Task[edit]
Compute all three of the Pythagorean means of the set of integers 1 through 10 (inclusive).

Show that 



A
(

x

1


,
…
,

x

n


)
≥
G
(

x

1


,
…
,

x

n


)
≥
H
(

x

1


,
…
,

x

n


)


{\displaystyle A(x_{1},\ldots ,x_{n})\geq G(x_{1},\ldots ,x_{n})\geq H(x_{1},\ldots ,x_{n})}

 for this set of positive integers.

 The most common of the three means, the arithmetic mean, is the sum of the list divided by its length:
 



A
(

x

1


,
…
,

x

n


)
=




x

1


+
⋯
+

x

n



n




{\displaystyle A(x_{1},\ldots ,x_{n})={\frac {x_{1}+\cdots +x_{n}}{n}}}


 The geometric mean is the 



n


{\displaystyle n}

th root of the product of the list:
 



G
(

x

1


,
…
,

x

n


)
=




x

1


⋯

x

n




n





{\displaystyle G(x_{1},\ldots ,x_{n})={\sqrt[{n}]{x_{1}\cdots x_{n}}}}


 The harmonic mean is 



n


{\displaystyle n}

 divided by the sum of the reciprocal of each item in the list:
 



H
(

x

1


,
…
,

x

n


)
=


n



1

x

1




+
⋯
+


1

x

n









{\displaystyle H(x_{1},\ldots ,x_{n})={\frac {n}{{\frac {1}{x_{1}}}+\cdots +{\frac {1}{x_{n}}}}}}




See also
Tasks for calculating statistical measures



in one go
moving (sliding window)
moving (cumulative)

Mean
Arithmetic

Statistics/Basic
Averages/Arithmetic mean
Averages/Pythagorean means


Averages/Simple moving average




Geometric

Averages/Pythagorean means




Harmonic

Averages/Pythagorean means




Quadratic

Averages/Root mean square




Circular

Averages/Mean angle
Averages/Mean time of day




Median

Averages/Median




Mode

Averages/Mode




Standard deviation

Statistics/Basic



Cumulative standard deviation






#include <vector>
#include <iostream>
#include <numeric>
#include <cmath>
#include <algorithm>
 
double toInverse ( int i ) {
   return  1.0 / i  ;
}
 
int main( ) {
   std::vector<int> numbers ;
   for ( int i = 1 ; i < 11 ; i++ ) 
      numbers.push_back( i ) ;
   double arithmetic_mean = std::accumulate( numbers.begin( ) , numbers.end( ) , 0 ) / 10.0 ;
   double geometric_mean =
      pow( std::accumulate( numbers.begin( ) , numbers.end( ) , 1 , std::multiplies<int>( ) ), 0.1 ) ;
   std::vector<double> inverses ;
   inverses.resize( numbers.size( ) ) ;
   std::transform( numbers.begin( ) , numbers.end( ) , inverses.begin( ) , toInverse ) ;  
   double harmonic_mean = 10 / std::accumulate( inverses.begin( ) , inverses.end( ) , 0.0 ); //initial value of accumulate must be a double!
   std::cout << "The arithmetic mean is " << arithmetic_mean << " , the geometric mean " 
      << geometric_mean << " and the harmonic mean " << harmonic_mean << " !\n" ;
   return 0 ;
}