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






class PythagMeans {
  function : Main(args : String[]) ~ Nil {
    array := [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0];
    arithmetic := ArithmeticMean(array);
    geometric := GeometricMean(array);
    harmonic := HarmonicMean(array);
 
    arith_geo := arithmetic >= geometric;
    geo_harm := geometric >= harmonic;
 
    "A = {$arithmetic}, G = {$geometric}, H = {$harmonic}"->PrintLine();
    "A >= G is {$arith_geo}, G >= H is {$geo_harm}"->PrintLine();
  }
 
  function : native : ArithmeticMean(numbers : Float[]) ~ Float {
    if(numbers->Size() = 0) { return -1.0; };
 
    mean := 0.0;
    each(i : numbers) {
      mean += numbers[i];
    };
 
    return mean / numbers->Size();
  }
 
  function : native : GeometricMean(numbers : Float[]) ~ Float {
    if(numbers->Size() = 0) { return -1.0; };
 
    mean := 1.0;
    each(i : numbers) {
      mean *= numbers[i];
    };
 
    return mean->Power(1.0 / numbers->Size());
  }
 
  function : native : HarmonicMean(numbers : Float[]) ~ Float {
    if(numbers->Size() = 0) { return -1.0; };
 
    mean := 0.0;
    each(i : numbers) {
      mean += (1.0 / numbers[i]);
    };
 
    return numbers->Size() / mean;
  }
}