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

The cubic mean is used to predict the life expectancy of machine parts.^[3][4][5][6]

The cubic mean wind speed has been used a measure of local potential for wind energy.^[7]

The cubic mean is also used in biology to measure the mean dimensions of spherical bacteria (cocci)^[8] and of larger animals that are approximately spheroidal in shape.^[9] In this case using the conventional arithmetic mean will not give an accurate result because the size of a spherical bacterium increases as the cube of the radius.