The cubic mean (written as ) is a specific instance of the generalized mean with .
Definition
For real numbers 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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b779561c8e64a5bfda052e9a4847cb7683e25b7c)
For example, the cubic mean of two numbers is:
- .
![{\displaystyle {\sqrt[{3}]{\frac {x_{1}^{3}+x_{2}^{3}}{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e600e24ebb4cc85ef2dc73bafdab5366ff343693)
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
- ^ "calculation formulas" (PDF). Archived from the original (PDF) on June 13, 2018.
- ^ Svarovski, Ladislav (31 October 2000). Solid-Liquid Separation. Elsevier. ISBN 9780080541440. Retrieved 2015-01-20.
- ^ a b "Equivalent Load". Creative Motion Control. Archived from the original on 2015-01-20. Retrieved 2015-01-19.
- ^ "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.
- ^ Babu & Sridhar (2010). Design of Machine Elements. McGraw-Hill Education (India) Pvt Limited. ISBN 9780070672840. Retrieved 2015-01-20.
- ^ Harris, Tedric A.; Kotzalas, Michael N. (9 October 2006). Essential Concepts of Bearing Technology, Fifth Edition. CRC Press. ISBN 9781420006599. Retrieved 2015-01-20.
- ^ Da Rosa, Aldo Vieira (2013). Fundamentals of renewable energy processes. Academic Press. p. 696. ISBN 9780123972194.
- ^ Rodina, Antonina Gavrilovna (1972). Methods in aquatic microbiology. University Park Press. p. 158. ISBN 083910071X.
- ^ 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.
