Since, owing to the insufficient number of cases, the numerical results of the Worcester series are not quite satisfactory, we decided to calculate the same data from a more extended series, from which we selected the fifth, seventh, tenth, thirteenth, and sixteenth years as representative values. The calculations were made from measurements taken in Toronto and Milwaukee. The results are given in the following tables: TABLE VIA.-Averages and variabilities of measurements of boys, TABLE VID.-Averages and variabilities of measurements of girls. Stature. Height sitting. Average. σ. Cases. Average. σ. Cases. Average. σ. Weight. Cases. 5 10 13 16 The averages for each year give us the following results: It will be noticed that the more extensive material agrees well with the measurements collected in Worcester. The averages for the values of Ru", and of did (see Table III, p. 28) give us the following results: We may say, therefore, that increases of the values of correlations take place during the period of growth, in accordance with the demands of our theory of variability in period of development, although the correspondence between theory and observation can not be definitely determined unless we know the amount of the variability in period 67, which, as appears from equation (1), determines the value of Ru12. It seems, therefore, necessary to determine the values of 67. Considering this problem from a general point of view, it will be seen that 6, is the variability of period at which a certain physiological stage of development is attained. We may investigate this variability by means of various groups of physiological data. We may, for instance, determine the variability of the period of pregnancy, of the phases of first and second dentition, of puberty, and of the appearance of the wisdom teeth. In later life we may also include the menopause and death by arterial diseases. Unfortunately it is not easy to obtain satisfactory statistics relating to these phenomena; statisties relating to the exact period of teething are particularly unsatisfactory. We have collected data of this description, and the following pages contain the results of our inquiries. Pregnancy.—Schlichting gives the following table for the period of ges tation: Archiv für Gynecologie, XVI, p. 2210. Dentition.-Whitehead gives the following table for the appearance of the inner pair of incisors in well-nourished children. Most of the other tables are given in a form which makes it difficult to determine the average age and its variability directly. In these cases Pearson's method of curve fitting has been applied in a somewhat simplified form, which seems satisfactory owing to the small number of available observations. The method adopted will be illustrated by our treatment of Woronichin's tables.c He gives the following data for well-nourished children, indicating the percentile frequency of the appearance of the inner incisors at various ages. TABLE IX.--Percentage of children that have teeth. cor Assuming that the actual variability of time follows the laws of chance, we can determine from the tables of the probability integral the multiples of responding to their frequencies. Using W. F. Sheppard's tables, we find, for instance, for the age of 7 months, that the value of 0.3368 corresponds to the sum total of frequencies under the value of 1-0.426, where A is the average, 6 the standard variability. We may, therefore, write the equation— 7= A -0.426. In the same way we find— 6-A-0.966. Any two equations of this kind would give us a pair of values of A and 6, from all of which the most probable value may be calculated. For the purpose of this calculation we will write, in place of the actual month, the middle month-in our case 8.0-plus or minus such differences as will give us each particular month. If we call these differences z, the middle month M, and the multiples of 6 found from the table of the probability integral y, we can write our equations or by taking the average of these equations for all our observations we find, sincə [x] = 0 (3). (A -- M) x = x2 - yx 6, and taking the average, [yx]ó = [x2]. (4). (3) and (4) give us satisfactory values for A and 6. Applying this to Table IXb (p. 32), we find the following multiples of 6 corresponding to the percentile frequencies. If the table signifies that each group embraces children of from 3 to 4, 4 to 5, 5 to 6, etc., months of age, this would give us an average of 8.83 months. The values for y which correspond to the lowest and highest percentages are, of course, very uncertain, and must be given in an exact calculation a very low weight, in accordance with the uncertainty of the values of y. The available material for the calculation of the time and variability of the appearance of the first molars is still more unsatisfactory. Fleischmanna gives the following data: Dr. L. Fleischmann, Klinik der Pädiatrik, II. Der erste Zahndurchbruch des Kindes, p. 82. ED 1904 M-3 6-7 8-9 10-11 12-13 14-15 16-17 18-19 TABLE X.-Number of children having four molar teeth. Calculated in the same manner as the preceding table, we find the average age 18.83.7 months. For the second dentition we have utilized a series of plaster casts of palates of normal children collected by Dr. Walter Channing. These give for the upper teeth the following results: TABLE XIa.-Number of children having permanent teeth. Cases. Inner in cisors. Outer incisors. Canines. Bicuspids. First mo- Second molars. Cases. Inner in- Outer in cisors. མ བད??? 1 5 12 11 18 11 15 12 TABLE XIb.-Percentage of children having permanent teeth. |