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Biometrics & Biostatistics International Journal

Research Article Volume 10 Issue 2

AHM as a measure of central tendency of sex ratio

Dhritikesh Chakrabarty

Department of Statistics, Handique Girls’ College, Gauhati University, India

Correspondence: Department of Statistics, Handique Girls’ College, Gauhati University, India

Received: May 03, 2021 | Published: May 25, 2021

Citation: Chakrabarty D. AHM as a measure of central tendency of sex ratio. Biom Biostat Int J. 2021;10(2):50-57. DOI: 10.15406/bbij.2021.10.00330

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Abstract

In some recent studies, four formulations of average namely Arithmetic-Geometric Mean (abbreviated as AGM), Arithmetic-Harmonic Mean (abbreviated as AHM), Geometric-Harmonic Mean (abbreviated as GHM) and Arithmetic-Geometric-Harmonic Mean (abbreviated as AGHM) have recently been derived from the three Pythagorean means namely Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM). Each of these four formulations has been found to be a measure of central tendency of data. in addition to the existing measures of central tendency namely AM, GM & HM. This paper focuses on the suitability of AHM as a measure of central tendency of numerical data of ratio type along with the evaluation of central tendency of sex ratio namely male-female ratio and female-male ratio of the states in India.

 Keywords: AHM, sex ratio, central tendency, measure

Introduction

Several research had already been done on developing definitions/formulations of average,1,2 a basic concept used in developing most of the measures used in analysis of data. Pythagoras3 is the first mathematician to introduce the concept of average and develop its formulation. He had developed three formulations/definitions of average which were later named as Pythagorean means4,5 as a mark of honor to him. The three Pythagorean means are Arithmetic Mean (AM), Geometric Mean (GM) & Harmonic Mean (HM). A number of definitions/formulations of average have already been developed in continuation to the three Pythagorean means.6-19 The next attempt had been initiated towards the development of generalized formulation/definition of average. Kolmogorov20 formulated one generalized definition of average namely Generalized f - Mean.7,8 It has been shown that the definitions/formulations of the existing means and also of some new means can be derived from this Generalized f - Mean.9,10 In an study, Chakrabarty formulated one generalized definition of average namely Generalized fH – Mean.11 In another study, Chakrabarty formulated another generalized definition of average namely Generalized fG – Mean12,13 and developed one general method of defining average15-17 as well as the different formulations of average from the first principles.19

In many real situations, observed numerical data

x1,x2,.........,xNx1,x2,.........,xN

are found to be composed of some parameter µ and respective errors

ε1,ε2,......,εZε1,ε2,......,εZ

usually of random in nature i.e

xi=μ+εi  ,  (i= 1 , 2 ,  ,N)xi=μ+εi  ,  (i= 1 , 2 ,  ,N) (1.1)21-29

The statistical methods of estimation of the parameter developed so far namely least squares estimation, maximum likelihood estimation, minimum variance unbiased estimation, method of moment estimation and minimum chi-square estimation,31-52 cannot provide appropriate value of the parameter µ.21-23 Therefore, some methods have recently been developed for determining the value of parameter µ in the situation mentioned above.21-30,53-60 These methods, however, involve huge computational tasks. Moreover, these methods may not be able to yield the appropriate value of the parameter if observed data used are of relatively small size (and/or of moderately large size too) In reality, of course, the appropriate value of the parameter is not perfectly attainable in practical situation. What one can expect is to obtain that value which is more and more close to the appropriate value of the parameter. Four methods have therefore been developed for determining such value of parameter. These four methods involve lighter load of computational work than respective load involved in the earlier methods and can be applied even if the observed data used are of small size.61-64 The methods developed are based on the concepts of Arithmetic-Geometric Mean (abbreviated as AGM),61,62,67,68,69 Arithmetic-Harmonic Mean (abbreviated as AHM),63 Geometric-Harmonic Mean (abbreviated as GHM)64 and Arithmetic-Geometric-Harmonic Mean (abbreviated as AGHM)65,66 respectively. Each of these four formulations namely AGM, AHM, GHM & AGHM has been found to be a measure of parameter µ of the model described by equation (1.1). In other words, each of these four formulations can be regarded as a measure of the central tendency, in addition to the usual measures of central tendency namely AM, GM & HM of the observed values x1,x2,.........,xN,x1,x2,.........,xN, since the values can be expressed by the model (1.1) if µ is the central tendency of them and vice versa. However, for different types of data different measures are suitable. This paper focuses on the suitability of AHM as a measure of central tendency of numerical data of ratio type along with the evaluation of central tendency of sex ratio namely male-female ratio and female-male ratio of the states in India.

Four formulations of average from pythagorean means

Let

x1,x2,.........,xNx1,x2,.........,xN

N positive numbers or values or observations (not all equal or identical)

and

a0=AM(x1,x2,.........,xN)=1NNi=1xi,a0=AM(x1,x2,.........,xN)=1NNi=1xi, (2.1)

g0=GM(x1,x2,.........,xN)=(Ni=1 xi)1/Ng0=GM(x1,x2,.........,xN)=(Ni=1 xi)1/N (2.2)

 & h0=HM(x1,x2,.........,xN)=(1NNi=1x1i)1h0=HM(x1,x2,.........,xN)=(1NNi=1x1i)1 (2.3)

i.e.a0, g0a0, g0 & h0h0 are respectively the Arithmetic Mean (AM), the Geometric Mean (GM) & the Harmonic Mean (HM) of x1,x2,.........,xNx1,x2,.........,xN

which satisfy the inequality 4 ,5namely

 AM > GM > HM i.e. h0< g0<a0h0< g0<a0 (2.4)

 Arithmetic-geometric mean (AGM)

 The two sequences {an}&{gn}{an}&{gn} respectively defined by

an+1=12(an+gn)an+1=1/2(an+gn) (2.5)

 & gn+1=(an gn)12gn+1=(an gn)1/2 (2.6)

where the square root assumes the principal value,

converge to a common point MAGMAG which can be termed as the Arithmetic-Geometric Mean (abbreviated as AGM) of x1,x2,.........,xNx1,x2,.........,xN 61 , 62, 66 , 67 , 68.

Thus,

 AGM(x1,x2,.........,xNx1,x2,.........,xN ) = common converging point of {an}&{gn}{an}&{gn} (2.7)

Arithmetic-harmonic mean (AHM)

The two sequences {a/n}&{h/n}{a/n}&{h/n} respectively defined by

a/n+ 1=½(a/n + h/n) (2.8)

h/n+ 1= ½(a/n  1  + h/n   1)}  1 (2.9)

converge to common pointMAH which can be termed as the Arithmetic-Harmonic Mean (abbreviated as AHM) of x1,x2,.........,xN .63,66

Thus,

 AHM (x1,x2,.........,xN ) = common converging point of {a/n} & {h/n} (2.10)

Geometric-harmonic mean (GHM)

The two sequences {g//n} & {h//n} defined respectively by

 g//n+1=(g//n. h//n)12 (2.11)

 & h//n+1={12(g//n1+h//n1)}1 (2.12)

where the square root takes the principal value,

converge to common pointMGH which can be termed as the Geometric-Harmonic Mean (abbreviated as GHM) of x1,x2,.........,xN .64,66

Thus,

 GHM(x1,x2,.........,xN ) = common converging point of {g//n} & {h//n} (2.13)

Arithmetic-geometric-harmonic mean (AGHM)

The three sequences{a///n},{g///n}&{h///n} defined respectively by

a///n=1/3(a///n1+g///n1+h///n1), (2.14)

g///n=(a///n1  g///n1  h///n1)1/3 (2.15)

 & h///n={1/3(a///1n1 + g///1n1 + h///1n1)}1 (2.16)

converges to a common limit MAGH which can be termed as the Arithmetic-Geometric-Harmonic Mean (abbreviated as AGHM) of x1,x2,.........,xN .65,66

Thus,

 AGHM(x1,x2,.........,xN ) = common converging point of {a///n},{g///n}&{h///n}, (2.17)

AHM as measure of central tendency of sex ratio

Let

x1,x2,.........,xN

be observed values (which are strictly positive and not all identical) on the Ratio Male/Female.

Also let µ be the central tendency of the observed values.

Then xi can be expressed as

xi=μ+εi (3.1)

where εi is the error associated to xi for (i = 1 , 2 , ………… , N) which is random in nature

 i.e. each εi assumes either positive real value or negative real value with equal probability.

Again since µ is the central tendency of the observed values

x1,x2,.........,xN

therefore, µ  1   will be the central tendency of reciprocals

x11,x12,.........,x1N

of the observed values.

Accordingly, the reciprocals can be expressed as

xi1=μ1+εi/    ,    (I= 1 , 2 ,  ,N)  (3.2)

where

ε1/,ε2/,........εN/

are the random errors, which assume positive and negative values in random order, associated to are the random errors associated to

x11,x12,.........,x1N respectively.

Let us now write

 AM(x1,x2,.........,xN ) = a0 (3.3)

 & HM(x1,x2,.........,xN ) = h0 (3.4)

and then define the two sequences {a/n} & {h/n} respectively by

a/n+ 1 = ½ (a/n + h/n) (3.5)

h/n+ 1 = ½(a/n  1  + h/n  1)}  1 (3.6)

Then, both of {a/n} & {h/n}  converges to some common real numberC.

Let us now search the relation betweenC and µ.

Equation (3.1) together with (3.3) & (3.4) implies that a0=μ+δ0  &  h0=μ+e0

By inequality (2.4), h0<a0 i.e. e0<δ0 

Therefore, a/1 =µ+δ1  where δ1 =½(δ0+)e0

Since ½(δ0+e0)<δ0

Therefore,δ1<δ0

At the nth step, one can obtain that

δn+1=½(δn+en)<δn

which implies, δn+1<δn since½(δn+en)<δn

This implies, δn becomes more and more smaller as n becomes more and more larger.

This means. a/n becomes more and more closer to µ as n becomes more and more larger.

Since {h/n } converges to the same point to which {a/n } converges,

Therefore,  also becomes more and more closer to µ as n becomes more and more larger.

Accordingly, the AHM(x1,x2,.........,xN ) can be regarded as the value of  i.e. the value of the central tendency of x1,x2,.........,xN.

Example

Data on the population of India (state-wise) in 2011, published in “Census Report” by Register General of India, have been shown in the following table (Table 1):

State

Number of Persons

Number of Males 

Number of Females

Jammu & Kashmir

1,25,41,302

66,40,662

59,00,640

Himachal Pradesh

68,64,602

34,81,873

33,82,729

Punjab

2,77,43,338

1,46,39,465

1,31,03,873

Chandigarh

10,55,450

5,80,663

4,74,787

Uttarakhand

1,00,86,292

51,37,773

49,48,519

Haryana

2,53,51,462

1,34,94,734

1,18,56,728

Delhi

1,67,87,941

89,87,326

78,00,615

Rajasthan

6,85,48,437

3,55,50,997

3,29,97,440

Uttar Pradesh

19,98,12,341

10,44,80,510

9,53,31,831

Bihar

10,40,99,452

5,42,78,157

4,98,21,295

Sikkim

6,10,577

3,23,070

2,87,507

Arunachal Pradesh

13,83,727

7,13,912

6,69,815

Nagaland

19,78,502

10,24,649

9,53,853

Manipur

28,55,794

14,38,586

14,17,208

Mizoram

10,97,206

5,55,339

5,41,867

Tripura

36,73,917

18,74,376

17,99,541

Meghalaya

29,66,889

14,91,832

14,75,057

Assam

3,12,05,576

1,59,39,443

1,52,66,133

West Bengal

9,12,76,115

4,68,09,027

4,44,67,088

Jharkhand

3,29,88,134

1,69,30,315

1,60,57,819

Odisha

4,19,74,218

2,12,12,136

2,07,62,082

Chhattisgarh

2,55,45,198

1,28,32,895

1,27,12,303

Madhya Pradesh

7,26,26,809

3,76,12,306

3,50,14,503

Gujarat

6,04,39,692

3,14,91,260

2,89,48,432

Daman & Diu

2,43,247

1,50,301

92,946

Dadra & Nagar Haveli

3,43,709

1,93,760

1,49,949

Maharashtra

11,23,74,333

5,82,43,056

5,41,31,277

Andhra Pradesh

8,45,80,777

4,24,42,146

4,21,38,631

Karnataka

6,10,95,297

3,09,66,657

3,01,28,640

Goa

14,58,545

7,39,140

7,19,405

Lakshadweep

64,473

33,123

31,350

Kerala

3,34,06,061

1,60,27,412

1,73,78,649

Tamil Nadu

7,21,47,030

3,61,37,975

3,60,09,055

Pondicherry

12,47,953

6,12,511

6,35,442

Andaman & Nicobar

3,80,581

2,02,871

1,77,710

India

1,21,08,54,977

62,32,70,258

58,75,84,719

Table 1 Population of India in 2011 (State-wise)

From the data in the observed values on the two ratios

Male/Female & Female/Male have been computed which have been shown in Table 2.

State

Value of the Ratio
Male/Female

Value of the Ratio
Female/Male 

Jammu & Kashmir

1.1254138534125111852273651671683

0.88856201384741461016988968870875

Himachal Pradesh

1.0293088804926436613751796256809

0.97152567023553127871119940330966

Punjab

1.11718611741734676457868601138

0.89510600284914783429585712319405

Chandigarh

1.2229968385823537712700642603947

0.81766360177934533455722165869015

Uttarakhand

1.0382445737805593956494862402266

0.96316419584905755859591305415792

Haryana

1.1381499179200197558719403869263

0.878618874592118673847146598073

Delhi

1.1521304409972803426396508480421

0.86795727672502366109786158864161

Rajasthan

1.077386518469311558714857879884

0.92817200035205763708961523638845

Uttar Pradesh

1.0959666766496911194331303675474

0.91243650131493423988837726768373

Bihar

1.0894569681498644304609103396449

0.91788847952225054362107394324387

Sikkim

1.1236943796151050235298618816238

0.88992168879809329247531494722506

Arunachal Pradesh

1.0658345961198241305435082821376

0.93823188292114434272011116216004

Nagaland

1.0742210801874083323111632505218

0.93090707159232088256563955071444

Manipur

1.0150845888535768920299631387912

0.98513957455445833617176866728857

Mizoram

1.0248621894302476437945104610541

0.97574094381990099740878994632108

Tripura

1.0415856043291039214999824955364

0.96007471286444128606000076825568

Meghalaya

1.0113724418785172369610123540989

0.98875543626896326127874988604615

Assam

1.0441048168517855831597956077024

0.95775824788858682201128358123932

West Bengal

1.0526667948213744061675457587868

0.94996821873695430584361430969287

Jharkhand

1.0543346515488809532602154750904

0.9484654597389357492757813425208

Odisha

1.0216767277963741786589610810708

0.97878318336258074151514020087369

Chhattisgarh

1.0094862433659738915914763831542

0.99060289981333128651017560729672

Madhya Pradesh

1.0741921997293521487367677330733

0.93093209972289388478334723747063

Gujarat

1.0878399216924771607664276945985

0.9192528974705997791133158851059

Daman & Diu

1.6170787338884943945947109074086

0.61839907918110990612171575704752

Dadra & Nagar Haveli

1.29217267204182755470193199021

0.77389037985136251032204789430223

Maharashtra

1.0759593940486569345112623151307

0.92940310343605596519523288750508

Andhra Pradesh

1.0072027731513157131279371653056

0.99284873578258743089946488568226

Karnataka

1.0278146308628600560795309711955

0.97293808627776643762353811714322

Goa

1.0274323920462048498411882041409

0.97330005141109938577265470682144

Lakshadweep

1.0565550239234449760765550239234

0.94647223983334842858436735802916

Kerala

0.92224729321594561234305382426448

1.0843078720382305015931455433978

Tamil Nadu

1.0035802105886977594941050244168

0.99643256159206485698216349975338

Pondicherry

0.96391330758747454527714567183158

1.0374376949964980220763382208646

Andaman & Nicobar

1.1415846041303246862866467840864

0.87597537351321775906857066806002

India

1.0607325851848778252519531570732

0.94274467850509882664736426425148

Table 2 Central tendency of the ratio Male/Female

 Central tendency of the ratio Male/Female

From the observed values on the ratio Male/Female in Table 3 it has been obtained that

AM of Male/Female = 1.0835068016450523020161865887443 & HM of Male/Female = 1.0740468088974845410059550737324

n

Value  of  a/n

Value  ofh/n

0

1.0835068016450523020161865887443

1.0740468088974845410059550737324

1

1.0787768052712684215110708312384

1.0787560661660274789282541031017

2

1.0787664357186479502196624671701

1.0787664356189714883012948072843

3

1.0787664356688097192604786372272

1.078766435668809719258176146917

4

1.0787664356688097192593273920721

1.0787664356688097192593273920721

Table 3 Values of {a/n} & {h/n}  of the Ratio Male / Female
The digits in a/n & h/n , , which are agreed, have been underlined in the above table.

The following table (Table 3) shows the values of a/n  &h/n , in this case, for n = 1 , 2 , 3 , ………… :

It is seen that the values of a/n & h/n  become identical at n = 4 which is

1.0787664356688097192593273920721

Therefore, this value can be regarded as the AHM and consequently the central tendency of the Ratio Male/Female.

Central tendency of the ratio female/male

From the observed values on Female/Male in Table 3 it has been obtained that AM of Female/Male = 0.9310581175009550726813265197974 & HM of Female/Male = 0.92292913942185992242619179784686

The computed values of {a/n } & {h/n }, in this case, have been shown in the following table Table 4:

n

Value  of  a/n

Value  ofh/n

0

0.9310581175009550726813265197974

0.92292913942185992242619179784686

1

0.92699362846140749755375915882213

0.92697580733443813334996246257971

2

0.92698471789792281545186081070092

0.92698471781227076522756233102558

3

0.92698471785509679033971157086325

0.92698471785509679033773303940364

4

0.92698471785509679033872230513345

0.92698471785509679033872230513345

Table 4 Values of {a/n } & {h/n } of the Ratio Female/Male
The digits in a/n & h/n , which are agreed, have been underlined in the above table

It is seen that the values of a/n & h/n become identical at n = 4 which is

0.92698471785509679033872230513345

Therefore, this value can be regarded as the AHM and consequently the central tendency of the Ratio Female/Male.

Results and discussions

If µ is the central tendency of x1,x2,.........,xN then the central tendency of x11,x12,.........,x1N

should logically be µ  ̶ 1 . Similarly, the central tendency of x11,x12,.........,x1N should logically be  ̶ µ .

It is seen in the in the above example that the AHM of the ratio Male/Female is 1.0787664356688097192593273920721 and of the ratio Female/Male is 0.92698471785509679033872230513345

These two values are reciprocals each other i.e.

(1.0787664356688097192593273920721)  ̶ 1 = 0.92698471785509679033872230513345

& (0.92698471785509679033872230513345)  ̶ 1 = 1.0787664356688097192593273920721

Moreover, it is found that the AHM of the additive inverses of the observed values of the ratio Male/Female, is

̶ 1.0787664356688097192593273920721 and of the ratio Female/Male is

̶ 0.92698471785509679033872230513345

Thus, AHM can logically be regarded as an acceptable measure of central tendency of data of ratio type.

It is to be noted that each of AM & HM does not satisfy these two properties of central tendency and therefore cannot logically be regarded as acceptable measure of central tendency of data of ratio type.

Of course, GM satisfies the first property but not the second property of central tendency. Thus, is to be studied further on the acceptability of GM as a measure of central tendency of data of ratio type.

Regarding accuracy, it is to be noted that a0=μ+δ0  &  δn+1<δn

This means, δn becomes more and more smaller as n becomes more and more larger which means, a/n becomes more and more closer to µ as n becomes more and more larger which further means, AHM (x1,x2,.........,xN ) becomes more and more closer to µ as n becomes more and more larger.

Since δn<δ0 for all n > 1 therefore, the deviation of AHM (x1,x2,.........,xN ) from µ is more than that the deviation of  . But,  = AM (x1,x2,.........,xN )

Hence, AHM (x1,x2,.........,xN ) is more accurate measure of central tendency than AM (x1,x2,.........,xN ) in the case of data of ratio type.

Similarly, AHM (x1,x2,.........,xN ) can be shown to be more accurate measure of central tendency than HM (x1,x2,.........,xN ) in the case of data of ratio type.

Therefore, AHM can be regarded as a measure of central tendency of data of ratio type which is more accurate than each of AM and HM. However, it is yet to be studied on the comparison of accuracy of AHM with that of GM as measure of central tendency of data of ratio type.

It is to be noted that the GM of AM of the Ratio Male/Female & HM of the Ratio Male/Female is found to be 1.0787664356688097192593273920721 which is nothing but the AHM of the observed values of the Ratio Male/Female.

Similarly, the GM of AM of the Ratio Female/Male & HM of the Ratio Female/Male is found to be 0.92698471785509679033872230513345 which is nothing but the AHM of the observed values of the Ratio Female/Male.

Thus, AHM of the observed values can be regarded as the GM of AM of the observed values and HM of observed values. In general, AHM(x1,x2,.........,xN ) can be defined as the GM of AM(x1,x2,.........,xN ) and HM(x1,x2,.........,xN ) in the instant case. However, it is to be established for general case.

On the whole, the two values 1.0787664356688097192593273920721 and 0.92698471785509679033872230513345 can be regarded as the respective values of central tendency of the Ratio Male/Female and the Ratio Female/Male of the states in India which are very close to the respective actual values while the overall values of these two ratios in India (combing the states) are 1.0607325851848778252519531570732 and 0.94274467850509882664736426425148 respectively.

However, it is yet to be determined the size of errors or discrepancies in values obtained by AHM. It is also to be assessed the performance of AHM by applying it in the data with various sample sizes.

Acknowledgments

None.

Conflicts of interest

None.

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