On the Sex-Ratios of Births in the Registration Districts of England and Wales, 1881-90 Author(s): H. D. Vigor and G. Udny Yule Source: Journal of the Royal Statistical Society, Vol. 69, No. 3 (Sep., 1906), pp. 576-582 Published by: Wiley for the Royal Statistical Society Stable URL: http://www.jstor.org/stable/2339346 . Accessed: 25/06/2014 02:16 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Wiley and Royal Statistical Society are collaborating with JSTOR to digitize, preserve and extend access to Journal of the Royal Statistical Society. http://www.jstor.org This content downloaded from 62.122.77.48 on Wed, 25 Jun 2014 02:16:22 AM All use subject to JSTOR Terms and Conditions 576 Miscellanea. [Sept. Crime is also fomentedby the contestbetween capital and labour. The action of Socialists in Germany,and, in a more intenseform, that of the Nihilistsin Russia,and the Irishagrarian disturbance(1881, Cd-4439), are violent endeavoursto settle a difficultproblem. In Germanythe numberof Socialist trials amountedin one yearto 237. It is worthnoting that the greater numberof the accused were between20 and 30 years of age; 2 were under 15, and 32 between 16 and 18. Only 4 women figurein the return. One-fourthof the whole criminalityof the countrywas locatedin Berlin. III.-On theSex-Ratios of Bir ths in theRegistration ZDistr icts of England and WT'ales,1881-90. By H. D. VIGOR, BicardoScholar, and G. UDNY YULE, NewmarchLecturer in Statistics,University College, London. THE purposeof the presentnote is to exhibit,somewhat more fully than has yet been done, the amount and nature of the variationin the sex-ratiosof birthsin the differentregistration districtsof England and Wales on the basis of the data for the period 1881-90 given in the Decennial Supplementto the 55th Report of the Registrar-General.In this Supplementthe total numbersof births of each sex are given for each of the 632 registrationdistricts. The sex-ratiosare not stated,however, so the necessaryfirst step in the arithmeticwas carriedout by one of us, theproportions of males per i,ooo totalbirths being worked out withthe aid of a large slide-rule,to the nearestunit. Occasional errorsof a unit are possible,but are of no practicalconsequence whatever. From the values so obtained,a correlationtable (Table 1) was compiledbetween (1) the proportioinof males per i,ooo birthsin each district,(2) the totalnumber of birthson whichthe ratio was based. The sex-ratioswere grouped by intervalsof threeunits, the numbersof birthsby intervalsof 4,000, the range of the latter being,it may be noted,very large indeed,viz., from564 in the ScillyIslands and 1,000 or 2,000 in numeroussmall rural districts, to 105,000 in West Ham and over 148,0o0 in the West Derby divisionof Liverpool. The meanproportion of male birthsis 5092, witha standard deviationof 7 46. The meannumber of births in a districtduring the decadeis 14,500, witha standarddeviation of i8,Ioo. The coefficientof correlationis obviously,from the appearanceof the table,small, its actual value being - 014, with a probableerror of oo27. A carefulexamination shows, however, that the means of successiverows are hardlycollinear, the very small value of the This content downloaded from 62.122.77.48 on Wed, 25 Jun 2014 02:16:22 AM All use subject to JSTOR Terms and Conditions 1906.] OnSex-Ratios of Birtlhs in Englandand Wales,1881-90. 577 correlationcoefficient arising partly from the fact that the mean sex-ratiofirst increases and then decreasesas we pass fromthe smallto the largerdistricts. In Table 2, column3, are given the meansof successiveindividual rows up to the fifth,and means of groupsof rows thereafter,with theirprobable errors. It will be seen that the increaseis uniformup to the fourthrow, the pro- portionof males risingfrom 5o8g2 to 51I-I, and this increaseis followedby an equally uniformdecrease-back to the original value-for the very large urban districtswhich constitutethe tail ofthe table. The generalrun of the pointsis well shownin the figure,the smallcircles indicating the above means,and RR the line thatis the " bestfit " theretoas givenby the theoryof correlation, the " line of regression" as it has been termed. The mean of the firstrow lies to theleft of this line, all the nextfive lie to the right, and thelast threeto theleft again. The generaltrend is so smooth that we are inclined to accept it as significant,though, perhaps, it maybe rashto do so until the data for the decade 1891-1900are available for comparison.The districtswhich exhibit the largest proportionsof malebirths are, according to thisresult, for the most part the provincialtown or semi-urbandistricts; the morepurely rural districts,and the purely urban districtsof large towns, both exhibiting a distinctly smaller proportion. Only two London districts,viz., Hampstead and St. Giles, fall into the fourth row, and amongst others we find such districts as Guildford,Farnham, Tunbridge, Maidstone,Thanet, Hastings, Wycombe,Wellingborough, Peterborough, Bedford, Yarmouth, Redruth, Penzance, Gloucester, Cheltenham, Kidderminster, Warwick,Runcorn, Fylde, Lancaster,Wharfedale, Scarborough, Guisborough,Morpeth, Kendal, Pontypool, Carnarvon, &c. In column4 of Table 2 is giventhe standarddeviation of each rowor group of rows; as is evidentfrom the appearanceof the table, the dispersionsof the rows decreasevery rapidly as the numberof birthsis increased. This is, of course,a very well knownand expectedresult, but so far as we are aware it has not hithertobeen exhibited in quiteso strikinga fashion. Further,the ninestandard deviations of Table 2 give a widebasis for comparing the actualdispersions with those given by the theoryof artificial chance. It willbe familiarto most statisticiansthat if p be, say, the chanceof throwing " heads" witha (possiblysomewhat biassed) coin,and q the chanceof throwing"tails," the standarddeviation of the proportionof "heads" obtainedin a series of throwsof n coins is ,/ , and it has been shownby Lexis, Edgeworth, and othersthat the standarddeviation of the proportionof male births(say in a given districtduring successive equal intervalsof time)is givenwith surprising accuracy by the same"combinatorial" rule,the fluctuationsthat usually occur from month to monthnot being significantof any definitecauses, but only the inevitable resultsof the complexcauses analogousto those that act in the tossingof a seriesof coins. The combinatorialstandard deviations VOL. LXIX. PART III. 2 P This content downloaded from 62.122.77.48 on Wed, 25 Jun 2014 02:16:22 AM All use subject to JSTOR Terms and Conditions 578 Mliscellanea. [Sept. are given in column 5 of the table, the values of p and q being taken throughoutas the mean values for England and Wales, viz., 0o5092 and o049o8. As regards the value of n, however,there is a certain difficulty. For the individual rows, it makes very little difference whetherwe take the mid-valueof the row (2,000, 6,ooo, 10,000, &C.), or use any moreelaborate process,but whereseveral rows are grouped togetherthe mid-valuecan hardly be used. Suppose then that we have to deal with a large numberof districts for which it may be assumed that p and q are constant,but in fi of which the births are ni, in f2 the birthsare n2,and so on. The firstgroup will contribute to the sum of the squares of deviations a total amount fpq; the nl second will contributef2Pq, and so on. The standard deviation for n2 the whole series of districtswill thereforebe- 3l2 + . *) = where N is the whole numberof districts,and H the harmonicmean of the numbers of births. The standard deviations for the groups of rows in Table 2 have been calculated by means of this formula from the data of Table 1. Now it will be seen that while the actual standard deviation a-, neveY differsfrom the combinatorial value ( by more than once or twice the probable error, in every case, except that of row 5, r is greater than (r2. If rows 5, 6, and 7 be all pooled together,the rule holds without exception,the actual standard deviation for these three rows being 3-97 ? *25 and the combinatorial 3-47. This result appears to indicate clearly that in every group of districts there is a certain small amount of physically significantvariation of the sex-ratio, on which that due to the mere "fluctuations of sampling" is superposed. The actual measure of significantfluctuation is easily obtained, as has been indicated by ProfessorEdgeworth. If any variable is subject to a standard deviationz due to definitecauses on whichthere is imposed a standard deviation ( due to errors of sampling (or indeed any other source), the total resultant standard deviation a-, will be ,/y2+ (T2 . But it is a1 and 2 which we know, and therefore we have the physically significantstandard deviation given by the relation z = x/'oi2 - 22 The value of this is given in column 6 of Table 2, and it ranges from I to 3 units. Possibly the value for the firstrow may be somewhat too large, as the harmonicmean of the birthsmay be less than 2,000. This does not seem very likely,however, seeing that there are many more districts near the upper limit of the class interval than the lower. Further,if the harmonicmean number of birthsbe reckoned for the whole of England and Wales from the grouped distributionof Table 1, as may be very readily done, the value is 5,071, correspondingto a combinatorial standard deviation of 7 02 in the sex-ratio as compared with the actual standard This content downloaded from 62.122.77.48 on Wed, 25 Jun 2014 02:16:22 AM All use subject to JSTOR Terms and Conditions 1906.] OnSex-Ratios of Births in Englandand WVales,1881-90.