In combining these, we must give each the weight corresponding to the number of cases, n1, N2, Ng from which it is derived. Let n be the total number of Then we have the average for the whole year. cases. As a, is the average of all the values of the period 1, we have a1 = is the sum of all the values of the period 1. Therefore The sum of all the s is evidently equal to the sum total of all the observations during the year, which we will call S. The last quotient in the equation is the average of all the periods, which is multiplied by the annual increment d. We have therefore the average value for the year equal to the average of all the observations, plus a correction which is equal to the annual increment multiplied by the difference between the average period for all the observations and the full or half year, as the case may be. While the average may be corrected in this manner without much difficulty, the variability of the series for the whole year is affected in a much more complex manner. We will suppose that the variability did not change much in the course of one year, which at certain periods of life is, however, not the case. Since the values of the average increase from month to month, it is clear that the range of variation for the early periods must begin at a lower point than for the later periods, so that the variation for the total year covers a wider range than the variations at a given moment do. As an example I will give here the distribution of observations of 8-year-old girls, first in periods of three months, then for the whole year, with their averages and the means of the squares of deviations. Distribution of observations of the height of 8-year-old girls. Distribution of observations of the height of S-year-old girls—Continued. The average of the variability of the four quarters is 5.34, while that for the total year is +5.50, a very considerable difference, which will be the greater, the more rapid the growth or the more rapid the change of variability during the year. Previous investigations have shown that variability decreases very rapidly in the period of adolescence. During this time it is imperative to divide the series according to intervals shorter than years in order to obtain results that bring out the physiological relations clearly. We will call the variability at any given period t of a certain year μ; the average value of the measurement for the same period, At. The sum of the squares of all the deviations for this period, divided by the number of observations n1 for this period, will then be (At-x)? μ. The variability for the whole year is computed according to the formula where A is the general average, and n the total number of cases. substitute (A-At+At-x) For this we can (At-x)? + Ent Nt n +2 Em (A—A1)at¬~ Nt At being the average of all the values of the measurement at the period t, then We will assume that nt can be represented by the formula If we assume t as continuous, and carry out the addition between the limits, +0.5-0.5, thus covering the whole year, we find + 0.5 ] = f (C +at+bt3) [ μ22 (1 + a, t + b, t2) + a;t + b; l2 ) at. μ2=f(C+at+i -0.5 μ2 =μo2 [C++1⁄2 (b + b ̧ С + aa1)+bb1] + (C b2+aa)+3% bbq. When a, b; a,, b1; da, b; are computed from the values of the year under consideration, and the preceding and following years, which may be designated by the marks -1, 0, +1, we find From these data the final corrected values of average statures and of their variabilities have been computed (see also pp. 1555, 1556.) Average statures and variabilities.1 Age. 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 16.5 17.5 18.5 Boys: Average stature.. 105.90 111.58 116. 83 122. 04 126. 91 131.78 136.20 140.74 146. 00152. 39 159. 72 164. 90 168. 91 171.07 Variability (4.40) Girls: Average 4.62 4.93 5.34 5.49 5.75 6.19 6.66 7.51 8.49 8.78 7.73 7.22 (6.74) stature. 104. 88 110. 08 116. 08 121. 21 126. 14 131. 27 136. 62 142. 52 148. 69 153. 50 156. 50 158. 03 159. 14 Variability It might seem that this correction could be better made by adding the proportionate amount of growth to the measurement of each individual, i. e., for those of 6 years 0 months, for instance, the amount of 6 months' growth if the measurements are to be reduced to the period of 6 years 6 months. This, however, must not be done, as small children grow differently from tall children, and therefore the amount of growth to be added differs for the various values of the measurement. That this is the case has been proved by Dr. Henry G. Beyer. I collected some statistics on this subject in Worcester, Mass., the results of which are briefly given here. I am indebted to Dr. G. M. West for many of the measurements, while others were taken by myself. The first series was taken in May, 1891. The second series was repeated in May, 1892. I give first the series of annual increases which were obtained in Worcester. Figures in parentheses denote approximate values. The Growth of United States Naval Cadets" (Proc. U. S. Naval Institute. Vol. XXI, No. 2, whole No. 74). Increase in stature of boys. Number of boys whose increase in stature was observed between the ages of Increase in centimeters. Increase in stature of girls. Increase in centimeters. Number of girls whose increase in stature was observed between the ages of— I next divided the series into two equal parts, the first embracing the short, the second the tall, individuals. The following amounts of growth were found for these two groups: Average annual increase (d+4) in stature of short and tall children between the following years: Class of children and differences. BOYS. Years. 6 and 7.7 and 8.8 and 9. 9 and 10 and 11 and 12 and 13 and 14 and 15 and 10. 11. 12. 13. 14. 15. 16. |