Cubic mean

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

Definition

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

x ¯ c u b i c = 1 n ∑ i = 1 n x i 3 3 = x 1 3 + x 2 3 + ⋯ + x n 3 n 3 . {\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}}.} {\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:

x 1 3 + x 2 3 2 3 {\displaystyle {\sqrt[{3}]{\frac {x_{1}^{3}+x_{2}^{3}}{2}}}} {\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). Archived from the original (PDF) on June 13, 2018.
  2. Svarovski, Ladislav (31 October 2000). Solid-Liquid Separation. Elsevier. ISBN 9780080541440. Retrieved 2015-01-20.
  3. "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. Academic Press. p. 696. ISBN 9780123972194.
  8. Rodina, Antonina Gavrilovna (1972). Methods in aquatic microbiology. University Park Press. 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.