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

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 means^4,5 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 f_H – Mean.¹¹ In another study, Chakrabarty formulated another generalized definition of average namely Generalized f_G – Mean^12,13 and developed one general method of defining average^15-17 as well as the different formulations of average from the first principles.¹⁹

In many real situations, observed numerical data

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

${\epsilon }_1,{\epsilon }_2,\mathrm{......},{\epsilon }_Z$

usually of random in nature i.e

$xi=\mu +\epsilon i , \left(i{=}_{ }1,2,\dots \dots \dots \dots ,N\right)$ (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 $x_1,x_2,\mathrm{.........},x_N,$ 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.

Let

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

and

$a_0=AM\left(x_1,x_2,\mathrm{.........},x_N\right)=\frac{1}{N}{\displaystyle {\sum }_{i=1}^N{x}_i},$ (2.1)

$g_0=GM\left(x_1,x_2,\mathrm{.........},x_N\right)={\left({\prod }_{i=1}^N{x}_i\right)}^{1/N}$ (2.2)

 & $h_0=HM\left(x_1,x_2,\mathrm{.........},x_N\right)={\left(\frac{1}{N}{\displaystyle {\sum }_{i=1}^N{x}_i^{-1}}\right)}^{-1}$ (2.3)

i.e.$a_0,g_0$ & $h_0$ are respectively the Arithmetic Mean (AM), the Geometric Mean (GM) & the Harmonic Mean (HM) of $x_1,x_2,\mathrm{.........},x_N$

which satisfy the inequality ^{4 ,5}namely

 AM > GM > HM i.e. $h_0<g_0<a_0$ (2.4)

 Arithmetic-geometric mean (AGM)

 The two sequences $\left\{a_n\right\}\&\left\{g_n\right\}$ respectively defined by

$a_{n+1}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left(a_n+g_n\right)$ (2.5)

 & $g_{n+1}={\left(a_n g_n\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ (2.6)

where the square root assumes the principal value,

converge to a common point $M_{AG}$ which can be termed as the Arithmetic-Geometric Mean (abbreviated as AGM) of $x_1,x_2,\mathrm{.........},x_N$ ^{61 , 62, 66 , 67 , 68}.

Thus,

 AGM($x_1,x_2,\mathrm{.........},x_N$ ) = common converging point of $\left\{a_n\right\}\&\left\{g_n\right\}$ (2.7)

Arithmetic-harmonic mean (AHM)

The two sequences $\left\{a^{/}n\right\}\&\left\{h^{/}n\right\}$ respectively defined by

$a^{/}{{}_n}_{+1}=\mathrm{½}(a^{/}{}_n + h^{/}{}_n)$ (2.8)

$h^{/}{{}_n}_{+1}= \mathrm{½}(a^{/}{{}_{n }}^{ -}{}^1 + h^{/}{{}_{n }}^{ -}{}^1)\}{ }^{ -}{}^1$ (2.9)

converge to common point$M_{AH}$ which can be termed as the Arithmetic-Harmonic Mean (abbreviated as AHM) of $x_1,x_2,\mathrm{.........},x_N$ .^63,66

Thus,

 AHM ($x_1,x_2,\mathrm{.........},x_N$ ) = common converging point of $\left\{a^{/}{}_n\right\} \&\left\{h^{/}{}_n\right\}$ (2.10)

Geometric-harmonic mean (GHM)

The two sequences $\left\{g^{//}n\right\} \&\left\{h^{{}^{//}}{}_n\right\}$ defined respectively by

 $g^{//}{}_{n+1}={\left(g^{//}{}_n. h^{//}{}_n\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ (2.11)

 & $h^{//}{}_{n+1}={\left\{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left(g^{//}{{}_n}^{-1}+h^{//}{{}_n}^{-1}\right)\right\}}^{-1}$ (2.12)

where the square root takes the principal value,

converge to common point$M_{GH}$ which can be termed as the Geometric-Harmonic Mean (abbreviated as GHM) of $x_1,x_2,\mathrm{.........},x_N$ .^64,66

Thus,

 GHM($x_1,x_2,\mathrm{.........},x_N$ ) = common converging point of $\left\{g^{//}{}_n\right\} \&\left\{h^{{}^{//}}{}_n\right\}$ (2.13)

Arithmetic-geometric-harmonic mean (AGHM)

The three sequences$\left\{a^{///}{}_n\right\},\left\{g^{///}{}_n\right\}\&\left\{h^{///}{}_n\right\}$ defined respectively by

$a^{///}{}_n=1/3\left(a^{///}{}_{n-1}+g^{///}{}_{n-1}+h^{///}{}_{n-1}\right),$ (2.14)

$g^{///}n={\left(a^{///}{}_{n-1} g^{///}{}_{n-1} h^{///}{}_{n-1}\right)}^{1/3}$ (2.15)

 & $h^{///}n={\left\{1/3\left(a^{///}{}_{n-1}^{-1} + g^{///}{}_{n-1}^{-1} + h^{///}{}_{n-1}^{-1}\right)\right\}}^{-1}$ (2.16)

converges to a common limit $M_{AGH}$ which can be termed as the Arithmetic-Geometric-Harmonic Mean (abbreviated as AGHM) of $x_1,x_2,\mathrm{.........},x_N$ .^65,66

Thus,

 AGHM($x_1,x_2,\mathrm{.........},x_N$ ) = common converging point of $\left\{a^{///}{}_n\right\},\left\{g^{///}{}_n\right\}\&\left\{h^{///}{}_n\right\},$ (2.17)

Let

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

$x_i=\mu +{\epsilon }_i$ (3.1)

where ${\epsilon }_i$ is the error associated to $xi$ for (i = 1 , 2 , ………… , N) which is random in nature

 i.e. each ${\epsilon }_i$ assumes either positive real value or negative real value with equal probability.

Again since µ is the central tendency of the observed values

therefore, ${\mu }^{ -}{}^{ 1}$ will be the central tendency of reciprocals

of the observed values.

Accordingly, the reciprocals can be expressed as

^{$x_i{}^{-1}={\mu }^{-1}+{\epsilon }_i{}^{/}{ }_{ } ,{ }_{ } \left(I{=}_{ }1,2,\dots \dots \dots \dots ,N\right)$}  (3.2)

where

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

$x_1^{-1},x_2^{-1},\mathrm{.........},x_N^{-1}$ respectively.

Let us now write

 AM($x_1,x_2,\mathrm{.........},x_N$ ) = $a_0$ (3.3)

 & HM($x_1,x_2,\mathrm{.........},x_N$ ) = $h_0$ (3.4)

and then define the two sequences $\left\{a^{/}{}_n\right\} \&\left\{h^{/}{}_n\right\}$ respectively by

$a^{/}{{}_n}_{+1 }= \mathrm{½}(a^{/}{}_n + h^{/}{}_n)$ (3.5)

$h^{/}{{}_n}_{+1} = \mathrm{½}(a^{/}{{}_{n }}^{ -}{}^1 + h^{/}{{}_{n }}^{-}{}^1)\}{ }^{ -}{}^1$ (3.6)

Then, both of $\left\{a^{/}{}_n\right\} \&\left\{h^{/}{}_n\right\}$ converges to some common real number$C.$

Let us now search the relation between$C$ and µ.

Equation (3.1) together with (3.3) & (3.4) implies that $a_0=\mu +{\delta }_0{}_{ } \& h_0=\mu +e_0$

By inequality (2.4), $h_0<a_0$ i.e. $e_0<{\delta }_0{}_{ }$

Therefore, $a^{/}{{}_1}_{ }=\mu +{\delta }_1{}_{ }$ where ${\delta }_1{}_{ }=\mathrm{½}\left({\delta }_0+\right)e_0$

Since $\mathrm{½}\left({\delta }_0+e_0\right)<{\delta }_0$

Therefore,${\delta }_1<{\delta }_0$

At the n^th step, one can obtain that

${\delta }_{n+1}=\mathrm{½}\left({\delta }_n+e_n\right)<{\delta }_n$

which implies, ${\delta }_{n+1}<{\delta }_n$ since$\mathrm{½}\left({\delta }_n+e_n\right)<{\delta }_n$

This implies, ${\delta }_n$ 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($x_1,x_2,\mathrm{.........},x_N$ ) can be regarded as the value of  i.e. the value of the central tendency of $x_1,x_2,\mathrm{.........},x_N.$

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):

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.

 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

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:

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.

If µ is the central tendency of $x_1,x_2,\mathrm{.........},x_N$ then the central tendency of $x_1^{-1},x_2^{-1},\mathrm{.........},x_N^{-1}$

should logically be µ  ^{̶ 1} . Similarly, the central tendency of $-x_1^{-1},-x_2^{-1},\mathrm{.........},-x_N^{-1}$ 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 $a_0=\mu +{\delta }_0 \& {\delta }_{n+1}<{\delta }_n$

This means, ${\delta }_n$ 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 ($x_1,x_2,\mathrm{.........},x_N$ ) becomes more and more closer to µ as n becomes more and more larger.

Since ${\delta }_n<{\delta }_0$ for all n > 1 therefore, the deviation of AHM ($x_1,x_2,\mathrm{.........},x_N$ ) from µ is more than that the deviation of  . But,  = AM ($x_1,x_2,\mathrm{.........},x_N$ )

Hence, AHM ($x_1,x_2,\mathrm{.........},x_N$ ) is more accurate measure of central tendency than AM ($x_1,x_2,\mathrm{.........},x_N$ ) in the case of data of ratio type.

Similarly, AHM ($x_1,x_2,\mathrm{.........},x_N$ ) can be shown to be more accurate measure of central tendency than HM ($x_1,x_2,\mathrm{.........},x_N$ ) 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($x_1,x_2,\mathrm{.........},x_N$ ) can be defined as the GM of AM($x_1,x_2,\mathrm{.........},x_N$ ) and HM($x_1,x_2,\mathrm{.........},x_N$ ) 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.

None.

None.

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