l'lE· . ilI .:J rI 1 _ . R: ()N _ 1

l'lE· . _{ilI .:J} rI ¹ _{_ .} R: ()N _{_ 1}

~IVI5TPt

INTERNPtZIONf\LE DI 5Tf\Tl5T1Cf\ - REVllE INTERNlHIONf\LE DE 5Tf\TI5TIQllE INTERNf\TIONf\L REVIEW Of 5TflTl5TIC5 - INTERNRTIONf\LE 5Tf\Tl5TI5CHE RllND5CHF\1l

DIRETTORE PROPHIET ARIO - DIRECTEUR ET PROPRIÉT AIRE EDITOR AND PROPRIETOR - HERAUSGEBER UND EIGENTH'UMER

Dott.

Cor~ado

Gini, pro{.

M'd. di Statistica

nella n.

Univcrsitù

di Padut'a

(I

talia).

COMITATO DIRETTIVO - COMITÉ DE DIRECTION - EDITORIAL COMMITTEE - DIREKTIONS-KOMITEE

Prof. A. Andréadès (Athènes) - Prof. A. E. Bunge IBuenos Ay1'CS) - Dott. F. P. Cantelli

(Roma)

- Prof. C. V. L. Charlier

(Lund) -

Prof. E. Czuber

(TVien) -

Prof. F. v. Fellner

(Budapest) -

Prof. A. Flores de Lemus

(Mad1'id) -

Dr. M. Greenwood

(London) -

Sir G. H. Knibbs

(Melbourne)

- Ing. L. March

(Paris) -

Dr. A. W. Methorst

(La

llaye) -Dr. A. Julin

(Bruxelles) -

Prof. R. Pearl

(Ba.ltimo1'c) --

Prof. H. Westergaard

(Copenhagen)

SEGRETARIO DI REDAZIONE - SECRÉTAIRE DE RÉDACTION

EDITORIAL SECRETARY - REDACTIONSSECRETAER

Prof. Gaetano Pietra,

istituto di Statisft'ca delta R. L'niven'ilù di ['adora

(Italia) Prof. Jacopo Tivaroni,

[sfituto Tecnico di Cdine (I

t.

ti liti)

Vol. IV. N. 3-4. 1 - VI - 192&

SOMMARIO SOMMAIRE CONTENTS INHALT

G. Pietra, The theO'J'Y of statistical }'elations Icith special

'reference to cyclical series . pago 383

F. Savorgnall, La Fecondità delle Al'istoc1'az.ie » 558 E. Lindelof, Les cornrnunes suedoises rurales de la Pinlande. » 576 G. Findlay Shirras, A Statistical Study or India' s Population » 590 H. Bunle, Notes StatiStiques sw' la Demogl'aphie des Colonies

Francaises . » 605

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f\IVISTA INTERNAZIONALE DI STATiSTICA f\EVUE INTERNfHIONRLE DE 5 f,cHl5TIQUE ìNTE'f\NATIONAL REVIEW Of STATISTICS

'~li~TERNATIONRLe

STATI5TISCME

fHJ~jD5CHfìU

DIRETTORE PROPRIETM~IO - DìRECTElm ET PROPRJÉTAIRE EDITOR ANO PROPRIETOR -- HEI~AUSGEBEI~ UNO EIGENTHÙMER

Dott. Corrado Gini,

Imi(. (ln/, di Stahl'li('(1 Ilei/I[ /(, {"lIi/'el'si/i! di fJut!o/'u ill,lIiil^l, COMITATO DIRETTIVO - COMlTÉ DE DIRECTIQN - EDITORIAL COJ\\MITTEE - DIREKTIONS-KOMiTEE

Prot. A. Andréadès,

de Sàcnce des f/nalltcS

lì ['

{·nl/'';7'.I'/'{' II' U/II'ne.\'

«;

l'(\Ct~).

Prof. A. E. Bunge,

Ililedu1' !/l'/te/'(l[ de gstlUlistil'lr di' /11 SI(n'1I11.. 1ii/l'lIli.\' ,1/II't'.\' (,\r~"tlljf1Ll),

Prof. F. p, Cantelli,

inwl'imtu (Li S{lItisLim I/IIIÙIì/IILiI'u i' Iii J/Il!':l!llllim 11l/'iUliil.!!; /l1'i!U

n,

Univelsitù Ili !iUIJW (I t:dia).

Dr. C. V. L Charlier,

j»)'u(csso)' Ile)' ..]st"IJJJillllÌ/j fin III;)' f'!Iil'i'l'l'i!tu ritI/t! (SI')I\\l'df'III,

Dr. E. Czuber,

J!J'I}j'I'S,\'(J)' an liDI' Fa}lllirl'IlDn JJIII'l!fl.JIII!,o; iII n'i,'l/

il

)1"ll~,'11 I ì('-i(I'! r":I'III,

Dr. F. von Fellner,

li.

il;/'.

Unil',;}',rWi/r-l'}'II/i',\'\'IIi' iII /1111/1,[II:\t n'llg:tl'lll,

Prof. A. Flores de Lemus,

.Jete de FS/Ilil/sliw (Id llillil/,'}'() !ll' Ilrll'I"lllit1.1/"illl,II!-:~!,uìal.

Or. M. Greenwood,

l'Cadé/' in IIIC,{icai ,\'{UbStil:S ÙL f/I/.: (iL/l'enitl!

III

j.,)/II(I/I \ 1':II~l:tlltl).

Sir G. H. Knibbs,

dircl'tl)/' u( {Ile ('Ulillilì/fiU'('O{!f1 IlI.srUui/' Il! Sf'l{:!I'~" IInd 1/,,111\'/;"1, ,~ll'III,)/JI'ne.

(Allslrallal.

Ing. L. March,

tlil'el'le'/l1' !wno),llire (il~ la Slati,l'(il!lic fI,IIl':"ltll5 tle {Il ""'{(fU'I' 1'(1/ i.1 ! \"1';1111'1').

Or. H. W. Methorst,

dil'él'tl'll), di:

t'

()/lil'l! IJI:nuUJ/I:n! III; l'Il/stilli! !n!fTllo{i'J//lI! di StatistillU!;

ct dii /luJ'I:a/l anlJ'al tle Stati,l'filj/lc. 1~ll 1/1/.'11} i/ll>iLlllI1pl,

Prof. A. Julin,

se''l'rI[oiJ'c !Jh1h'ul dII ,l/inil'!iT(; III' j' //111/1.\'11'11' DI dii f"'IIi'!lil. I:r, ^1i'1! i! ^{I \} iHIII~H{Uel,

Dr. R. Pearl,

p]'o!'

or

JJ/())/ldl'!/ (llll! j 'i/a! SliI!i.\'LiI'.\' in llli:.I. Ilo!Jltitl,1' CItt'i'i:i'.I'il!!, 1:1111/'/11111"; !' ,S, A,).

Or. H. Westergaard,

/I1'o/,e,\'\'(}/' iII tlte /ilÌl.'(n'l'il!l fJ( ('u/II:/tlw!/ett Il )t'!1 111;(1'1, ì, SEGRETAmO DI REDAZIONE .- SEC!~f~TAWE DE f~ÉDACT!ON

EDITORIAL SECRETARY ~-rmDACT!ONSSECRETAER

Prof. Gaetano Pietra,

Islitulu di .... 'ta{i.ltil'll dellu.

IL

f.llù'el',I'i(lì di flUi/llllli (\ td 11a)

Prof. Jacopo Tivaroni,

/.l'utlilu Tel'fll,'tl Ili Cililll; ilt.;di:l)

Vol. IV. N. 3-4. l - VI - 1925

~O:Ml\1AHIO

SOMMAIH.E CONTENT:-; INHAf.T

H. Pietra, T/w the()J'l} 0/ statistical J'eltttions Il'ilh s/J(xial

re(f;j'L'Il('e to

cf/dù:({l series pago

:;S;~

F. Savorgn~tn, La Fecondità delle Aì'istoCTrt::,;e

E.LindeIUf, l,p,') comlHltnes swjdoises l'w'(l/es de 1ft Fi,tlrlnrle G. Fin(lIay Shirras~ A Statistical 8tud,lJ 0(' Indi(t's l )o/Iula!ion

H. UUIlJC, Notes ,\'tatistù[ues

sW'

La

Ih;jl'().rJj·uJ!llie

des Co {lrnies

Fì'{(.llcu;ses

j)/l!){;fiCfCilJlIL

}'ù;('?)ute - j)ulJlicati{Jì/s'

)'('~:un; - jJu!)/i,;rttifjJIS

)'f!ceincd EllwlLnw lTe1'ril/,e1ltlù:/mn[!ell

FEHl{:\UA (IT.\LlA)

CASA

~:-:l

\'Ìa de' l\.o

^III

e i

~> ;);)~

» :)/ti

?> ;)!H)

CHE VERRANNO PUBBLICATI NEI GELANGT SIND UND WELCHE IN DEN NACR-

PROSSIMI NUMERI. FOLGENDEN NUMMERN ERSCHEINEN WERDBN.

(Secondo l'ordine d'arrivo) (Nach del' Reihenfolge des Ez'ngangs)

ARTICLES REçUS PAR I.A REVUE ARTICLES RECEIVED BY TRE REVIEW WHICII ET À PARAITRE PROCHAINEMENT. WII.L BE PUBLISHED IN FUTURE ISSUBS.

(D'apl'ès la date de l'éception) (According to date or receipt)

c. ^Gini, Sulle legg'i della frequenza e delle comb'inazioni sessuali dei parti plw'irni.

C. Gini e M. Boldrini, Il Cflnt1'O della popolazione italiana.

W. F. Willcox, Methods of estirnal'ing tlle population of the Uni- ted States. .

A. Henry, La consommation des pl'oduits aliJìtentaires en Belgique avant et après la guerre.

C. Gini, La riehesse et les ?'evenus nationaux des Indes B1'itanniques.

K. Popoff, La prèdom'inance des naissances masculines (D'apl'ès les donnees de la Statistique du Royaume de Bulgarie).

G. Findlay Shirras, Pl'oduetion in India

bef01~e

and after the' War.

J. H. van Zanten, Quelques donnèes demographiques sur les lsraé:- lites à A7Jtstel'dam,

H. Westergaard, On

Pe1~iods

in Economie Life.

L., Hersch, La m01'talité eausee l'la?' la gue1're 11'wndiale.

E. C. Ithodes, On Sarnpling.

W. R. Dunstan, Height and TVeight of Sehool Childl'en in an English R'll1'at Area.

S. Novossclski and V. Paevski, Life Tables of the City of Leningrad

(forllt. Pel1'og1'ad) for the years 1910-11, 1918, 1920 and 1923.

Tbe theory of st~ltistieal relations

.

,

with speeial referenee to cyelical series.

The theory of the measure of the statistical relatiolls hetween quantitative or qualitative modalities or two chal'acters may be considered as one of the most extensive branches or modèrn me- thodology; however, far from aHaining its fulfilment, it is stilI

811-

sceptihle of further ad vancement.

1"01'

the quantitative chal'actel's we may pal'ticularly refer to the well known wOl'ks of BRAVAIS, GALTON, PEARSON, amI LIPPs;

1'01' the q lIalitative ones to those ol'

PEARSO~, ,BENINI,

anct YULE, but it is to be remarked that a systematical settlement of the thpory has been glven only recent.1y by GINI' s works published dUl'illg

tht~

period 1914 - 1918. Fil'sUy, this aulhor eslablished a dear distinction between the diffel'ent meallillg"s or connection and concordance which by the precedi Ilg alI thol's were rreq uently considered under the same expl'essioll of correlation; he also introduced a new principle of dissimilarity between two st." t.istical distributions, and put it as the basis of t.he theory of statislical relatiolls. Thell, considering the ctifferent aspects accordillg to which the qllantitative, or qualitative modalities of two charact.el's rnay be presented, he sllggesls fol' each of them a suitable me(\sure.

l'hus, we obtaill a complete system of indices in which a pl'Oper pIace is given 10 BRAVAIS' s correlation coefficient, lo PEARSON' s correlation ratio, and to BENINI' s attraclion index.

Howevel', thA researches of GINI l'efer to t.he quantitative or

qualitative modalities or t:wo chal'actm's, which may ue distl'ilHlfed

according to

1~ectilinear>

or disconnected series; those fol' cyclical

series have not. J)een - as far as I kllow - beglln; hellce the

theory of statistical relatiolls stilI remains to be attelllpted.

It is here my inten tion to complete the subject; for a better unrlerstanding or this wOl'k, I give inmy Introduction a summary of the actual situation of the theory, according io GIN I 's publica- tions (l) toge/her with applications and examples; in the first Part I shali deal with the dissimilarity between cyclical series anrl in the se,:()nd with the connection and concordance; I shall submit t'or each oC these relations, applications and examples.

[ ntroduction

§ I

I. - Series and seriations - Refore illllstl'ating the relations between qualii.ative and qualltitative modalities of two

~haraeters,

lei. us briefly summarize the most important definitions and meanings of the theory of statistical series.

First oC all, we call

se1~iation

a succession of quantities which measure the intensity of a character classi fied according to the intensities of another chal'acter. E. g. the . number of recruits ac- cording to theil' size; the amount of assets accol'ding to classes of incomes, and so ono

On the cOfltrary) we call series a succession of quantities, which measure the intensity of a chal'actel', classified accol'ding to qualities of another charader .. E. g. the number or recruits accor- ding to the colollr of hai r, the· amoullt of assets according 1.0 the employment, and so ono

In the particular case that the first characìer indicates how many times the cOl'l'esponding rnodalit.y is vel'ified, we have to refer to the

f1~eq'Uency

ser'iation or frequency sedes. 'l'hus, in the pre-

(1) Cfr. C.

^GINI:

VariaMl#à e Mutabilità - Bologna Cuppini 1912 - Sulla misura della concentra:;iune e della variabil#à dei caratteri, Atti del R. Istituto Veneto di S. L. A. 1913-J4. T. LXXIII P. II

~

Di una misura della dissomiglian:;a tra due gruppi di quantità e della sua applica.

~ione

allo studz'o delle relazioni statisNche, id. 1014-15 T. LXXIV P. II - Indicz' di omofilia e di rassomiglianza e loro relazioni col cot/ficz"ente di con'ela:;z'one e con gli indici di altra:;ione, id, id. 1914-:15 T. LXXIV P. II - Nuovi contrz'buti alla teoria delle relazioni statistiche id. id.

1914-15 T. LXX IV P. II - Sul crite1'io di concordan:;a tra due carat-

teri, id. id. 1915-16 T. LXXV P. II. - Indz'cz' di concordan:;a id. id. 1915-

1916 '1\ LXX V P. II - ])elle rela:;ionz' tra le 'intensità congraduate dt

due caratteri, id. id. 1916-17, T. LXXVI P. II - Di una estensz'one del

concetto di scostamenio medio etc. id. id. 1917-18 'l'. LXXVII, P. II.

385 ceding examples, the numbers of recruits accordin g to the size, or according to the colonr of hair consti tute, respectively, a fre- q uency seriation and a frequency series.

Series of qualitatiDe characters may be rlistinguished according to the following definitions:

a) rectilinear series, presenting modalities accordirlg' to a naturai order of succession, in which t,wo are to be considel'ed the first and tbe last ones, the remaining t.he intermediate ones. E. g.

the number of the officel's accol'ding to the ranks of a Hierarchy;

b) cyclical sel'ies, presenti Ilg modali ties accordillg' to a na- turaI order of succession in which, howeyer, we cannot find two modalities that may be considered - except in the case of a particular agreement - respectively, as the first and the last one of the succession. E. g. tho seri es of weddillgs according to the days of the weck. In this regard the day on w hich the weck commences is immaterial;

c) disconnected series, which do not present any order of succession. E. g. the series of numbers of inhabitants according to religious confession.

2. - Cog1"aduation and oont1'ag1'aduation - COtTespondent 'intensities in statistical nomenclature are those presentèd by two different characters under the same unit, or those that a character present under different units which are to each olle confl'onted. E. g.

The assets and the income of a specified family are called cor- respondent intensities when assets and iucomes of families are considered. Yet correspondent intensities are the size of a husband and that of his wife, when the sizes of a certain Ilumber of husbands and wives are compared.

Considering the intensities of a charactel' according to an in- creasing or decreasing order of graduation the deg'ree of an in- tensity is given by the ordinaI numbel' which the said intensity has in the graduatian.

Two intensities wiU he termed cograduate when they present the sa me degree in two graduations iu which both are increasing or decreasing.

Contragraduate wiU be those that present the same degree in two graduations, one of which is increasing the other decreasing.

Thus, if e. g.

al 2: a2 > ... > a,. > ... > a

n - .. + l

> .... '. > a

_n

(l)

bl>b2> ... >b .. ~ ... >bn-r+l> ... /b" (2)

I

f

the quantities a

r

and b

r

may be conSldered as coqraduate; the quantities a

r

and b

n - r

+

1

or' a

n - r

+

1

alld b

r

will be called, on the con tl'al'y, contragraduate.

In this regard the following theorerns are to be considered:

« Two groups of ordered quantities (1) and (2) being given, let

Or

be paired to the

cog1~aduate

b

r ,

then the sum:

(3)

will be a 1ninimwn;

let, a r be paired to the contrograduate b _{n _}

r

+

1,

then the sum:

will be a rnaximum;

we shall ha ve. also :

n

n

~r ⁽ a

r -

b

1,

t ^{= minimwu}

1

~r (a

r

-b

n _ r+ ₁

t=maximun»

1

(4)

(5)

Of course, both the princi ples of cograduation and contragra- duation, and, consequently, the proceding theorerns may be ap- plied only to seriations and 10 those series of quantitative characters to tbe successive modalities of which a succession of number may correspolld.

'l'hen, under this aspect, we cannot consider cyclical series,

While in the following paragraphs of this introduction we wish

to show the principal results of the application of the above men-

tiolled principles and theorerns to the theory of the statistical re-

latiolls, we wiU give, in t.he fil'st part of this work, the analogous

principles and theorerns, which wiU fit the cyclical sel'ies.

387

~II

3. - The dissimUarity - A set or values

being given, indicate by F

xi

the freq uency of

Xi

iII a g-roup (a) or KI quantities and by

f1Jxi

the fl'equency or

.'ìh

in a gl'oup (fJ) of K% quant.ities.

If,

__ !Xi_ ⁼ _.~

f1J,'ri

K,!

(i=I,2 ... n)

then (a) and (fJ) wiU be calleò two similw' gl'aupS.

Again, where

that is, let (a) and (fJ) be cOl'nposed by the same number ol' quan- tities, then, in this case,

J1'xi = l q,xi

aH the quantities of (a) may be pail'ed, each other, to the quantities of (P) so that the difference between paired quantities will disappear.

We have the following theorem:

« Two similar groups (a) ant! (fJ) being given composed or a different Bumber of quantities we may form two similar g-roups composed of an equal nllmbel' of quantities ».

In fact (a) being composed of Kl quantities and (fJ) of K

2

quan-

tities it is always possible to fonn a gl'oup (y) similal' to (a) and

a group

(~)

similar to (fJ) both of K qllantities, K being a multiple

both or KI and K

2

,or of KI .' K

₂

quantities if Xl and K

2

do not have

a lower multiple. Then, speaking of similarity we may only considel'

group composed of the same numhel' of qoantities. Two groups

which are not similar wi Il he tpnned dissim.ila't'.

When comparing the qualltities of two groups, we shall find that the gl'eatel' the dissimilarity the greater will be the differences between the pait'ed quantities. (1)

4. - Index or dissimilarity - To get a measure of the dissimi- .Iarity of two gl'oups it seems natural to confront the quantities ac- cording to a correspondence which .makes a mininium the absolute value of the sum of differences between confronted quantities.

Thus, the arith1netical mean of these differences may be ta- ken as the sfmple index or dissimilarrity; ^the quadrratic nwan will be the quadratic index or dissimila1'ity. .

According to the theorems of n

^r

2 we immediately see that, considerillg the ordered groups (l) and (2), the simple index or dis- similarity . is :

and the quadratic index or dissimilarity is:

~ ^{D == 1/_1_.} ~i ^{(ai - b}

i

)2

V ⁿ

¹

(7)

(8)

It is easy to see that both the indices are nil if the dift'erences between cograduate quantities vanish, that is, wné'il' the two distribu:- tions are similar; wi th the increase of indices wilI follow a correspon- ding illcrease in the differences between cograduate quantities.

(1) 'l'he principie of dissimilarity has been recently recolIsidered by prof.

F.

BOA:;,

in Journal of the «American Statistical Association» (December' ]922) « The measurement or ditferences between variable quantt'ties ».

According to this author, if twoset'ies are so far apart that notwi ...

thstanding their variability they do Ilot overlap, they are elltil'ely dissimilar; if they do overlap they wil! be the more dissimilar, the less the amount of overlappillg. The minimum amount of dissimilarity is found whell the series are identical.

This mallller of conceiving dissimilarity appears to us incomplete and presents some incongruit.y: Let

IIS

e. g. consider two different groups of quan- tities which with a third group constit.ute two pairs of seI'ies the variabilities of each one of which do not overlap; then, according to Boas' s meaning these pairs of series are to be considered (lqually dissimilar.

It seems to us, on the contrary more resonable to cOllsider these pairs as presenting different dissimilarity according to differences between the correspouding quantities of each pail', that is, according to Gini' s meaning of dissimilarity.

Moreover I have to noti ce, heJ'e, that the method used by Boas did

not clearly show to me the logical way by which he has reached the for-

mula of his measure of dissimilarity.

1'wo groups being composed of a different number of qu-antities, according to the theorem of n

^r

3, we may calculate the indices of dissimilal'ity by reducing both the given groups to othel's which must be composed of the sa me number of q lIantities.

To simplify the calculation of the simple illdex of di:-:similarity it may be suggested, fir,st, to eliminate t.he qllantities which belong

"

to both the groups, then to apply (7) to the reduced groups.

'l'his method of simplification is noL suitable for the calcula- tion or the qlladratic index of dissimilarity.

It is intere8ting to notice also that presenting in a system or orthogonal coordillates the quantities of the two groups, if the curves of frequency meet

^III

no more thall one pojnt, the

~imple

index of dissimilarity corresponds to the diffel'ence Letween the ari- thmetical means of the groups; in any othe1' cases the index will be higher than the said differ'ence.

6. - Dissimilarity between the distribution of sizes or recruits according to the terdtorial divisions ol the Kingdom of ltaly and the whole Kingdom.

, As an application of the indices of dissimilal'ity let us confl'ont the distribution of sizes of recruits according to the territorial di- visions of the Kingdom of Italy to that of the whole Kingdom.

Then, it will be shown how actually the calculation of the above indices may be effected. First of all we have ,to notice that the originaI seriations, given by the Ant.hropometry of LIVI(l), present different totais foreach territorial divisioll and for the whole King- domo Therefore it wilI be necessary to substitute, first of all, to the originaI seriations similat' ones having the same total. For this purpose we may substitute to LIYI 's table another one in which the ftgures of each territorial division given by LIVI, are multi- plied by the total of the Kingdom and divided by the totai of the territoriai divisioll which has been considered.

Rere we l'eproduce the complete method for the calcùlation of the simple index of dissimilal'ity between the distributioll 01 the territorial division of Piedmont alld that of the Kingdom.

In the table which follows the 2

^d

and 3

^d

columns contain the original seriations, the 4

^th

one the seriation similar to that one of the 3

^d

column, obtained multiplying this last one by ~:::15

(1) cfr. Dr.

RODOLFO

LI

VI

Antropometria Militw'e - Roma, presso il

Giornale Medico del R. Esercito - 1896.

390

TABLE

originaI seriations positive differeIices between

- - -

Similar

I

seriation to

the:2<1 and the Ithe4

^{t h}

^{and the}

Sisez that one or the

Kingdom

I ^Piedmont sa column

.", columoo I "" colum ••

I

1$t ' 2^d

I

,3,1 I 4th r)th 6^th

I

154 202 I ^{24 214}

^-

¹²

155

I

2658 I

200

1785 873

^-

156 10219 833 ' 7434 2785

^-

157 11907 1155 10308 1599 -

158 14085 1384 12352 1733

^-

159

!

15473 1555 13878 1595

^-

160 19748 2097 18716 1032

I

-

161

^I

19484 2045 18252 1232

^-

162

ⁱ

22268 2408 21491 777

^-

163

ⁱ

21700 2411 21518 182 -

164

I

21436 2483 22161

- 725

165 ! 21917 2524 22527

-

610

166

I

19472 2271 20269

^-

797

167 17798 2182 19474

^--

1676

168

^I

15649 1913 17074 1425

1

-

169 12558 1591 14200

^-

1642

170

I

12428 1492 13316

-

888

171 _l 9276 1213 10826

^-

1550

172

^I

7672 952 8497

-

825

173

I ⁵⁶⁵⁰

I 708 6319

^-

669

174 4488

I

I 565 5043 - 555

175

[

3818

I 468 4177

-

359

176

I

2898 294 2624 274

^{- -}

177 2066 I 254 2267

-

201

178

^!

1522

I 198 1767

^-

245

179

_I

1005 127 1133

^--

128

180

I

714

!

67 598

I ^{116 -}

181

I

414 46 411 3

^-

182

I

298

I

28 250 48

^-

183 184 24 214

^-

30

184

ⁱ

130

!

14 125 5

^-

185

^!

64 2 18 46 -

186

I

48

^I

5 45

^{I I}

3 I

^-

187 42

I

3 27 15

188

i

29 2 18 11

^-

189 13

^I

2 18

^-

5

190

ⁱ

8

^I

1 9

^-

1

191

ⁱ i

5

1 - -

5

I -

192 I 5

^!

-

^-

5

I

^-

193

^I _I

2

^{- -}

:2

_I -

194

_I

1

^{- -}

1

I

^-

195 I 1

I

- -

1

_I ^-

---1

^~-- _ _ _ _ _ 1 _ _ _ _ _ '

I

1

^I

I

'l'olals

I

299355

^I

33541 I 299355 12343

i ¹²³⁴³

I I I

Calculating the differences between cograduated sizes of 5

^th

a~d 8

^th

columns we shall ohlain the following results:

12 x I 154 - 155 I = 725 x I 164 - 155 I =

136 x I 165 - 155 I =

474: x I 165 - 156 I = 797 x

I

166 - 156

I

= 1514 x I 167 - 156 I = 162 x I 167 - 157

!

=

1425 x

I

168 - 157

I

= 12 x I 169 - 157 I =

1630 x I 169 - 158 I =

103 x I 170 - 158 I =

785 x I 170 - 159 I =

810 x I 159 - 171 I = 740

X I

171 - 160

I

= 292

X

I 172 - 160

I

= 533

X ,

172 - 161

I

=

669

X

I 173 - 161 I =

30

X

I 174 - 161 I =

525

X I

174 - 162

I

= 252

X I

175 - 162

I

=

107

X I

163 - 175

I

= 75 X I 163 - 177 I =

12 6525 1360 4266 7970 16654 1620 15675 144 17930 1286 8635 9720 8140 3504 5363 8028 390 6300 3276 1284 1050

126 X I 177 - 176 I = 126 148

X I

178 - 176

I

= 296 97 X I 178 - 180 I =" ¹⁹⁴

19

X I

180 - 179

I

= 19 3

X I

181 - 179

I

= 6 48

X

I 182 - 179

I

= 144 5 X I 184 - 179 I = ²⁶

46

X I

179 - 180

I

= 276 3

X ~

186 - 179

I

= 21 4

X I

187 - 179 I = 32 11

X I

187 - 183

I

= ⁴⁴

11 X I 188 "'- 183 I = ⁵⁶

5

X I

191 - 183

I

= ⁴⁰

3

X

I 192 - 183

I

= ²⁷

2

X

I

192 - 189

I

=. 6

2 X I

193 - 189

I

= ⁸

1

X I

194 - 189

I

= 5 l X I 195 - 190 I = ⁵

Total 130911

Then, the simple index of dissimilarity wiU be according to (7) 130.911

D == 299.355 = 0,437

In l'able II we give the indices of dissimilarity fol' all territo·

riaI divisions of the Kingdom of ItaIy and also we compare them with the deviation of the aritmetical means of each territorial division from that of the Kingdom.

l'he diffel'ences between these deviations and the indices of dissimilarity are very small ^01' niI; we wilI see a liftle further that the above results are in relation also with a gl'aphic presentation of the dissimilarity.

As we have aIl'eady noticed, for the calculation of the qua- dratic index of dissimilarity, the elimination of the q llantites which be long to both the grOl1ps cannot be effected, then to establish, e. g.

the average difference between the cograduate sizes of Piedrnont and Kingdom, we have io consider in l'able I coIurnns 4

^th

anò 2

^d•

l'hus, we shall obtaill the results of Table lI bis •

TABLE II

Deviation Simple index Average from the

Territorial division

mean of the of size Kingdom Dissimilarity

Piedmont

\

164.947 0.431 0.437

Liguria 165.521 1.005 1.009

T,ombardy 165.347 0.831 0.831

Venetia 166.553 2.037 2.037

Emilia 165.294 0.778 0.778

Tuseany • 165.647 1.131

I

1.132

The Marches 163.819 0.697

I

0.697

Umbl'ia 164.226 0.290 0.318

Lazio. 164.254 0.262 0.280

Abruzzi nnd M. 163.167 1.349 1.349

Campania 163.505 1.011 1.011

Apulia 163.500 1.016 1.017

Basil:eata. 162.581 1.935 1.937

Calabria. 163.121 1.395 1.395

Sieily. • 163.524 0.992 0.992

Sal't1'inia • . . ^{161.893 2.623 2.625}

Mean of the Kingdom 164.516

TABLE IIbis

Absolute values Squal'es ofdif- Absolute values

s~uaresofdif-

Frequen- of differences ferences be- Freguen- of differences erences be- cies between cogra- tween cogra- Cles between cogra- tween cogra-

duate sizes duate sizes duate sizes duate sizes

12 I ^{1 1 75 1 1}

861 I

1

I

1 349 1 1

3646 1 1 148 1 1

5245 1 1 97 1 1

6978 I ^{1 1 225 1 1}

8573 1 1 109 1 1

9605 1 1 106 1 1

10837 1 1 58 1 1

11614 1 1 88 1 1

11796 1 1 64 1 1

11071 1 1 19 2 4

10461 1 1 18 1 1

9664 1 1 34 1 1

7988 1 1 19 1 1

6563 1 1 8 1 1

4921 1 1

⁸

1 1

4033 1 1 5 2 4

2483 1 1 5

^I

2 4

1658 1 1 2 3 9

989 1 1 1

⁴

16

434

I

1 1 1 5 25

In Table III we present the simple and quadratic indices of dissirriilal'ity betweell si zes of recruits of each territorial division and the whole Kingdom :

TABLE

III.

Indices of dissimilarity Territorial division

Simple Quadratic

-

Pipdmont. 0.437 0.662

Liguria 1.009 1.073

LOJIlbardy 0.831 0.912

V ent-.. tia . 2.037 2.053

Emilia. 0.778 0.884

Tuscany 1.132 1.418

'l'hp Marches 0.697 0.884

Umbria 0.318 0.568

Lazio 0.280 0.530

Abruzzi and M. 1.349 1.484

Campania. 1.011 1.140

Apulia 1.017 1.152

Basilicata . ^{1.937 2.084}

Calabria 1.395 1.540

Sicily 0.992 1.081

Sardinia 2.625 2.797

6. - Simple and quadratic indices of dissi'milal'Uy between the total distribution or the incomes of the Australia in 1915 and those or the incomes classified according to classes or assets. (1)

The following talJles give aH example of' the method that we have followed in calculating simple and quadratic indices of dissimilarity between the total distribution of the incomes of Au- stralia iII 1915 alld those of the incomes

cla~sified

according io classes of assets.

The example refers, particularly, 1.0 thè 5

^th

class of assets, cor- responding io an asset from L. 500 to L. 750.

(1)

^dI'.

1'he Private Wealth or Australia and lts Growth. Common-

wealth Bureau of Cellsus and Statistics - Melbourne 1917.

394

TABLE IV.

1 2 3 4 5

Number of Number of Similar seri- INCOME Average persons person in the ation to that Groups income for aH values asset group Oll.e of

of assets L 500 and column (

-

-,-

under L.750

Deficit anò nil 315.936 7.365 158.084

unuel' L. 50

^~

24.33 447.105

I

27.980 601.305

L. 50 and under L. 100 72.03 495.941

^I

16.857 362.266 ,. ₁₀₀

_» lt

150 122.40 501.124 17.633 378.943

,. ₁₅₀

1> lt

200 168.30 220.328 14.061 302.180

» 200

^{~ lt}

300 237.21 117.325 11.597 249.224

»

300

^{l> lt}

500 374.31 55.725 4.906 105.431

,. _{500 »}

_»

_{750 603.48 18.619 1.072 23.037}

,. ₇₅₀

l>

» 1000 853.15 7.458 297 6.382

» 1000

»

» 1500 1.213.20 5.838 153 3.288

»

1500 ,.

_»

_{2000 1.725.10 2.496 39 838}

}) 2000 ,. ,. 3000 2.431.37 2.024 25 537

»

3000 » » 4000 3.429.05 761 14 301

»

4000 » » 5000 4.488.83 433 2 43

»

5000 and upwarus 9.562.79 832 4 86

Mean and Total

109.60 2.191.945 101.996 2.191.945

Absolute and square values 01 the dilferences bet'ween cograduate te1'ms or the seriations contained in columns 4

^th

and 5

^th

or

TABLE

IV.

Frequencies Values Square values

157.852 24.33 591.95

3652 47.70 2.275.29

137.327 50.37 2.537.14

259.508 4590 2.106.81

177.656 68.91 4.748.59

45.757 137.60 18.93376

3.949 22867 52.289.97

7.458 249.67 62.335.11

909 609.72 371.758.48

4.929 360.05 129.636.00

1.453 871.95 760.296.80

1.043 511.90 262.041.61

2.024 1.218.07 1.483.694.52

221 2.215.85 4.909.991.22

540 1.703.95 2.903.445 60

298 2.763.73 7.638.203.35

135 2.057.56 4.223.553.15

402 7.131.52 50.858.577.51

301 6.133.74 37.622.766.39

43 5.073.96 25.745.078.08

l'he total of absolute value of the above ditferences, in round numbers, is

57.934.000

then the simple index of dissimilal'ity will be : D - 57.934.000 - ? - 69

- 2.191.945 - _I, '-

l'be total sq tiare value of the same differences is:

47.010.175.000

Hence, the quadl'atic index of dissimilal'ity will be:

2D == l / 47.010.175.000

V ^{2.191.945 146.45}

In the same manner we ha ve calculated the simple and q ua- dratic indices of dissimilal'ity oetween the total distributioll ol' t.he incomes of Australia dUl'illg 1915 aliti

I

hose of the incomes clas- sified accol'dillg 1.0 classes ol' assets.

TABLE

V.

J:..verage Deviation Indices

Assets lncomes of the of dissimilarity

corl'espon-

^~vprllge

ding to lncomes the assets from the

Groups Averages groups generaI simpie quadratic mean

l

Defìcit and nil . 0.- I 88.90 20.70 34.13 189.51 under 100 l

J.

30.- I ^{72.60 37.- 37.47 217.55} 100 and under 250 160.-

I

93.70 15.90 21.46 19404

250

»

500 354.- 106.02 3.58 2t.85 17474

500

»

750 608.- 120.70 1110 27.69 146.45

750

^lO

1.000 863.- 135.90 26.30

I

3841 ]28.02

1.000 ^lO

2.500 \ 1554.- 172.20 6:2,60 64.18 150.42 2500

li>

5.000 3.460.-

^l i

272.80 163.20 16430 237.H3 5.000

»

10000 6.874.- I ^{461.60 352,- 35841 501.74}

10.000 »

15.000 12.077.- I 74560 636.- 636.23 710.-

15.000

»

20.000 17 246.- 11.079.17 96957 969.79 1.413,20 20.000

" ^{25.000 22.405.-} 1.487,90 1,37830 1,37846 2.095.10 25.000

»

50.000 34098.- 2.11130 2.001.70 2.00170 2863.70 50000

^l>

75.000 60,778.- /3.58430 3.474.70 8.475.20 4.67250 75.000

»

100.000 85.8tl3 - 4.934.40 4824,80 4,82480 6.04980 100.000

»

upwards 197.693.- I 6.920.70 6.811.10 6.811.94 7.722.20

Means _555.- 109.60

I

7. - Graphic presentation - ^III pIace of the arithmetical methorl we may apply to the calculatioll of the indices of dis- similal'ity the method of graphic preselltation according 'to Prof.

GIN!' s suggestiono

Take e. g. the data contained in the columns 3 and 4 01' Table IV; then make, for each one, the followillg sums, where the r

^th

total is

o~tailled

adding to the preceding ones the r

^th

term t'espe- ctively of columns 3 and 4, so the Iast total will be equal to the sum of ali terms which ha ve been eonsidered:

TABLJ;J

Vbis

Surns of each terrn of colurnn 3 (Table

IV)

with th0 preceding ones '315.936

315.936

+

763.041

+

1.258.982 + 1.760.106 + 1.980.434 + 2.097.759

+

2.153.484 + 2.172.103 + 2.179.561 + 2.185.399 + 2.187895 + 2.189919 + 2.190.680

+

2.191.113 +

315.936 447.105 = 763.041 495.941 = ^1.258.982

501.124 = ^1.760106

220.328 = 1. 980 .434 117325 = 2.097.759 55.725 = 2.153.484 18.619 = 2.172103 7.458 = 2.179 561 5.838 = ^{2.185 399}

2.496 = 2.187.895 2.024 = 2.189 919 761 = 2.190680 433 = 2.191.113 832 = 2.191.945

Surns of each terrn ofcolurnn

4,

(Table

IV)

with the preceding ones 7.356

7.356

+

35.336

+

52.193 + 69.826

+

83.887 + 95.484 + 100.390

+

101.462' + 101.759

+

101.912 + 101.951 + 101.976 '+

101.990

+

101.992 +

27.980 = =

16.857 =

17.633 =

14.061 =

11 597 = 4.906 =

1.072 = 297 =

153 =

39 =

25 =

14 =

2

= 4 =

7.356 35.336 52.193 69.826 83.887 95.484 100.390 101.462 101.759 101.912 101.951 101.976 101.990 101.992 101.996 Let us now divide the totals, which have been ot;tained from column 3, by 2.191.945 and those obtained from colult11l4, by 101.996.

Both the seriatiolls so ollfained will be, therefol'p, enclosed in an interval (0,1).

'l'hell, ill an orthogonal system of cool'dinates, let us present

011

the axis of y the aver:lge iilcollles cOIlt.ained in the coluUln 2 of Table IV, and on the axis of x the values j\lst calculated.

Considerillg e. g. the indicated preselltation of the seriation which refers to thai of table V

^bis

to the left., let:

Pv~l

be the point in which two cOl'l'espolldillg cool'dinates meet and let us extend the ordi- nare to cut the abscissa of the subseq tlent poillt pv . 'l'hen, the l'eetan- gle dètel'mined by the diffel'ence be/,ween the abscissae of pv-l and Pv and the ordi nat.a of Pv will l'epresen t the portion of i Ilcomes, which belollgs 1.0 the tl'act.ioll of persons eont.ained in th8

Vth

class.

Afterthat, it will be easy to conclude that, by tlte sum of the

area' of alI analog-ous rectangles obtained lnakillg suecessively

v == l, 2, ... 15, we shall obtain the total amount of the

illCo-

397 mes divided by the total number of persons to which it belongs, that is, the arithmetical mean of the incomes.

This will be equally true of the data contained in

tabl~

vòis

to the right.

Thus, it will be easy to construct two graphs, each of which determines with the axis of x and the extreme ordinates an area cOl'responding to the arithmetical mean of the considered

seri~tion.

Since, now, the ordinates of both the graphs are ordereà ac- cording to an increasing order of succession, two ordinates, corre- sponding to the same abscissa, will be cograduate. Thel'efore, it will be easy to see that the differences between cograduate ordinates are contained in the area enclosed within the graphs and the extreme ordinatés, so it gives the index of dissimilarity between the two considered groups. (See fig. 1

^st,

in which only the centraI part of the graphs is represented).

Then, we may conclude that, if the above graphs .do not meet in any point, the index of .dissimilarity wlli be equal to the difference between the arithmetical means of the originaI seriations; if, in- stead, the above graphs meet in one or more points, then the index of dissimilarity will be higher than the difference between the said means.

In l'able VI we compare the results obtained by the graphi- cal and arithmetical calculations for the simple indices of dissimi-

larity of the seriations of Australia.

TABLE

VI

Simple index DitTerences between the gra1?hic and the of dissimilarity arithmetlCal methods Assets groups

by arithmeti- by graphic absolute

^%

cal method method value

Deficit and ni! 34.13 34.91 +' 0.78 2.3

under L. 100 37.47 37.55 + 0.08 0.2

100 and under L. 250 21.46 20.90 - 0.56 2.5 250 " " ^.500

^~21.85

^{22.- + 0.15 0.7}

500

_~~

750 27.69 27.55 - 0.14 0.5

"

750

_H

" ^{1.000 38.41 38.90 + 0.49 1.3}

1.000 2.500 64.18 75.- + 0.82 1.3

"

^,~

2.500

_{~~ ~~}

5.000 164.30 166.- + 1.70 1.0

5.000 10.000 353.41 360.- + 6.59 1.9

l ' ^~,

10.000 15.000 636.23 635.- - 1.23 0.2

" "

15.000 20.000 969.79 972.40 + 2.61 0.4

" "

20.000 25.000 1378.46 1.366.20 - 12.26 0.9

" "

25.000

,~

,,50.000 2.001.70 ' 2.006.20 + 4.50 0.2 50.000 " ,,75.000 I ^{3.475.20 3.483.80 + 8.60 0.2}

75.000 "

~,

100.000 4.824.80 4.843.60 + 18.80 0.3 100.000 and upwards . . 6.811.94 6.855.80 + 43.86 0.6

Metron - V,ol. IV, n. 3-4 2

398

r- - --~-- -

F'

4 ,

F

I I

J

• p\ l"

,-f ^o'

rA' 1'1

I

t"

,_ ... _ _ _ _ _ 1.--_...:...-_ ',r--

-1-](-

^-"/l'

, I

^{l '} L.

X' K ..

•

{OOO

100

399 The differences between the graphic and arithmetical methods are very small, so that both· the methocfs may be considered ap-

plicable. .

\Ve have to notice that both the arithmetical and graphic calculations assume in our particular case that alI the quanti ties contained in a class of assets

Ol'

incomes are equal to the corre- sponding mean of the considered class. In 'rabIe IV e. g. the num- bers of persons contained in cols. 3 and 4 will be consider,ed as "' fl'equencies of the corresponding average of incomes (col. 2).

It is evident that different distributions within the classes may oceur.

Let ua consider, e. g. MI' M

2 . . . .

and M~, M~ ... the mid- dle points of AB, CD .... respectively A' B', C' D' . .. of Fig. l;

Construct, then, the graphs MI M2 ... and. M~ M~ ... (see Fig. 2 in which onIy the centraI part of the graph is represented). In this case, therefore, we assume that the distributions within consecutive avel'age incomes are in arithmetical progression.

The caleulation of the index of dissimilarity by the graphic method gi ves in this case:

D == 30,00 in pIace of

D == 27,69

obtained according to the preeeding hypothesis that each incorne of a elass is equal to its mean.

Analogously, caleulating the index of dissimilarity between the total distribution of ineomes and that corresponding to the highest group of assets, that is L. 100.000 and upwar;'ds, we shall find

D == 6680,00 in pIace or

D == 6811,94 obtained by the former hypothesis.

Still speaking of graphic pl'esentation I desil'e to point out that

we may refer to it also for the caleulation of

th~

quadratic index

of dissimilarity. Keeping for x the sétme quantities which have been

calculated in the case of the simple iodex of dissimilarity, we

shall ha ve only to take for y the sq uares values ol the differen-

400 .

_f'j~o~"

_

".:

... ----t'l)::-.--- -- ^i---

I I I I~

I

,o

{

...

IAtI,' }Cj

, f

"',·0 "'i

,})~ l:

... i- _ - - -

~~

.... , -_-..;;0:..-_ _ _ _ - ' r,~ ^-

/II"

- ".

401 ces between the ~orresponding ordinates in the graphs of simple index. We -inay, then, construct a new graph and the area enclo- sed by it, the axis of x and the extreme ordinate gives the mea- sure of the required qua-dratic indeX' of dissimilarity.

8. - lndices of dissimilarity between deviations and 1'a- )'iations.

Let be

1 n

A(t ==-~k alt;

n

1

\

(9)

the arithmetical mean of the n intensities of a charader A; then the value

(lO) wiU be called the (positive or negative) deviation of the intensity

alt; from thè arithmetical mean.

If

la =! ~.I ^{a. -} A.I = ,~ ~.ll··1 ^(lI)

then, the val ue

(12) will be called va1'iation of A in the k

^th

case.

In pIace of the arithmetical mean we may use tbe quadra/ic ,nean ^or themedian.

It is easy to see that the simple index of dissimilarity be- tween two groups of deviations will be:

( 13)

The quadratic index:

(14)

"

402

Thus, the simple index of dissimilarity between two groups of va- ria ti ons wi 11 be:

(15) the quadrati c one:

1/ ^{l n (} )2

:ld == V n 1

^k V ah -1'bh

(16)

We shall have occasion to give further applications of the meaning and formulae or ^the deviations and vw'iations when speaking of the concordance.

9. - According to the contents of this paragraph t\lld n

^r

l and 2 of § I we infer that the indices of dissimilarity should be applied only to the relations between series of quantitative cha- racters or of those to the modalities of which a succession of

nu~bers

always corresponds.

1'herefore, the same method ii is not 'available for the cy- clical series; so that we could not yet establish, e.

g~,

the dissimÌ- larity of the distribution of we~,~i.l!g~ accordipg to the days of the week for differeni countries, or periods _9f time; of births of the mother and sons according to the months, or seasons; of ag-ri- cultural l'otations in different periods of tirne or different regions, and so ono

As we alreatfy have noticed, it is the object of the first part of the present work to state the theory of the dissimilarity betweell cyclical series. Let us consider in the following paragraphs the

m~aning

of the connection and concordance between rectilinear and disconnected ijeries.

§ III

l'O. - Meanings of connection and concordance - Under the common term c01"relation two very different meanings have frequently been considered, connection and conco}·dance.

Two characters may be regarded as connected each wi th the

other when the distribution of the modalities of the first one de-

pends on the modalities of the other. Thus, there wilI be connection

of distribution of incomes of families to their assets, when it will

be possible to find out that the said distribution varies in fun- dion of the assets. In the same way, we may sày that there is connection between social class and profession, between size and profession etc. Moreover, we may investigate in which direction modalities of two. connected chal'aèters are assocja ted, that is, if the modalites of the first character are more frequently associated to the concordant modalities, or to the discordant ones of tQe se- , cond

characte~

By· this method of research we may discover the exact meaning of concordance. E. g. let connection lJetween incomes and assets be known: we may then proceed to investigate the movement 'of income when assets increase or decrease.

It is· clear that there is not concordance if connection does not exisL

l'he concordance is, therefo1'e, a research subsequent to that of the connection. In the present paragraph we shatl deal.with the connectioll, while the concordance will be studied in the next one.

First of alI let us recall· some formulae of the pl'incipal indices of variability, to which we will refer during the COUI'se of this pa-

l'agraph.

Il. - lndices of variability. - Among the indices which may be employed to measure the variability of statistical chara- cters the simple and quadratic mean de1)iation (rom the arithme- tical lnean and the mean diffeJ"ence are of particular interest to us at present.

Let tlS indicate, respectively, by 18 and 28 the simple and the quadratic mean deviation from the arithmetical mean, that is:

18 = !: I a - ai I (17)

n

'8 = V ^{~ (a: - a)' (') (18)}

a being the arithmeticalmean ofag1'oup ofquantitiesa

i

(i==1,2, ... n).

1'he arithmetical mean of absolute values of all differences, which may be calculated among the terrns of a series is. called mean difference.

Let us, in this calculation, compare a tel'm also with itself, then we shall have the 1nean differ'ence with J"epetition; leaving out the preceding comparison we shall obtain the simple Jnean difference./

(l) This expression is also known as the standard deviation.

.,

.

f):

404

Representing by A the sirnple mean dift'erence anrl by A

R

that with l'epetition, whe shall h'ave evidently:

n - l A R'== - - - L1 "

n

n- being the numher of terms. of the series.

Let be

<a

_ n

n ordered quantities; make the sums :

Sn

= al + al . + ^a

n

S ==

S}

+

^S%

+ . . . +

^Sn

then we shall ha ve :

S'l = a

n

S' n

= a

n

+ ^a

^{n - l}

S'

ⁿ

== a

n

+ a

n ~ l • • •

+ al S' --

S' l

+

^{S' 2}

+ . . . +

^{S' n}

L1 == 2 (S' - 8) . n (n-l)

(19)

(20) The mealling of the mean d"ifference· may be· genel'alised by the form~la:

m L1

R

= l /_' ~_r_, -,--a_s_-_a_·

^r

-!.'_1_n'_

V ^{nnt (21)}

Then, put m = l we shall obtain, again, the sin~ple mean dir ference with repetition; put m = 2 we shall obtain the quadratic mean ditference with repetition, and so ono

It is demonstrate that it is .

(22) The methods used for the calculation of 18 and %S are very weIl known, so that it does not seem to us necessary to' descriL>e thern.

Let ·us onIy consider, therefore, an appIication 9f calculating the mean ditference.

To calculate practical1y the mean diffet'ence, when many of

ai quantities have the same value, or when the intensities of a cha-

racter are grouped in more orless extended c]asses, it is conve-

p;,ent to proceed according to the following example, which re-

fers to the mean ditference of the distribution 01' the incomes

of Australia in 1915 and which is self. explanatory.