Mean M(x1,x2,…xN) (no weighted):
N
N

1. M is a function: R  Rminxi   M  maxxi  or R  R minxi   M  maxxi 
2. Simmetry: for all permutation  M x1 , x2 ,.....x N   M xi , x N ,.....x2 
3. Fixed point: if xi  K i  1,...,N  M xi   K
4. Homogenity:  and xi  M   xi     M xi 
5. Monotony: xi , yi and xi  yi  M xi   M yi 
6. Continuity: xi , yi and xi  yi  lim M xi   M yi 
x y
So for the point 2, to make a mean, we have to use some
operation with commutative property; the simplest are:
N
N
x
i 1
x
i
i
i 1
Moreover for the fixed point property (3), if all number are
iguals the mean is the same value, so we have to add
something:
1
N
N
x
i 1
N
i
Arithmetic Mean
N
x
i
i 1
Geometric Mean
First method to build anothers means:
x1
N
1


f 1   f ( xi ) 
 N i 1

f (..)
Traditional Block
xi
xN
…………
f (..)
…………
1
N
N

i 1
N
N

f
1
(..)
i 1
f (..)
For example, if we choose f(x)=1/x and arithmetic
mean as traditional block, we discover the
Harmonic Mean:
N
Harmonic Mean
N
1

i 1 x i
Media
if we choose f(x)=xs and arithmetic mean as
traditional block, we find the Power Mean:
Power Mean
1

N

x 

i 1

N
s
i
1
s
Second method to build anothers means:
for two numbers, but is
possible to generalize
f ' ( x)

y
 f ( x)  f ( y ) 

 f ' 
x y


1
x
x
f ( x)  f ( y )   f ' ( x)dx
y
f ( x)  f ( y )
f ( x)  f ( y )  f ' ( )  ( x  y )  f ' ( ) 
( x  y)
 f ( x)  f ( y ) 

   ( f ' ) 
 ( x  y) 
1
Third form, to build anothers means:
N
p 1
x
 i
Lehmer mean
i 1
N
p
x
 i
i 1
First group
 f ( x)  f ( y ) 
f 1 

2


Harmonic Mean
f ( x) 
2
1 1

x y
1
x
f ( x)  ln(x)
xy
f ( x)  x p
Power Mean
x y

2

p
p  1
p  
p



p  
p 1
max(x, y)
p2
x y
2
2
x y
2
p0
1
p
2
min(x, y)
Power Mean of orden 2
(Root Mean Squares)
Quadratic Mean
Second group
 f ( x)  f ( y ) 

( f ' ) 
x y


1
f ( x)  x ln(x)
f ( x)  x p
Identric Mean
1 x y x x
e
yy
Power mean
orden 1/2
Logarithmic Mean
Stolarsky Mean
p 1
p
 x y




2


f ( x)  ln(x)
1
2
2
 xp  yp 


 p  ( x  y) 
p  1
1
p 1
p0
p  
min(x, y)
p2
xy
p  
max(x, y)
x y
2
x y
ln(x)  ln(y )
p3
Strange connection
geometric-quadratic
x 2  y 2  xy
3
Looks Heronian
x p 1  y p 1
xp  yp
Third group
p  
p 1
max(x, y)
x y
x y
2
p  
p  1
2
2
min(x, y)
1 1

x y
p0
Contraharmonic Mean
1
p
2
x y
2
p
1
2
xy
x  y  xy
Clic there is a
demostration
All means with negative p have the
the coniugate mean respect the
arithmentic mean, with positive p.
Equivalent Form of Lehmer Mean q=-p
yq x  xq y  yq
  q
q
q
q
x y
x

y

q  
q
q 1
min(x, y)
2
  xq
 x   q
q
x

y
 
xy
1 1

x y
1
2
q0
x y
2

 y

q  
max(x, y)
We can write the Lehmer Mean in this form too:
x p 1  y p 1 x p x  y p y  x p
 p
  p
p
p
p
p
x y
x y
x

y

Like weighted mean….
  yp
 x   p
p
x

y
 

 y

Another type
Heronian Mean
Heinz Mean
x  y  xy
3
x p y 1 p  x1 p y p
2
1

2

p
1
0
Logarithmic Mean

i /  1i / 
x
 y
i 0
 1
Average of
Means like
Heronian and
Heinz.
  
1
x
0
p
y
1 p
x y
dp 
ln(x)  ln(y )
Mixed Means
Arithmetic-Geometric Mean
Arithmetic-Harmonic Mean
Geometric-Harmonic Mean
an  g n
x y


a1 
a n 1 
2
2 

 g  xy
g  a g
n n
1

 n 1
These two sequences converge to the same number.
For the Arithmetic-Geometric Mean there is a closed for expression:

x y
AGM  
4
x y

K 
x y
Where K(x) is the complete elliptic integral of first kind.
Demostration of an expression:


( x  y )  x3  y3
x 2  y 2  yx 3  xy 3 ( x  y )  ( x  y )  yx x  y 


x y
x y
x y

( x  y )  ( x  y  xy )
 x  y  xy
x y
An observation:


( x  y )  x y  y x ( xy  xy x  xy y  xy)

 xy
x y
x y
Another observation:
xp
2p
 p    p
0
p
p
p
y x
2 3
3p
p    p
1
p
2 3
Moreover:
1. Elementary Symmetric Mean
And….
1. Median
2. Mode