Cubic mean

The cubic mean (written as ${\displaystyle {\bar {x}}_{\mathrm {cubic} }}$) is a specific instance of the generalized mean with ${\displaystyle p=3}$.

Definition[]

For ${\displaystyle n}$ real numbers ${\displaystyle x_{i}\in \mathbb {R} }$ the cubic mean is defined as:

${\displaystyle {\bar {x}}_{\mathrm {cubic} }={\sqrt[{3}]{{\frac {1}{n}}\sum _{i=1}^{n}{x_{i}^{3}}}}={\sqrt[{3}]{{x_{1}^{3}+x_{2}^{3}+\cdots +x_{n}^{3}} \over n}}.}$   ^[1][2][3]

For example, the cubic mean of two numbers is:

${\displaystyle {\sqrt[{3}]{\frac {x_{1}^{3}+x_{2}^{3}}{2}}}}$.

Applications[]

It is used for predicting the life expectancy of machine parts.^{[3][4] [5] [6]}

References[]