Averages/Pythagorean means

Compute all three of the Pythagorean means of the set of integers 1 through 10.

Show that $A(x_{1},\ldots ,x_{n})\geq G(x_{1},\ldots ,x_{n})\geq H(x_{1},\ldots ,x_{n})$ 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},\ldots ,x_{n})={\frac {x_{1}+\cdots +x_{n}}{n}}

The geometric mean is the $n$ th root of the product of the list:

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

The harmonic mean is $n$ divided by the sum of the reciprocal of each item in the list:

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

C.f. Averages/Root mean square

Clojure

<lang lisp>(use '[clojure.contrib.math :only (expt)])

(defn a-mean [coll]

 (/ (reduce + coll) (count coll)))

(defn g-mean [coll]

 (expt (reduce * coll) (/ (count coll))))

(defn h-mean [coll]

 (/ (count coll) (reduce + (map / coll))))

(let [numbers (range 1 11)

     a (a-mean numbers) g (g-mean numbers) h (h-mean numbers)]
 (println a ">=" g ">=" h)
 (>= a g h))</lang>

Factor

<lang factor>: a-mean ( seq -- mean )

   [ sum ] [ length ] bi / ;

g-mean ( seq -- mean )

   [ product ] [ length recip ] bi ^ ;

h-mean ( seq -- mean )

   [ length ] [ [ recip ] map-sum ] bi / ;</lang>

( scratchpad ) 10 [1,b] [ a-mean ] [ g-mean ] [ h-mean ] tri
               "%f >= %f >= %f\n" printf
5.500000 >= 4.528729 >= 3.414172

Haskell

This example is incorrect. Please fix the code and remove this message.

Details: Need to show the relationship between A,G and H

The general function given here yields an arithmetic mean when its first argument is 1, a geometric mean when its first argument is 0, and a harmonic mean when its first argument is -1.

<lang haskell>import Data.List (genericLength) import Control.Monad (zipWithM_)

mean :: Double -> [Double] -> Double mean 0 xs = product xs ** (1 / genericLength xs) mean p xs = (1 / genericLength xs * sum (map (** p) xs)) ** (1/p)

main = zipWithM_ f "agh" (map (flip mean [1 .. 10]) [1, 0, -1])

where f c n = putStrLn $ c : ": " ++ show n</lang>

J

Solution: <lang j>amean=: +/ % # gmean=: # %: */ hmean=: amean&.:%</lang>

Example Usage: <lang j> (amean , gmean , hmean) >: i. 10 5.5 4.528729 3.414172

  assert 2 >:/\ (amean , gmean , hmean) >: i. 10    NB. check amean >= gmean and gmean >= hmean</lang>

Lua

<lang lua>function fsum(f, a, ...) return a and f(a) + fsum(f, ...) or 0 end function pymean(t, f, finv) return finv(fsum(f, unpack(t)) / #t) end nums = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

--arithmetic a = pymean(nums, function(n) return n end, function(n) return n end) --geometric g = pymean(nums, math.log, math.exp) --harmonic h = pymean(nums, function(n) return 1/n end, function(n) return 1/n end) print(a, g, h) assert(a >= g and g >= h)</lang>

Oz

<lang oz>declare

 %% helpers
 fun {Sum Xs} {FoldL Xs Number.'+' 0.0} end
 fun {Product Xs} {FoldL Xs Number.'*' 1.0} end
 fun {Len Xs} {Int.toFloat {Length Xs}} end

 fun {AMean Xs}
    {Sum Xs}
    /
    {Len Xs}
 end

 fun {GMean Xs}
    {Pow
     {Product Xs}
     1.0/{Len Xs}}
 end

 fun {HMean Xs}
    {Len Xs}
    /
    {Sum {Map Xs fun {$ X} 1.0 / X end}}
 end

 Numbers = {Map {List.number 1 10 1} Int.toFloat}

 [A G H] = [{AMean Numbers} {GMean Numbers} {HMean Numbers}]

in

 {Show [A G H]}
 A >= G = true
 G >= H = true</lang>

PL/I

<lang PL/I> declare n fixed binary,

       (Average, Geometric, Harmonic) float;

declare A(10) float static initial (1,2,3,4,5,6,7,8,9,10);

n = hbound(A,1);

/* compute the average */ Average = sum(A)/n;

/* Compute the geometric mean: */ Geometric = prod(A)**(1/n);

/* Compute the Harmonic mean: */ Harmonic = n / sum(1/A);

put skip data (Average); put skip data (Geometric); put skip data (Harmonic);

if Average < Geometric then put skip list ('Error'); if Geometric < Harmonic then put skip list ('Error'); </lang>

Python

Works with: Python 3

<lang Python>from operator import mul from functools import reduce

def amean(num): return sum(num)/len(num)

def gmean(num): return reduce(mul, num, 1)**(1/len(num))

def hmean(num): return len(num)/sum(1/n for n in num)

numbers = range(1,11) # 1..10 a, g, h = amean(numbers), gmean(numbers), hmean(numbers) print(a, g, h) assert( a >= g >= h ) </lang>

Output:

5.5 4.52872868812 3.41417152147

Tcl

<lang tcl>proc arithmeticMean list {

   set sum 0.0
   foreach value $list { set sum [expr {$sum + $value}] }
   return [expr {$sum / [llength $list]}]

} proc geometricMean list {

   set product 1.0
   foreach value $list { set product [expr {$product * $value}] }
   return [expr {$product ** (1.0/[llength $list])}]

} proc harmonicMean list {

   set sum 0.0
   foreach value $list { set sum [expr {$sum + 1.0/$value}] }
   return [expr {[llength $list] / $sum}]

}

set nums {1 2 3 4 5 6 7 8 9 10} set A10 [arithmeticMean $nums] set G10 [geometricMean $nums] set H10 [harmonicMean $nums] puts "A10=$A10, G10=$G10, H10=$H10" if {$A10 >= $G10} { puts "A10 >= G10" } if {$G10 >= $H10} { puts "G10 >= H10" }</lang>