Editing Mean (section)

===Pythagorean means===
{{Main|Pythagorean means}}
In mathematics, the three classical '''Pythagorean means''' are the [[arithmetic mean]] (AM), the [[geometric mean]] (GM), and the [[harmonic mean]] (HM). These means were studied with proportions by [[Pythagoreans]] and later generations of Greek mathematicians<ref>{{cite book|first=Thomas|last=Heath|title=History of Ancient Greek Mathematics}}</ref> because of their importance in geometry and music.
==== Arithmetic mean (AM) ====
{{Main| Arithmetic mean }}
The [[arithmetic mean]] (or simply ''mean'' or ''average'') of a list of numbers, is the sum of all of the numbers divided by their count. Similarly, the mean of a sample <math>x_1,x_2,\ldots,x_n</math>, usually denoted by <math>\bar{x}</math>, is the sum of the sampled values divided by the number of items in the sample.

:<math> \bar{x} = \frac{1}{n}\left (\sum_{i=1}^n{x_i}\right ) = \frac{x_1+x_2+\cdots +x_n}{n} </math>

For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is:
:<math>\frac{4+36+45+50+75}{5} = \frac{210}{5} = 42.</math>

==== Geometric mean (GM) ====
The [[geometric mean]] is an average that is useful for sets of positive numbers, that are interpreted according to their product (as is the case with rates of growth) and not their sum (as is the case with the arithmetic mean):
:<math>\bar{x} = \left( \prod_{i=1}^n{x_i} \right )^\frac{1}{n} = \left(x_1 x_2 \cdots x_n \right)^\frac{1}{n}</math>  <ref name=":2">{{Cite web|title=Mean {{!}} mathematics|url=https://www.britannica.com/science/mean|access-date=2020-08-21|website=Encyclopedia Britannica|language=en}}</ref>

For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:
:<math>(4 \times 36 \times 45 \times 50 \times 75)^\frac{1}{5} = \sqrt[5]{24\;300\;000} = 30.</math>

==== Harmonic mean (HM) ====
The [[harmonic mean]] is an average which is useful for sets of numbers which are defined in relation to some [[Unit of measurement|unit]], as in the case of [[speed]] (i.e., distance per unit of time):
 
:<math> \bar{x} = n \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1}</math>

For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is

:<math>\frac{5}{\tfrac{1}{4}+\tfrac{1}{36}+\tfrac{1}{45} + \tfrac{1}{50} + \tfrac{1}{75}} = \frac{5}{\;\tfrac{1}{3}\;} = 15.</math>

If we have five pumps that can empty a tank of a certain size in respectively 4, 36, 45, 50, and 75 minutes, then the harmonic mean of <math>15</math>
tells us that these five different pumps working together will pump at the same rate as much as five pumps that can each empty the tank in <math>15</math> minutes.

==== Relationship between AM, GM, and HM ====
{{AM_GM_inequality_visual_proof.svg}}
{{Main|QM-AM-GM-HM inequalities}}

AM, GM, and HM of [[nonnegative]] [[Real number|real numbers]] satisfy these inequalities:<ref>{{Cite book |last1=Djukić |first1=Dušan |url=https://books.google.com/books?id=okx0d9jdM8oC&pg=PA7 |title=The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2009 Second Edition |last2=Janković |first2=Vladimir |last3=Matić |first3=Ivan |last4=Petrović |first4=Nikola |date=2011-05-05 |publisher=Springer Science & Business Media |isbn=978-1-4419-9854-5 |language=en}}</ref>

:<math> \mathrm{AM} \ge \mathrm{GM} \ge \mathrm{HM} \, </math>

Equality holds if all the elements of the given sample are equal.