Statistics/Summary/Averages/Relationships among Arithmetic, Geometric and Harmonic Mean

Relationships among Arithmetic, Geometric and Harmonic Mean

The Means mentioned above are realizations of the generalized mean

${\bar {x}}(m)=\left({\frac {1}{n}}\cdot \sum _{i=1}^{n}{|x_{i}|^{m}}\right)^{1/m}$

and ordered this way:

${\begin{alignedat}{2}&{\mathit {minimum}}&\;=\;&{\bar {x}}(-\infty )\\{\mathrel {<}}\;&{\mathit {harmonic\ mean}}&\;=\;&{\bar {x}}(-1)\\{\mathrel {<}}\;&{\mathit {geometric\ mean}}&\;=\;&\lim _{m\rightarrow 0}{\bar {x}}(m)\\{\mathrel {<}}\;&{\mathit {arithmetic\ mean}}&\;=\;&{\bar {x}}(1)\\{\mathrel {<}}\;&{\mathit {maximum}}&\;=\;&{\bar {x}}(\infty )\end{alignedat}}$