Transactions of The Academy of Science of St. Louis. VOL. VI. No. to. THE RELATION BETWEEN THE GROWTH OF CHILDREN AND THEIR DEVIATION FROM THE PHYSICAL TYPE OF THEIR SEX AND AGE. WM. TOWNSEND PORTER. Issued November 14, 1893. THE RELATION BETWEEN THE GROWTH OF CHILDREN AND THEIR DEVIATION FROM THE PHYSICAL TYPE OF THEIR SEX AND AGE. Wm. Townsend Porter. Quetelet induced from his measurements of children the law that the weights, heights or other physical dimensions at each age in the period of growth are approximations of a median value,1 about which they are grouped in the form of a probability curve, being related to the median value as the individual observations in a series of measurements of the same thing are related to its actual size. Quetelet assumed that the median value of an anthropometric series expressed the physiological type of the series and that each deviation from this value expressed the physiological difference between an individual and the type. Fifty years of research have placed the truth of Quetelet’s law beyond all doubt and have not weakened the reasonableness of his assumption, so that both law and hypothesis are rarely questioned and are re- garded as a secure base from which to explore the phenomena of growth. The degree of deviation of the individual measurements from the median value of an anthropometric series is meas- ured by the Probable Deviation, that value which, in the words of Lexis,2 is as often exceeded as attained. Hence, if Quetelet’s theory is true, the Probable Deviation is a measure of the degree of deviation of individuals from their Physical Type. The Probable Deviation from the median value of a 1 Moyenne of Quetelet, see Lettres sur la Theorie des Probability, Brux- elles, 1846, page 66; and mean of Sir John Herschel and other English writers, see Edinburg Beview, 1850, page 23. 2 Ueber die Theorie der Stabilitat statistischer Reihen. Hildebrand’s Jahrbiicher fur Nationaldkonomie und Statistik. Bd. 32, 1879, S. 60-98. See page 62. 234 Trans. Acad. Sci. of St. Louis. series containing many measurements may be calculated by the approximation formula (1) d dr 0.8453 ~ n where d = Probable Deviation, d = Deviation of individual from median value, I'd = Sum of individual deviations, n = Total number of observations. The Probable Deviation contains the Error of Observation, as well as the Physiological Difference of the Individual from the Type. The Error of Observation, in a large series of measurements, is always relatively small. Its insignificance can be made clear in several ways. If the height of one bo}' at any age is measured 1000 times, the Probable Deviation will be much smaller than when the heights of 1000 boys at that age are measured once. Compare, for example, the Probable Deviation from the average height of one boy aged 17 measured 78 times with that of 78 boys aged 17 measured once, the measurements being made under conditions as nearly alike as possible in both instances. One Boy aged 17 Measured 78 times. Average Height 176.28 cm. Probable Deviation zb 0.24 cm. 78 Boys aged 17 Measured once. 165.13 cm. zb 5.15 “ In the single boy, the Difference of the Individual from the Type is not present and the Probable Deviation is very small: in the 78 boys, the opposite is true. Yet the difficulty of cor- rect measurement and hence the Error of Observation in each measurement in the two series cannot differ greatly. It fol- lows that by far the greater portion of Probable Deviation is made up of the Physiological Difference of Individual from Type. Again, the Error of Observation is inversely as the square root of the number of observations and should, were it an important constituent of the Probable Deviation, cause the latter to increase as the number of observations decreased. Porter Relation of Growth to Probable Deviation. 235 Thus, in the following table, comparing the Eelative Probable Deviation from the average height standing of boys with the square roots of the number of observations, the Probable Deviation should be much greater at ages 17 and 18, in which the number of observations is small, than at age 10 or 11, in in which the observations are much more numerous. A look at the figures shows that the Probable Deviation is very little Age at nearest Number of Square Relation of Probable Birthday. Observations. Boot. Deviation to Average. 6 709 26,63 3.1 % 7 1850 43.01 3.2 8 2228 47.15 3.3 9 2205 46.95 3.0 10 2087 45.68 3.1 11 1819 42.63 3.2 12 1653 40.67 3.2 18 1268 35.62 3.5 14 925 30.42 3.8 15 490 22.14 4.1 16 189 13.75 3.7 17 78 8.85 3.1 18 29 5.40 2.8 influenced by variations in the number of observations, within the limits given here. The Probable Deviation may, therefore, without any error of importance, be considered as the Physi- ological Difference between the Individual and the Type. Not all observers have taken the Median Yalue as the Type. The arithmetic mean is frequently employed in Germany, Denmark and elsewhere. In a large series the difference between the two is so small that either may be safely used. The maximum and the mean Median minus Average values for the physical dimensions studied in this paper are as follows: MEDIAN MINUS AVERAGE VALUE. Dimension. Unit of Measurement. Maximum. Arithmetic Mean. Boys. Girls. Boys. Girls. Weight Height Standing... Height Sitting Span of Arms Girth of Chest Kilogram Centimetre Centimetre Centimetre Centimetre 0.73 1.00 0.94: 1.35 0.84 0.74 1.10 0.99 1.33 0.71 0.23 0.50 0.44 0.53 0.44 0.25 0.49 0.67 0.59 0.46 There can, therefore, be no objection to the use of the 236 Trans. Acad. Sci. of St. Louis. Average in place of Median Value in the series about to be studied. The accuracy of the average can be estimated by the formula:1 (2) d E ± y n where E = Probable Error of Average, d Probable Deviation from Average, n = Number of Observations. The values for E are given in Tables No. 6, 7 and 8. It has already been said that the Physiological Difference between the Individual and the Type is expressed by the Probable Deviation from the Average. According to Geissler and Uhlitzsch,2 the interval between Average minus d and Average plus d increases with the age. The values of d found by them in their measurements of height standing are as fol- lows (page 33): PKOBABLE DE VIATION (A-\~d). AGE. Boys. Girls. zb zb to 7 years 3.5926 7 “ 8 “ 3.7362 8 “ 9 “ 3.8293 9 “ 10 “ 4.0067 3.7785 10 “ 11 “ 4.2265 11 “ 12 “ 4.4125 12 u 13 “ 4.8013 13 “ Over 14 << 4.7844 5.2155 4.8520 14 “ The authors say concerning this table (page 34) : “ Hieraus geht hervor, dass diese Intervalle mit dem Alter “im Allgemeinen zunehmen und es diirfte dies auch ganz “ natiirlich erscheinen, da wohl anzunehmen ist, dass die das “ Wachsthum hemmenden bez, fordernden Ursachen bei den “ meisten Individuen ziemlich dieselben bleiben, den Unter- 1 Formulas (1) and (2) are contained in L. Stieda’s paper: Ueber die An- wendung der Wabrscheinlichkeitsrechnung in der anthropologischen Statis- tik. Archiv fur Anthropologic, Bd. xiv, 1882, S. 167-182, » Arthur Geissler and Richard Uhlitzsch. Die Grossenverhaltnisse der Schulkinder im Schulinspectionsbezirk Freiberg. Zeitschrift des kdniglichen Sdchsischen Statistichen Bureaus, xxxiv, Heft 1 and 2, 1888, S. 28-40. Porter Relation of Growth to Probable Deviation. 237 “ schied also in der Grosse der Individuen imraer merklicher “ hervortreten lassen. Audi hinsichtlich der Geschlechter ist “ ein Unterschied deutlich bemerkbar und zwar zeigen sich “ fiir die Madchen vom 11 Jahre an grossere Schwankungen “ als fiir die Knaben. Da aber beobachtet worden ist, dass “ die Madchen circa zwei Jahre vor Eintritt der Pubertats- “ periode verhaltnissmassig rascher wachsen, so dlirfte die “ Yerschiedenheit, rait welcher der Eintritt dieser Zeit erfolgt, “ wohl ein Grand mit sein fiir die grosseren Schwankungen.” It appears from this extract from the valuable work of Geissler and Uhlitzsch that they were very near discovering the law which it is the purpose of this paper to demonstrate and would certainly have done so had they not contented themselves with the Absolute Probable Deviation, in which the real Physiological Difference of Individual from Type lies hidden, and had the material furnished them been sufficiently extensive. The Absolute Probable Deviation is entangled with the size of the individual, and its true value can be known only when this disturbing factor is removed. It is the relation between Probable Deviation and size of individual that must be studied, if the character of the Probable Deviation would be known. But even if Geissler and Uhlitzsch had pursued the method just suggested, the limitations of their material would have prevented them from solving the problem completely. For the material given them to analyze extended no further than the fourteenth year, with some observations over, almost wholly missing the period of pre-pubertal acceleration in boys and the early pubertal years in girls and entirely omitting the early pubertal years in boys. It would seem from their results that the Probable Devia- tion increases with the age, whereas it shall be presently shown that the Absolute Probable Deviation in height standing as well as in weight, height sitting, span of arms and girth of chest does not increase with age during the whole period of growth. Moreover, the Eelative Probable Deviation does not increase during seven of the nine years of boys’ growth and five of the nine of girls’ growth included in the observations of Geissler and Uhlitzsch, and shows a merely secondary relation 238 Trans. Acad. Sci. of St. Louis. to age daring the remaining years. They state further that the amount of the Probable Deviation from age 11 on is greater in girls than in boys, meaning of course that this is true within the limits of their own observations. A wider experi- ence shows that the Absolute Probable Deviation in height standing of girls ceases to be greater than that of boys at age 14 (nearest birthday). Finally, it does not appear from the context that the statement “ Da aber beobachtet worden ist, “ dass die Madchen circa zwei Jahren vor Eintritt der Puber- “ tatsperiode verhaltnissmassig rascher wachsen, so diirfte die “ Verschiedenheit, mit welcher der Eintritt dieser Zeit erfolgt, “ wohl ein Grand mit sein fiir die grosseren Schwankungen J’ includes the conception of the relation of the Probable Deviation to the quickness of growth (to be demonstrated below), as distinct from the absolute difference in size of boys and girls at this period. Before applying to the present material the ideas which the foregoing paragraphs have attempted to state with some preci- sion, it will be interesting to compare the Probable Deviation of Height Standing of German children in the Freiberg district with that of the St. Louis children. TABLE No. 1. A Comparison of the Absolute Probable Deviation from the Average Height Standing of School Children in the Freiberg School District CGeissler & Uhlitzsch) and in the St. Louis Public Schools. St. Louis Age at Nearest Birthday. Boys. Girls. Freiberg Age. St. Louis. Freiberg. St. Louis. Freiberg. rh =h dr dc 6 8.40 cm. 3.4488 era. 3.42 cm. 3.5926 cm. to 7 yrs. 7 3.61 “ 3.5841 £- 3.75 “ 3.7362 £‘ 7 - 8 £t 8 3.89 “ 3.8546 “ 3.70 “ 3.8293 <£ 8 - 9 £< 9 3.75 “ 4.0067 “ 3.83 “ 3.7785 ,£ 9 - 10 £‘ 10 3.98 “ 4.2181 £i 4 06 « 4.2265 “ 10 - 11 ££ 11 4 23 “ 4.2434 “ 4.48 “ 4.4125 ££ 11 - 12 ££ 12 4.47 “ 4.5984 ££ 5.23 “ 4.8013 ££ 12 - 13 ££ 13 4.98 ££ 4.7844 “ 5.46 ££ 5.2155 ££ 13 - 14 (£ 14 5.58 “ 5.1479 “ 5.15 “ 4.8520 ££ over 14 ££ 15 6.33 “ 4.01 “ 16 5.87 “ 4.05 “ 17 5.15 “ 3.45 “ 18 4.98 “ 3.39 “ 19 4.04 “ 20 3.14 ££ 21 4.27 £‘ Porter Relation of Growth to Probable Deviation. 239 The agreement between the series is satisfactory and dem- onstrates the stability of the method as well as the accuracy with which it has been employed in these particular instances. The coincidence is the more significant because the St. Louis children are taller than the Freiberg children. TABLE No. 2. A Comparison of the Average Height Standing of School Children in the Frei- berg School District (Geissler and Uhlitzsch) and in the St. Louis Public Schools. St. Louis Age at Nearest Birthday. Boys. Girls. Freiberg Age. St. Louis. Freiberg. St. Louis. Freiberg. 6 108.94 cm. 108.6 cm. 107.67 cm. 107.9 cm. 64 to 7 yrs. 7 114.03 cc 112.6 a 112.95 Ci 112.0 CC 7 - 8 “ 8 119.13 Ci 117.6 cc 118.36 <4 116.7 Ci 8 - 9 “ 9 124.35 cc 122.1 cc 123.67 cc 121.5 u 9 - 10 “ 10 128.87 a 126.7 cc 128.43 cc 126,1 cc 10 - 11 “ 11 133.84 a 130.6 cc 133.19 cc 131.0 cc 11 _ 12 “ 12 138.21 a 135.5 cc 139.11 cc 135.5 c i 12 - 13 “ 13 142.91 ic 140.1 cc 146.53 Ci 141.6 cc 13 - 14 “ 14 148.58 cc 144.1 Ci 150.84 cc 145,5 cc over 14 15 154.90 cc 155.04 cc 16 160.27 cc 157.52 cc 17 165.13 a 159.33 Ci 18 170.41 cc 159.42 cc 19 158.46 cc 20 159.41 Ci 21 159.98 cc It follows from table No. 1 that The Physiological Differ- ence between individual school children and the Physical Type of their sex and age is essentially the same, where the differ- ences between the children compared are not greater than those existing between the St. Louis and the Freiberg children. The material now to be discussed is presented in Tables No. 6, 7 and 8 and consists of the Number of Observations, Aver- age, Probable Error of Average, Probable Deviation, Rela- tive Annual Increase of Average and Relation of Probable Deviation to Average of Weight in indoor clothing, Height standing without shoes, Height Sitting, i. e., height from the crown of the head to the chair on which the child sits erect, Span of Arms, or distance between the tips of the middle fingers when the arms are extended in a plane with the shoulders, and Girth of Chest, obtained by adding the girth of the chest 240 Trans. Acad. Sci. of St. Louis. on a level with the nipples at full inspiration to the girth at full expiration and dividing by 2, the measurement being made over the boys’ shirts and the girls’ dresses, the corsets occasionally worn by American school girls being previously removed. The manner of making the measurements is fully described in the author’s work On the Growth of St. Louis Children, about to be published by the Academy of Science of St. Louis. The Absolute Probable Deviation from the Average is given in Table No. 3, extracted from Tables No. 6, 7 and 8. The total deviation of the five dimensions meas- Porter Belation of Growth to Probable Deviation. 241 TABLE No. 3. S 8 The Absolute Probable Deviation prom the Average; d — dr 0.8453 n Dimension. Sex. Age at nearest Birthday and Absolute Probable Deviation from Average. 6 7 8 9 10 | 11 12 13 14 15 16 17 18 19 20 21 Weight Boys. Girls. dr 1.43 1.44 dr 1.08 1.88 dr 1.96 1.95 dr 2.09 2 23 dr 2.23 2.31 dr 2.60 2.91 dr 2,46 3.31 dr 3.88 4.22 dr 4.56 4.67 dr 5.06 4.05 dr 6.16 4.24 dr 4.38 3.70 dr 3.60 dr 3.76 dr 3.76 rh Height Standing. Boys. Girls. 3.40 3.42 3.61 3.75 3.89 3.70 3.75 3.83 3.98 4.06 4.23 4.48 4.47 5,23 4.98 5.46 5.58 5.15 6.33 4.01 5.87 4.05 5.15 3.45 4.98 3.39 4.04 3.14 4.27 Height Sitting Boys. Girls. 2.S2 2.03 2.64 2.19 2.26 2.04 2.34 2.11 2.42 2.19 2.56 2 37 2.72 2.61 2.74 2.87 3.15 3.11 8.59 2.54 3.48 2.36 3.77 2.17 2.89 1.72 1.82 2,03 1.86 Span of Arms. Boys. Girls. 3.85 3.87 4.16 4.18 4.18 4.28 4.25 4.18 4.70 4.69 4.84 4.87 4.57 4,51 5.71 5.55 ~37fi 3,54 6.03 5.29 ~3758 3.65 7.15 4.58 7.89 4.41 5.03 4.05 4.31 4.28 4.71 4.13 4.33 Girth of Chest* Boys. Girls. 2.22 2.48 2.38 2.47 2.35 2.40 2.51 2.53 2.72 2.67 2.61 3 04 2.94 8,24 3.77 3.70 4.19 3.27 3.15 3.34 2.94 3.23 Total .. Boys. Girls. 13.72 13.24 14.47 14.47 14.64 14.37 14.94 14,88 15.05 15.92 16.84 17.67 17.16 18.90 20.42 21.64 22.90 21.87 25.90 18.88 27,59 18.33 21.48 16.71 16.22^ * Obtained by adding Girth of Chest at full Inspiration to Girth of Chest at full Expiration and dividing by 2. 242 Trans. Acad. Sci. of St. Louis. ured at each age is also stated, in order to secure a more accurate general view. It is seen that the total Absolute Probable Deviation increases continuously from age 6 to age 16 in boys and from age 6 to age 15 in girls (except at age 8), after which periods there is a fall. The increase is not uniform, becoming suddenly much greater at age 13 in boys and age 11 in girls, the accelerated increase extending over four years in both sexes and ending as suddenly as it began. It has been already said that the Absolute Probable Devia- tion is entangled with the size of the individual. In Table No. 4 this obstacle has been removed and the Physiological Difference of Individual from Type appears. A comparison Porter Relation of Growth to Probable Deviation. TABLE No. 4. d The Relation of Probable Deviation (d) to Average (A). X1 Dimension. Sex. Age at Nearest Birthday and Relation of Probable Deviation to Average. 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Weight Boys. Girls. 7.2 7.fi 7.8 9.0 8.2 8.5 8.0 8.9 7.9 8.4 8.4 7.3 9.6 9.8 10.6 11.0 11.3 11.0 11.0 8.7 12.0 8.4 7.9 7.0 6.9 6.9 7.2 7.0 Height Standing. Boys. Girls. 3.1 3.2 3.2 3,8 3.3 3.1 3.0 3.1 3.1 3.1 3.2 3.2 3 2; 3.4 3.5 3.8 3.8 3.7 4.1 3.4 3.7 2.6 3.1 2.2 2.8 2.1 2.6 2.0 2.7 Height Sitting. Boys. Girls. 3 4 4.2 3.5 3.5 8.2 3.5 3.2 3.6 3.2 3.6 3.4 3.8 8.6 3.7 3 8 4.1 4.0 4.5 3.1 4.2 2.8 4.4 2.6 3.3 2.0 2.1 2.4 2.2 Span of Arms. Boys. Girls. 8.5 3.6 3.0 3.7 3.5 3.6 3.4 3.4 3.6 3.6 3.6 3.6 3.2 8.2 3.9 3.8 4.0 3.5 4.6 2,9 4.8 2.8 3.0 2.5 2.5 2.7 2.9 2.6 2 7 Girth of Chest* Bovs. Girls. 3.8 4.8 3.9 4.2 3.8 3.9 3.9 4.0 4.1 4.2 3.9 4.6 4.3 4.7 4.4 5.0 4.9 4.9 4.9 4.7 5.3 4.1 3.9 4.2 3.5 4.0 Total.... Boys. Girls. 22.2 22.1 22.7 23.7 22.3 22.3 21.8 22.6 22.2 22.5 22.2 24.4 21.8 24.7 26.1 27,4 28.1 27.1 29.Oj 30 0 22.8] 20.7 22.3 18.5 19.0 17.7 * Obtained by adding Girth of Chest at full Inspiration to Girth of Chest at full Expiration and dividing by 2. 244 Trans. Acad. Set. of St. Louis. of this table with the preceding one is very instructive. The continuous increase observed in the Absolute Probable Devi- ation has disappeared Indeed the total Deviation at age 12 is less than at age 7, An important increase occurs in the years 13, 14, 15 and 16 in boys and 11, 12, 13 and 14 in girls. Dur- ing these four years in each sex, the Physiological Difference of the Individual from the Type is greater than at any other time. Hence this difference does not increase with the age of the Type or with its size. The true relation of the Physi- ological Difference is made plain by a comparison of Relative Probable Deviation with Relative Annual Increase of Average Porter Relation of Growth to Probable Deviation. TABLE No. 5. The Relative Annual Increase of Average.* Dimension. Sex. Age at nearest Birthday and Relative Annual Increase of Average. 6 to 7 7 to 8 8 to 9 9 to 10 10 to 11 11 to 12 12 to 13 13 to 14 14 to 15 15 to 16 16 to 17 17 to 18 Weight Boys. Girls. 9.7 10.0 9.7 9.9 9.6 9.6 8.7 9.6 9.5 9.7 8.1 11.6 9.3 14.3 10 5 9.9 14.3 10.4 11.6 7.6 7.9 4.7 5.3 Height Standing.. Boys. Girls. 4.7 4.9 4.7 4.8 4.4 4.5 3.6 3.9 3.9 3.7 3.3 4.4 3.4 5,3 4.0 2.9 4.3 2.8 3.5 1.6 3.0 1.1 3.2 0.06 Height Sitting Boys. Girls. 3.8 4.1 2.2 3.5 8.1 8.4 3.8 3.1 2.1 2.7 2.7 3.8 2.3 4.6 3.6 3.5 3.8 3.0 3.2 2.9 4.1 1.1 3.1 0.6 Span of Arms Boys. Girls. 5.0 5.0 4.9 5.3 4.3 4.5 4.0 4.1 3.8 4.3 4,i 4,3 3.2 5.1 4.3 3.7 4.7 2.5 3.5 1.4 2.8 3.2 4.0 0.9 Girth of Chestf... Boys. Girls. 2.7 1.9 2.6 2.3 2.8 2.8 2.6 0.8 2.5 4.5 2.3 3.8 2.7 4.3 3.8 4.0 4.5 3.6 3.5 2.7 2.7 2.0 3.8 0.1 Total Boys. Girls. 25.4 26.9 24.1 25.8 24.2 24.8 22.7 21.5 21.8 24.9 20.5 27.9 20.9 33.6 26.2 24.0 31.6 22.8 25.3 16.2 20.5 12.1 19.4 1.7 * Obtained for age 6 to 7 by dividing the increase from age 6 to age 7 by the average at age 6; for age 7 to 8 by dividing the in- crease from age 7 to age 8 by the average at age 7; and so on. t Obtained by adding Girth of chest at full inspiration to Girth of chest at full expiration and dividing by 2. Tram. Acad. Sci. of St. Louis. 246 or quickness of growth, given in Table No. b. The compar- ison is made easy by arranging the totals side by side. The WH H-* O CO 6 7 8 ► Q K tO tO »—1 to to to to to to w o Eelative Probable bwOOMMooiotobo oo -a to QD Deviation. loWesMto,, 10 § m. os © J2 cd o Eelative CO to to to to to Ot Increase .8 .5 9 .2 6 e .5 A