American Journal of Computation, Communication and Control 2018; 5(3): 101-108 http://www.aascit.org/journal/ajccc ISSN: 2375-3943 Generalized fG -Mean: Derivation of Various Formulations of Average Dhritikesh Chakrabarty Department of Statistics, Handique Girls’ College, Gauhati University, Guwahati, India Email address Citation Dhritikesh Chakrabarty. Generalized fG -Mean: Derivation of Various Formulations of Average. American Journal of Computation, Communication and Control. Vol. 5, No. 3, 2018, pp. 101-108. Received : September 25, 2018; Accepted : September 26, 2018; Published : November 22, 2018 Abstract: Average is a basic concept and averaging is a basic technique/tool behind the most of the measures associated to characteristics of data. A number of definitions/formulations of average have been developed so far. These definitions/formulations have been found suitable to be used in different situations. There are still many situations where there are scarcities of definitions/formulations of average to deal with the respective situations. It has been found possible to compose general definition of average so that the definitions/formulations of average, already developed, can be obtained/derived from it as well as more new definitions/formulations of average can be obtained/derived from it. One general definition of average, termed in this paper as Generalized fG -Mean, has been introduced with a view to obtain different definitions/formulations of average. Applying the technique, some definitions/formulations have been derived for various types of averages. This paper describes this general definition and the derivation of a number of definitions/formulations of average. Keywords: Average, Generalized fG -Mean, Existing Means, Derivations be n numbers of a list and/or the values assumed by a 1. Introduction variable x. Average [1, 2] is basic concept and averaging is a basic Some of the definitions /formulations of averages, already technique/tool behind the most of the measures associated to constructed [6, 7, 8, 9, 10, 11], are as follows: characteristics of data. A number of definitions/formulations Arithmetic Mean = + . + ) (1) of averages had primarily been developed by Pythagoras [3, + 4, 5] who constructed the definitions / formulations of the Geometric Mean = . / (2) three most common averages namely Arithmetic Mean, Geometric Mean & Harmonic Mean which are known as provided the n numbers are non-negative. Pythagorean means [6, 7]. Harmonic Mean = { . ) (3) Let , ,.............., provided the n numbers are all different from 0. Quadratic Mean = { . )/ (4) Square Root Mean = { / / . /) (5) Cubic Mean = { . )/ (6) Cube Root Mean = { / / . /) (7) Generalized p-Mean = { . / (8) American Journal of Computation, Communication and Control 2018; 5(3): 101-108 102 pth Root Mean = { ( / + / + .+ /) (9) e-Mean = log e ( + + .+ (10) Scale s-Mean = { ( + + . + )} (11) Shift a-Mean = ( ̶a) + ( ̶ ) +........+ ( ̶ a)} + a (12) ̶ ̶ ̶ shift a– Inverse Scale s -Mean = ( + + + ) + a (13) These definitions/formulations have been found suitable to the arithmetic mean of the functional values be used in different situations. There are still many situations , , …………, where there are scarcities of definitions/formulations of () () () average to deal with the respective situations. It has been is taken first and then the inverse functional value of the found possible to compose general definition of average so arithmetic obtained is taken to obtain the generalized f-mean that the definitions/formulations of average, already defined by developed, can be obtained/derived from it as well as more new definitions/formulations of average can be [ () + () + . + ()}] obtained/derived from it. Kolmogorov constructed one definition/formulation of average known as the Generalized f Here, f is an invertible function. -Mean [12-17]. Let us now take the geometric mean of the functional The Generalized f - Mean of values , ,.............., , is defined by (), (), …………, () and then the inverse functional value of the geometric mean [ ( ) + ( ) + . + ( )}] (14) obtained. where f is an invertible function. In doing this we will obtain the formulation It has been shown that the Generalized f - Mean can be [( )( )........ ( )/ used to derive the definitions/formulations of the existing means and also of new means [9, 10, 11, 18]. This can be regarded one generalized definition of However, the Generalized f -Mean due to Kolmogorov has average. also been found to be not suitable for some situations. Let this average be termed as the Generalized fG - Mean in Recently, one more generalized definition of average has this article. been constructed which has been termed as Generalized f H – Thus, the Generalized fG - Mean of Mean [18]. In this paper, attempt has been made on the construction of one more general definition of average in a , ,.............., similar manner as the construction of the Generalized f - denoted by fG (, ,........,) can be defined as Mean by Kolmogorov. The definition constructed here has / been termed as Generalized fG -Mean. It has been shown that fG (, ,........,) = [()()........ () (15) the existing definitions/formulations of average can be where f is any invertible function. obtained from the Generalized fG -Mean. This paper describes this general definition of average and the derivation of the existing definitions/formulations of average. 3. Properties of Generalized fG - Mean The Generalized fG - Mean satisfies the following 2. Generalized fG - Mean: One properties: Technique of Definition of Average 1. Partitioning: The computation of the Generalized fG - Mean can be split into computations of equal k - sized sub- In constructing the generalized f-mean (also known as blocks i.e. Kolmogorov mean) of , ,..........., fG ( , , ………., ) = fG { fG ( , , ……, ), fG ( , , ……, ), ………, fG ( (), (), ……, )} Proof: We have 103 Dhritikesh Chakrabarty: Generalized fG -Mean: Derivation of Various Formulations of Average / fG ( , , ……, ) = [()()........ () , fG ( , , ……, ) = / [()()........ () , / fG ( (), (), ……, )= (()!. (())........ ( ) Accordingly, fG { fG ( , , ……, ), fG ( , , ……, ),.........., fG ( (), (), ……, )} / = [()()........ ( ) = fG ( , , ………., ) 2. Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. Thus, with m = fG ( , , ……, ) it holds that fG ( , , ……, , , , ……, )= fG (", ", ….,", , , ……, ) Proof: We have / m = fG ( , , ……, ) = ()()........ () Therefore, / (") = f [()()........ () ] / = ()()........ () Hence, fG (", ", ….,", , , ……, ) / = [ (") ("). (")()()........ () ] / = [()()........ ()()()........ () ] = fG ( , , ……, , , , ……, ) 3. The Generalized fG - Mean is invariant with respect to scaling of f i.e. for all real b, ϕ(t) = b f(t ) implies ϕG () $ fG () Proof: We have / ϕG ( , , ……, ) = % [%(). %()........ %() ] / = % [%(). %()........ %() ] Now, ϕ(t) = b f(t ) ⇒ t = %{ b f(t ) } ⇒ t = %{ b f(t )} & & ⇒ %(y) = ( ), putting y = b f(t ) ⇒ t = ( ) ' ' ⇒ %: x → ( ) i.e. p % maps x to ( ) ' ' Thus / ϕG ( , , ……, ) = % [%(). %()........ %() ] / = % [)(). )()........ )() ] American Journal of Computation, Communication and Control 2018; 5(3): 101-108 104 / = ['*.'*()........ '*() ] ' / = [()()........ () ] = fG ( , , ……, ) 4. If f is monotonic, then fG is monotonic. Proof: We have / fG ( , , ……, ) = [()()........ () ] If f is monotonic then either () + ()+ + () or () , (), , () This implies, / /() either [()()........ () + [()()........ () / /() or [()()........ () , [()()........ () for all k, which implies, either fG ( , , ……, ) < fG ( , , ……, ) or fG ( , , ……, ) < fG ( , , ……, ) Hence, fG is monotonic.. 5. Generalized fG - Mean of two variables has the mediality property namely fG [fG {( x, fG (, -), fG {( y, fG (, -)] $ fG (, -) Proof: We have / fG (, -) $ (). (-) Thus, fG {( x, fG (, -) $ [(). (). (-)]/ = [(). (). (-)]/ Similarly, fG {( y, fG (, -) $ [(-). (). (-)]/ Now, f f G {( x, fG (, -) = f [(). (). (-)] = [(). (). (-)]/ Similarly, f f G {( y, fG (, -) $ [(-). (). (-)]/ = [(-). (). (-)]/ Therefore, fG [fG {( x, fG (, -), fG {( y, fG (, -)] 105 Dhritikesh Chakrabarty: Generalized fG -Mean: Derivation of Various Formulations of Average = [(). (-)] = fG (, -) 6. Generalized fG - Mean of two variables has the self-distributive property namely fG {x, fG (-, .) $ fG { fG (, -), fG (, /) Proof: We have / fG (, -) $ (). (-) Accordingly, / / fG (, .) $ (). (.) & fG (-, .) $ (-). (.) Now, 1/2 fG { fG (, -), fG (, /) = [()(-). ()(.)] 1/2 = [()(-). ()(.)] = [(). (-)1/2.(.)1/2]1/2 Thus, 1/2 fG {x, fG (-, .) = { (). f fG (-, .) 1/2 = [(). (-)(.)] = [(). (-)1/2.(.)1/2]1/2 = fG { fG (, -), fG (, /) 7. fG - Mean of two variables x & y has the balancing property namely fG [fG {x, fG (, -), fG {( y, fG (, -)] $ fG (, -) Proof: We have / / fG {x, fG (, -) = [(). (). (-) ] = [(). (). (-)/]1/2 / / & fG {y, fG (, -) = [(). (). (-) ] = [(-). (). (-)/]1/2 Therefore, fG [fG {x, fG (, -), fG {( y, fG (, -)] 1/2 $ [fG {x, fG (, -). f fG {y, fG (, -)] = [ (). (-)//] = fG (, -) Remark: Then apply the function selected in the definition of This definition / formulation can be applied in searching Generalized fG - Mean defined by (15). for / constructing of a number of definitions / formulations for average. The technique can be