57 | 3.50 | 3.01 |
| Mean No. disc-florets | 56.43 | 51.71 | 49.16 | 45.78 |
| Standard deviation | 3.99 | 4.99 | 4.88 | 4.78 |
Notwithstanding this differentiation, the mean character of a series of repeated organs is often constant through a considerable region of the body or a considerable period of time; and the standard deviation of an “array” of repeated parts, chosen from such an area, or within such limits of time, may be taken as a measure of the individual variability of the organism which produces them. If such an array of repeated organs be chosen from the proper region of the body, within proper limits of time, in each of a large series of individuals belonging to a race, and if all the arrays so chosen be added together, a series will be formed from which the racial variability can be determined. Thus a series of arrays of beech leaves, gathered, subject to the precautions indicated, from each of 100 beech trees in Buckinghamshire by Professor Pearson, gave 16.1 as the mean number of veins per leaf, the standard deviation of the veins in the series being 1.735. The number of leaves gathered from each tree was 26, and the frequency of leaves with any observed number of veins in the whole series of 2600 leaves was as follows:—
Table IV.
| No. of veins. | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
| No. of leaves. | 1 | 7 | 34 | 110 | 318 | 479 | 595 | 516 | 307 | 181 | 36 | 15 | 1 |
The whole series contains 2600 leaves. If a leaf from this series be chosen at random, it is clearly more likely to have sixteen veins than to have any other assigned number; but if a first leaf chosen at random should prove to have some number of veins other than sixteen, a second leaf, chosen at random from the same series, is still more likely to have sixteen veins than to have any other assigned number. If, however, a series of leaves from the same tree be examined in pairs, the fact that one leaf from the tree is known to possess an abnormal number of veins makes it probable that the next leaf chosen from the same tree will also be abnormal—or, in other words, the fact that leaves are borne by the same tree establishes a correlation between them. Professor Pearson has measured this correlation. Taking each leaf of his series, with an assigned number of veins, he has determined the array of pairs of leaves which can be formed by pairing the chosen leaf with all others from its own tree in succession. The pairs so formed were collected in a table, from which the correlation between the first leaf and the second leaf of a pair, chosen from one tree, could be determined by the methods indicated in the article Probability. The mean and standard deviation of all first leaves or of all second leaves will clearly be the same as those already determined for the series of leaves; since every leaf in the series is used once as a first member and once as a second member of a pair. The coefficient of correlation is 0.5699, which indicates that the standard deviation of an array is equal to that of the leaves in general multiplied by ; and performing this multiplication, we find 1.426 as the standard deviation of an array. The variability of an array of such a table—that is, of any line or column of it—is the mean variability of pairs of leaves, each pair chosen from one tree, and having one leaf of a particular character; it may therefore be taken as a fair measure of the variability of such a tree. We see therefore that while leaves, gathered in equal numbers from each of 100 trees, are distributed about their mean with a standard deviation of 1.735 veins, the leaves gathered from a single tree are distributed about their mean with a standard deviation of 1.426 veins, the ratio between variability of the race and variability of the individual tree being = 0.822.
The correlation between undifferentiated sets of serial homologues, produced by a single individual, is the measure of what Pearson has called homotyposis. In an elaborate memoir on the homotyposis in plants (Phil. Trans., vol. 197 A., 1901), from which the foregoing statements about beech leaves are taken, Pearson has given the correlation between such sets of organs in a large number of plants: he and his pupils have subsequently determined the correlation between structures repeated in the bodies of individual animals. The results obtained are sometimes puzzling, because it is
