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.

## References

1. ^ "calculation formulas" (PDF).
2. ^ Svarovski, Ladislav (31 October 2000). Solid-Liquid Separation. Elsevier. ISBN 9780080541440. Retrieved 2015-01-20.
3. ^ a b "Equivalent Load". Creative Motion Control. Archived from the original on 2015-01-20. Retrieved 2015-01-19.
4. ^ "ISO 4301-1-1986 Cranes and lifting appliances; Classification; Part 1 : General". www.iso.org. 1986. also available from "freestd.us". Retrieved 2015-01-20.
5. ^ Babu & Sridhar (2010). Design of Machine Elements. McGraw-Hill Education (India) Pvt Limited. ISBN 9780070672840. Retrieved 2015-01-20.
6. ^ Harris, Tedric A.; Kotzalas, Michael N. (9 October 2006). Essential Concepts of Bearing Technology, Fifth Edition. CRC Press. ISBN 9781420006599. Retrieved 2015-01-20.
7. ^ Da Rosa, Aldo Vieira (2013). Fundamentals of renewable energy processes. p. 696. ISBN 9780123972194.
8. ^ Rodina, Antonina Gavrilovna (1972). Methods in aquatic microbiology. p. 158. ISBN 083910071X.
9. ^ Rice, Dale W.; Wolman, Allen A. (1971). The life history and ecology of the gray whale (Eschrichtius robustus). Stillwater, Oklahoma: American Society of Mammalogists. p. 34.