The 6th International Days of Statistics and Economics, Prague, September 13-15, 2012
A MATHEMATICAL PROPERTY OF THE HARMONIC MEAN
Mohamed Sadeg Zayed Ogbi
‫ـ‬-------------------------------------------‫ـ‬----------------------------------------------Abstract:
The harmonic mean is one of the important measures of central tendency, which is applied more
frequently in averaging speeds when the distances for which each speed is applicable are the same
.The harmonic mean is based on reciprocals of the numbers averaged .It is the reciprocal of the
arithmetic mean of the reciprocal of the numbers averaged .Experiment shows that whenever we
average a group of values the arithmetic mean will be larger than the geometric mean ,and the later
will be larger than the harmonic mean (unless all the values averaged are of the same size ,in which
case the three averages will be identical) , Properties of the harmonic mean are mentioned in many
elementary statistical books. In this paper, the author will give a proof concerning the adequacy of
the harmonic mean to mathematical manipulation, that is because, the author has not seen this proof
in any of the books or papers.
Key words: Central tendency, Harmonic mean, Mathematical property.
JEL Code: CO1, CO2
-----------------------------------------------------------------------------------------------------------------------Introduction
The harmonic mean is one of the measures of Central tendency, which is useful when averaging
rates, and it is desired to keep constant in the average the factor that is variable in the rate.
Some properties of the harmonic mean are mentioned in many elementary statistical books.
In this paper, the author will add another property concerning the susceptibility of the harmonic
mean to mathematical operations; this property is stated by following theorem.
Theorem:
(a)
if a set of n1 variates , say ,
y1 , y 2 ,...., yn2
x1 , x2 ,......, xn1
and a set of n2 variates , say ,
with harmonic means H1 and H2 respectively , then the
combined harmonic mean of the two sets of variates ,say , Hc2 is given by ;
Hc2 
(n1  n2 ) H1 H 2
n1 H 2  n2 H1
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The 6th International Days of Statistics and Economics, Prague, September 13-15, 2012
(b)
If m sets of variates , say , n1,n2,….nm with harmonic means H1,H2,….Hm
respectively, then the combined harmonic mean of the m sets of variates Hcm is
given by ;
H cm 
m
m
i 1
m
i 1
( ni )( H i )
 n ( H
i 1
i
j
;m  2
)
j i
Where
m
H
i
 H1H 2 ......H m
i 1
Proof:
The harmonic mean of the n set of Variates x1,x2,….,xn is defined as:
H 
n
n
1
x
c 1
c
(a) For the proof of part (a); consider the n1 set of variates
variates
x1 , x2 ,......, xn1
and the n2 set of
y1 , y 2 ,...., yn2 . Let H1 and H2 denote the harmonic means of the two sets of
variates respectively, then by definition we have;
n1

 H1  n1 1


i 1 xi


 H 2  n2
n2

1


i 1 yi

And hence we can write
874
.....(1)
The 6th International Days of Statistics and Economics, Prague, September 13-15, 2012
 n1

 i 1
 n2


 i 1
1
n
 1
xi
H1
.....( 2)
1
n
 2
yi
H2
Now Let Hc2 denotes the harmonic mean of the combined two sets of variates n1 and n2, Then by
definition , we have:
H C2 
1
1 1
1
1
1
1
1

  ... 
 
 .... 
x1 x 2 x3
x n1 y1 y 2
y n2
From (2) ,and after few steps, we can write:
HC2 
n  n H H
n1  n2
 1 2 1 2
n1 n2 n1 H 2  n2 H 1 

H1 H 2
Or
HC2

 2  2
  ni   H i 
i 1
 i 1 

2
2
n H
i 1
i
..........(3)
j
j i
Which completes the proof for part(a).
(b)
Suppose we have “m” sets of variates n1,n2,…,and nm with harmonic means H1,H2,…..,Hm
respectively, Let Hcm be the harmonic mean for the combined n sets of variates .For m=2.
the proof is given in (a).Now let us consider the case when m=3, that is we have three sets
of variates n1,n2, and n3 which are (
x1 , x2 ,......xn1 ),( y1 , y 2 ,...., yn2 ) and
( z1 , z 2 ,...., z n3 ) with harmonic means H1,H2 and H3 respectively, then the harmonic mean
of the combined. three sets of variates is:
H C3 
n1  n2  n3
1
1
1
1
1
1
 ... 

 .... 
  ... 
x1
x n1 y1
y n 2 z1
z n3
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The 6th International Days of Statistics and Economics, Prague, September 13-15, 2012

n1  n2  n3
n3
1 n2 1
1
 

i 1 xi
i 1 y i
i 1 z i
n1
From (2), and after few steps we can write:

 3  3
  ni   H i 
H C 3   i 1  i 1 
3
 3



n
H


i
i

i 1
 j i 
If we repeat the same procedure for m=4,5,…,etc, then we finally obtain the Harmonic
mean of the combined m sets of Variates n1,n2, …, nm with the harmonic means H1,H2,…,Hm
respectively, say, Hcm is as follows:
H cm 
n1  n2  .....  nm
m3
n1
1 n2 1
1
   .......  

i 1 xi
i 1 yi
i 1 z i
Again from (2), and after few steps we get:
H cm 
n1  n2  ......  nm H1 H 2 .....H m 
n1 H 2 H 3 ......H m  ..........  nm H1........H m1

 m  m
  ni   H i 
i 1
 i 1 
 
m
 m



n
H


i
i

i 1
 j i 
Which completes the proof.
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The 6th International Days of Statistics and Economics, Prague, September 13-15, 2012
References:
1. Remington D.R and Schork M.A, Statistics with application to Biological Health Sciences,
Printce—Hall Inc. Englewood Cliffs.N.J.,1970.
2. Blair M.M., Elementary Statistics, Henry-Holt and Company, Inc. USA 1944.
3. Mode B.C, Elements of statistics, Prentice-Hall Inc. New York,1951.
4. Alder L.H. and Roessler B.E., Introduction to Probability and Statistics, W. H
Freeman and Company ,USA,1960.
5. Mohamed S. Z . Ogbi ,Introduction to Descriptive Statistics ,Dar Libya for publication
Tripoli ,Libya,2004 ,
6. Robert G.D. Steel, and James H . Torrie . ,Megraw-Hill Book Company Inc. USA 1960.
Contact
Mohamed Sadeg Zayed Ogbi
University.of Tripoli
Tripoli ,Libya
Phone: +218-21-4627910
Fax : +218-21-4628839
Email : m.ogbi@yahoo.com
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