几何平均数

（重定向自几何平均

${\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}}$

計算

${\displaystyle \left(\prod _{i=1}^{n}a_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}}}.}$

${\displaystyle a_{n+1}={\frac {a_{n}+h_{n}}{2}},\quad a_{0}=x}$

${\displaystyle h_{n+1}={\frac {2}{{\frac {1}{a_{n}}}+{\frac {1}{h_{n}}}}},\quad h_{0}=y}$

{\displaystyle {\begin{aligned}{\sqrt {a_{i}h_{i}}}&={\sqrt {\frac {a_{i}+h_{i}}{\frac {a_{i}+h_{i}}{h_{i}a_{i}}}}}\\&={\sqrt {\frac {a_{i}+h_{i}}{{\frac {1}{a_{i}}}+{\frac {1}{h_{i}}}}}}\\&={\sqrt {a_{i+1}h_{i+1}}}\end{aligned}}}

與對數的關係

${\displaystyle a_{1},a_{2},\dots ,a_{n}>0}$ ，則
${\displaystyle \left(\prod _{i=1}^{n}a_{i}\right)^{\frac {1}{n}}=\exp \left[{\frac {1}{n}}\sum _{i=1}^{n}\ln a_{i}\right]}$

${\displaystyle \left(\prod _{i=1}^{n}a_{i}\right)^{\frac {1}{n}}=\left(-1\right)^{m}\exp \left[{\frac {1}{n}}\sum _{i=1}^{n}\ln \left|a_{i}\right|\right]}$

${\displaystyle b^{{\frac {1}{2}}\left[\log _{b}2+\log _{b}8\right]}=4}$

在恆定時間內計算

${\displaystyle \left({\frac {a_{n}}{a_{0}}}\right)^{\frac {1}{n}},}$

${\displaystyle n}$ 是從初始狀態到最終狀態的次數。

${\displaystyle \left({\frac {a_{1}}{a_{0}}}{\frac {a_{2}}{a_{1}}}\cdots {\frac {a_{n}}{a_{n-1}}}\right)^{\frac {1}{n}}=\left({\frac {a_{n}}{a_{0}}}\right)^{\frac {1}{n}}}$

屬性

${\displaystyle \operatorname {GM} \left({\frac {X_{i}}{Y_{i}}}\right)={\frac {\operatorname {GM} (X_{i})}{\operatorname {GM} (Y_{i})}}}$

電腦 A 電腦 B 電腦 C

電腦 A 電腦 B 電腦 C

電腦 A 電腦 B 電腦 C

電腦 A 電腦 B 電腦 C

應用

長寬比

Kerns Powers使用的縱橫比的相等面積比較推導出電影電視工程師協會16：9標準。[10] 　 TV 4:3/1.33 紅色，　 1.66 橘色，　 16:9/1.77 藍色　 1.85 黃色，　 潘那維申/2.2 洋紅色和 　 CinemaScope/2.35 紫色。

參考文獻

1. ^ Matt Friehauf, Mikaela Hertel, Juan Liu, and Stacey Luong On Compass and Straightedge Constructions: Means (PDF). UNIVERSITY of WASHINGTON, DEPARTMENT OF MATHEMATICS. 2013 [14 June 2018]. （原始内容存档 (PDF)于2018-06-14）.
2. ^ Euclid, Book VI, Proposition 13
3. ^ TPC-D – Frequently Asked Questions (FAQ). Transaction Processing Performance Council. [9 January 2012]. （原始内容存档于2011-11-04）.
4. ^ 幾何平均數只適用于同一個符號的數位，以避免取負乘積的根，從而產生虛數，也能滿足某種方法的某些性質，本文稍後將對此進行解釋。定義是明確的，如果你允許0（產生幾何平均數0），但可能被排除，因為你經常想要採取幾何手段的對數（在乘法和加法之間轉換），並且你不可能採取對數0。
5. ^ Crawley, Michael J. Statistics: An Introduction using R. John Wiley & Sons Ltd. 2005. ISBN 9780470022986.
6. ^ Mitchell, Douglas W. More on spreads and non-arithmetic means. The Mathematical Gazette. 2004, 88: 142–144.
7. ^ Fleming, Philip J.; Wallace, John J. How not to lie with statistics: the correct way to summarize benchmark results. Communications of the ACM. 1986, 29 (3): 218–221. doi:10.1145/5666.5673.
8. ^ Smith, James E. Characterizing computer performance with a single number. Communications of the ACM. 1988, 31 (10): 1202–1206. doi:10.1145/63039.63043.
9. ^ Frequently Asked Questions - Human Development Reports. hdr.undp.org. [2018-10-06]. （原始内容存档于2011-03-02）.
10. TECHNICAL BULLETIN: Understanding Aspect Ratios (PDF). The CinemaSource Press. 2001 [2009-10-24]. （原始内容 (PDF)存档于2009-09-09）.
11. ^ US 5956091,「Method of showing 16:9 pictures on 4:3 displays」,发行于September 21, 1999
12. Rowley, Eric E. The Financial System Today. Manchester University Press. 1987. ISBN 0719014875.