Statistics/Summary/Averages/Relationships among Arithmetic, Geometric and Harmonic Mean
From Wikibooks, open books for an open world
Jump to navigation
Jump to search
Relationships among Arithmetic, Geometric and Harmonic Mean
[edit | edit source]
The Means mentioned above are realizations of the generalized mean
![{\displaystyle {\bar {x}}(m)=\left({\frac {1}{n}}\cdot \sum _{i=1}^{n}{|x_{i}|^{m}}\right)^{1/m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fff4329c5542ad5b2262f6c6e6ac3f06cb30dee)
and ordered this way:
![{\displaystyle {\begin{alignedat}{2}&{\mathit {minimum}}&\;=\;&{\bar {x}}(-\infty )\\{\mathrel {<}}\;&{\mathit {harmonic\ mean}}&\;=\;&{\bar {x}}(-1)\\{\mathrel {<}}\;&{\mathit {geometric\ mean}}&\;=\;&\lim _{m\rightarrow 0}{\bar {x}}(m)\\{\mathrel {<}}\;&{\mathit {arithmetic\ mean}}&\;=\;&{\bar {x}}(1)\\{\mathrel {<}}\;&{\mathit {maximum}}&\;=\;&{\bar {x}}(\infty )\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a457ae89ec9faa58a4b77b5ad7b457977cfce104)