Jun 11, 2018 - three definitions/formulations of averages namely arithmetic mean, geometric mean and harmonic mean. Several definitions/formulations of ...
June 2018 - August 2018; Section C; Vol.7. No.3, 212-225.
E-ISSN: 2278–179X
[DOI: 10.24214/jecet.C.7.3.21225.]
Journal of Environmental Science, Computer Science and Engineering & Technology An International Peer Review E-3 Journal of Sciences and Technology Available online at www.jecet.org Section C: Engineering & Technology Research Article
One Generalized Definition of Average: Derivation of Formulations of Various Means Dhritikesh Chakrabarty Department of Statistics, Handique Girls’ College, Guwahati University, Guwahati, India Received: 19 May 2018; Revised: 11 June 2018; Accepted: 20 June 2018
Abstract: Average is the vital concept and averaging is the vital tool of most of the statistical measures. A number of definitions/formulations of average are available to be used in different situations. On the other hand, it is possible to construct more and more definitions/formulations of average by a single technique. One such technique has been introduced here for construction of definition/formulation of average. A number of definitions/formulations of average have been derived from the technique introduced. This paper describes the technique introduced and the derivations of a number of definitions/formulations of average. Keywords: Average, definitions, formulations, construction,
-Mean
1. INTRODUCTION The term “average” as a commonly speaking term, is used to infer a characteristic of an aggregate of individuals but not of any individual. Average has become a vital concept behind and averaging has become a vital tool of most of the measures used in analysis of data,1-,3. It is Pythagoras4 who constructed three definitions/formulations of averages namely arithmetic mean, geometric mean and harmonic mean. Several definitions/formulations of average have already been developed independently, which can also be formulated from the concept of Pythagorean mean. Different formulations of average are suitable for handling data in different situations 1, 2.5. The existing formulations of average are not sufficient to handle data of different types. There exist many types of data for which formulations of average are not available to handle with them. Thus, there is necessity of searching for / constructing of more and more definitions / 212
JECET; ; June 2018 - August 2018; Section C; Vol.7. No.3, 212-225. DOI: 10.24214/jecet.C.7.3.21225.
One Generalized …
Dhritikesh Chakrabarty.
formulations of average that can be suitable for handling data of various characteristics. In this connection, it can be thought that if some technique can be found for defining average in different situations then lots of definitions / formulations can be constructed for average. Kolmogorov introduced one such technique, known as the generalized f-mean which is also called Kolmogorov mean after Russian mathematician Andrey Kolmogorov6-11 , from which it is possible to derive several definitions/formulations of average. In the current study, an attempt has been made on searching for one more such technique, which has in this paper been termed as ϕ-Mean, of defining average. A number of definitions/formulations have been derived from the -Mean. This paper describes the construction of the number of definitions/formulations of average.
-Mean and the derivations of a
2. Existing Averages Here , denote the n numbers of a list and/or the values assumed by a variable x. 2.1. Pythagorean Means: The three most common averages, due to Pythagoras, are Arithmetic Mean, Geometric Mean & Harmonic Mean12- 15. In statistics, these three means are used as measures of central tendency of numerical data16,17 . The arithmetic mean of the n numbers, as defined by Pythagoras, is +
+
+
)
(1)
The geometric mean of them is (2) provided the n numbers are non-negative. Taking the arithmetic mean of the logarithms of the numbers and then taking the antilogarithm of the arithmetic mean obtained, the geometric can also be defined / formulated as +
+
+
)}
(3)
Thus, geometric mean can be thought of as the antilog of the arithmetic mean of the logs of the numbers. The Harmonic mean of the n numbers is
(4) or equivalently {
)
provided the n numbers are all different from 0. 213
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(5)
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Dhritikesh Chakrabarty.
2.2. Some Other Existing Averages Quadratic Mean (Also termed as Root Mean Square abbreviated as RMS): Quadratic mean18 is defined by
1 2 2 .......... .. x 2 x x n 2 n 1
(6)
which in another form of expression is as {
)
(7)
Cubic Mean: Cubic mean19 is defined by {
)
(8)
Generalized p-Mean: The expression of this mean is {
(9)
Note: If p = 1, the generalized p-mean becomes the arithmetic mean. If p = 2, the generalized p-mean becomes the quadratic mean. If p = ̶ 1, the generalized p-mean becomes the harmonic mean. 3. Generalized
-Mean: One Technique of Defining Average
Let us define mean termed as Generalized
-Mean of
,
denoted by M ϕ (
Mϕ (
,
) as
,
where
) =
(10)
is any invertible function.
The Generalized
-Mean satisfies the following properties:
1. Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks i.e. Mϕ (
,
, ………. ,
……… , M ϕ ( 214
) = Mϕ { Mϕ ( ,
,
, …… ,
, …… ,
) , Mϕ (
)}
JECET; ; June 2018 - August 2018; Section C; Vol.7. No.3, 212-225. DOI: 10.24214/jecet.C.7.3.21225.
,
, …… ,
), (11)
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Dhritikesh Chakrabarty.
2. Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. Thus, with
,
, …… ,
, …… ,
,
m = Mϕ (
)
it holds that Mϕ ( = Mϕ ( 3. The
,
, …. ,
,
,
, …… ,
, ,
, …… ,
)
)
(12)
-Mean is invariant with respect to offsets and scaling of f i.e.
for all real a & all non-zero real b , g (t) = a + b f(t) implies M g (
Mf(
(13)
is monotonic, then M ϕ is also monotonic20 .
4. If 5. Any
-Mean M ϕ of two variables has the mediality property namely
M ϕ [M ϕ {(x , M ϕ (
, M ϕ {(y , M ϕ (
Mϕ (
(14)
and the self-distributive property namely M ϕ {x , M ϕ (
Mϕ { Mϕ (
, Mϕ (
(15)
Moreover, any of those properties is essentially sufficient to characterize quasi-arithmetic means. 6. Any
-Mean M ϕ of two variables x & y has the balancing property namely
M ϕ [M ϕ {(x , M ϕ (
, M ϕ {(y , M ϕ (
Mϕ (
(16)
Remark: This definition -Mean can be applied in searching for / constructing of a number of definitions / formulations for average. The technique can be summarized in the following steps: (1) Select suitable function which is invertible. (2) Then find out the inverse function of the function selected. (3) Then apply the function selected in the definition of
-Mean defined by equation (10).
4. Construction of Some Definitions/Formulations of Average from the Generalized Arithmetic Mean: Let the invertible function
be such that :x → ex
maps x to e x )
( i.e.
=ex
i.e.
215
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-Mean
One Generalized …
Here,
Dhritikesh Chakrabarty.
( e x)
(x) =
which implies,
(e x)
I(x) =
where I(.) is the identity map (function) i.e.
I(x) = x (e x) = x
Accordingly, i.e. i.e. Thus
(y) = log e y (x) = log e x
(.) is a function with : x → log e x
( i.e.
maps x to log e x )
Accordingly, if in the
-Mean,
is selected as (x) = e x
then the
-Mean becomes +
+
+
)
which is nothing but Pythagorean arithmetic mean defined by equation (1). Geometric Mean: Let the invertible function
(.) be such that :x→x
(i.e.
maps x to x)
i.e.
(x) = x ̶ 1
Then Thus
̶ 1
(x) = x
(.) is a function with ̶ 1
̶ 1
(i.e.
:x→x
maps x to x)
Accordingly, if in the Generalized
-Mean, the function
(.) is selected as
(x) = x then the 216
-mean becomes JECET; ; June 2018 - August 2018; Section C; Vol.7. No.3, 212-225. DOI: 10.24214/jecet.C.7.3.21225.
One Generalized …
Dhritikesh Chakrabarty.
which is nothing but Pythagorean geometric mean defined by equation (3) which is same as with that defined by equation (2). Harmonic Mean: Let the invertible function
(.) be such that : x → exp (
i.e.
maps x to exp (
i.e.
)
(x) = exp ( ̶ 1
Then
{ exp (
)
)} = x
i.e.
̶ 1
(y) = (log e y )
̶ 1
i.e.
̶ 1
(x) = (log e x )
̶ 1
̶ 1
Thus
putting y = log e y
(.) is a function with ̶ 1
̶ 1
i.e.
)
maps x to (log e x )
: x → (log e x )
̶ 1
Accordingly, if the function
(.) is selected as (x) = exp (
then the Generalized
̶ 1
)
-Mean becomes
{
)
which is nothing but Pythagorean harmonic mean defined by equation (5). Quadratic Mean: Let the invertible function
(.) be such that
: x → exp (x2) i.e.
maps x to exp (x2)
i.e.
(x) = exp (x2)
Then
̶ 1{exp (x2)} = x
217
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Dhritikesh Chakrabarty.
i.e.
̶ 1(y) = (log e y)1/2
i.e.
̶ 1(x) = (log e x)1/2
putting y = exp (x2)
̶ 1 (.) is a function with
Thus
̶ 1 : x → (log e x)1/2 i.e.
̶ 1 maps x to (log e x)1/2
Accordingly, if the function
(.) is selected as
(x) = exp (x2) then the Generalized
-Mean becomes
{
)
which is nothing but the quadratic mean defined by equation (6). Square Root Mean: Let the invertible function
(.) be such that
: x → (√exp x) i.e.
(x) = (√exp x)
In a similar manner it can be shown that if this Mean becomes
(.) is substituted in the Generalized
1 x x .......... . x 2 n n 1
-Mean then the
-
2 (17)
This can also be a definition / formulation of average. This definition / formulation of average can be termed as Square Root Mean. Cubic Mean: Let the invertible function
(.) be such that
: x → exp (x3) i.e.
(x) = exp (x3)
In a similar manner it can be shown that if this Mean becomes 218
(.) is substituted in the Generalized
JECET; ; June 2018 - August 2018; Section C; Vol.7. No.3, 212-225. DOI: 10.24214/jecet.C.7.3.21225.
-Mean then the
-
One Generalized …
Dhritikesh Chakrabarty.
{
)
which is the cubic mean defined by equation (8). Cube Root Mean: Let the invertible function
(.) be such that
: x → exp (x1/3) i.e.
(x) = exp (x1/3)
In a similar manner it can be shown that if this Mean becomes
(.) is substituted in the Generalized
{
-Mean then the
)
-
(18)
This can also be a definition / formulation of average. This definition / formulation of average can be termed as cube root mean. Generalized p-Mean: Let the invertible function
(.) be such that
: x → exp (xp) i.e.
(x) = exp (xp)
In a similar manner it can be shown that if this Mean becomes
(.) is substituted in the Generalized
{
-Mean then the
)
-
(19)
which is nothing but the Generalized p-Mean. Generalized pth Root Mean: Let the invertible function
(.) be such that
: x → exp (x 1/p) i.e.
(x) = exp (x 1/p)
In a similar manner it can be shown that if this Mean becomes
(.) is substituted in the Generalized
{
)
This can also be a definition / formulation of average. 219
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-Mean then the
-
(20)
One Generalized …
Dhritikesh Chakrabarty.
This definition / formulation of average can be termed as Generalized pth Root Mean. Scale Mean or simply s-Mean: Let the invertible function
(.) be such that
: x → exp (s.x) i.e.
(x) = exp (s.x)
In a similar manner it can be shown that if this Mean becomes {
+
(.) is substituted in the Generalized
+
+
-Mean then the
)}
-
(21)
This can also be definition/formulation of average. This average can be termed as s-Mean. Note: Selecting the function
(.) as
(x) = exp ( .x) , for non-zero real number s one can obtain the
-Mean as
( This can be termed as
+
(22)
-Mean.
a -Shift Mean or simply a-Mean: Let the invertible function
(.) be such that
: x → exp (x – a) (x) = exp (x – a)
i.e.
In a similar manner it can be shown that if this Mean becomes ̶ a) + (
(.) is substituted in the Generalized
̶ a)} + a
This average can be termed as a-Mean. Note:
220
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-Mean then the
-
(23)
One Generalized …
Dhritikesh Chakrabarty.
Selecting the function
(.) as
( x – a)} , for real a non-zero real s
(x) = exp {
one can obtain that the (
-Mean comes down to be
+
+a
This can be termed as shift (a) - scale (
(24)
) -Mean.
5. Some Definitions/Formulations of Average of Variable From the Generalized
-Mean, as defined above, one can define the Generalized
-Mean of a function
(25) of x by [
{
]
(26)
where. =
,
=
, ………………. ,
=
From this definition, one can obtain the definitions/formulations of various means as mentioned above, for a function of variable, as follows: Arithmetic Mean: Arithmetic Mean of g(x) can be obtained as {
+
+
)}
(27)
In particular, Arithmetic Mean of x 2 is
(28) Similarly, the Arithmetic Mean of |x| is
(29) Arithmetic Mean of x 3 is
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Dhritikesh Chakrabarty.
(30) Arithmetic Mean of
x p is
(31) Arithmetic Mean of x 1/p is
(32) Arithmetic Mean of e x is
(33) Arithmetic Mean of log x is
(34) Geometric Mean: From this definition given by equation (26), Geometric Mean of g(x) can be obtained as (35) Harmonic Mean: Similarly, Harmonic Mean of g(x) can be obtained as
(36) Quadratic Mean: Quadratic Mean of g(x) can be obtained as {
)
(37)
Cubic Mean: Cubic Mean of g(x) can be obtained as {
)
Generalized p Mean: Generalized p Mean of g(x) can be obtained as
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(38)
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Dhritikesh Chakrabarty.
{
)
(39)
Generalized pth Root Mean: Generalized pth Root Mean of g(x) can be obtained as {
)
(40)
Scale s Mean or simply s Mean: Scale s Mean or simply s Mean of g(x) can be obtained as
{
+
+
+
)}
(41) Scale
Mean or simply
-Mean:
Scale
Mean or simply
-Mean of g(x) can be obtained as (
+
(42)
a - Shift Mean or simply a-Mean: a - Shift Mean or simply a-Mean of g(x) can be obtained as ̶ a) + ( Shift (a) - scale (
) – Mean:
Shift (a) - scale (
) - Mean of g(x) can be obtained as
(
+
̶ a)} + a
+a
(43)
(44)
6. CONCLUSION. One can conclude that the Generalized -Mean introduced here can be regarded as a source from where lots of definitions/formulations can be derived for various types of averages. Different types of formulations of average are necessary for handling different types of data. That is why we need more and more formulations of average. The types of average formulated here have been derived from the Generalized -Mean introduced here. However, the Generalized -Mean is not sufficient to generate many types of averages to deal with many types of data. Thus, there is necessity of further study on searching for more and more techniques of defining / formulating of more types of averages
223
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18. S Nastase Adrian, How to Derive the RMS Value of Pulse and Square Waveforms, MasteringElectronicsDesign.com.2015. 19. Svarovski Ladislav, Solid-Liquid Separation, Google book, 2015. 20. John Bibby, “Axiomatisations of the average and a further generalisation of monotonic sequences”, Glasgow Mathematical Journal, 1974, 15, 63 – 65.
Corresponding author: Dhritikesh Chakrabarty Department of Statistics Handique Girls’ College, Guwahati University Guwahati, India.
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