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






(function () {
    'use strict';
 
    // arithmetic_mean :: [Number] -> Number
    function arithmetic_mean(ns) {
        return (
            ns.reduce( // sum
                function (sum, n) {
                    return (sum + n);
                },
                0
            ) / ns.length
        );
    }
 
    // geometric_mean :: [Number] -> Number
    function geometric_mean(ns) {
        return Math.pow(
            ns.reduce( // product
                function (product, n) {
                    return (product * n);
                },
                1
            ),
            1 / ns.length
        );
    }
 
    // harmonic_mean :: [Number] -> Number
    function harmonic_mean(ns) {
        return (
            ns.length / ns.reduce( // sum of inverses
                function (invSum, n) {
                    return (invSum + (1 / n));
                },
                0
            )
        );
    }
 
    var values = [arithmetic_mean, geometric_mean, harmonic_mean]
        .map(function (f) {
            return f([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
        }),
        mean = {
            Arithmetic: values[0], // arithmetic
            Geometric: values[1], // geometric
            Harmonic: values[2] // harmonic
        }
 
    return JSON.stringify({
        values: mean,
        test: "is A >= G >= H ? " +
            (
                mean.Arithmetic >= mean.Geometric &&
                mean.Geometric >= mean.Harmonic ? "yes" : "no"
            )
    }, null, 2);
 
})();
 