Cubic mean explained

The cubic mean (written as

\bar{x} _{cubic} ) is a specific instance of the generalized mean with p =3 .

Definition

For

n real numbers x _{i} \in R the cubic mean is defined as:\bar{x} _{cubic} = \sqrt[ 3 ]{ (1)/(n) \sum ^{3 }}} _{i} = \sqrt[ 3 ]{{x ^{3} _{1} + x ^{3} _{2} + … + x ^{3 }} _{n} \over n}.    ^{[1] [2] [3]}

For example, the cubic mean of two numbers is:

\sqrt[ 3 ]{ (x ^{3} _{2})/(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.

Notes and References

1. Web site: calculation formulas.
2. Book: Svarovski. Ladislav. Solid-Liquid Separation. 31 October 2000. Elsevier . 9780080541440. 2015-01-20.
3. Web site: Equivalent Load. Creative Motion Control. 2015-01-19. https://web.archive.org/web/20150120013504/http://www.creativemotioncontrol.com/home-copy/cmc-roller-screws/calculation-and-selection/equivalent-load/. 2015-01-20. dead.
4. Web site: ISO 4301-1-1986 Cranes and lifting appliances; Classification; Part 1 : General. 1986. www.iso.org. also available from Web site: freestd.us. 2015-01-20.
5. Book: Babu & Sridhar. Design of Machine Elements. 2010. McGraw-Hill Education (India) Pvt Limited . 9780070672840. 2015-01-20.
6. Book: Essential Concepts of Bearing Technology, Fifth Edition. 9781420006599. 2015-01-20. Harris. Tedric A.. Kotzalas. Michael N.. 9 October 2006. CRC Press .
7. Book: Da Rosa, Aldo Vieira. Fundamentals of renewable energy processes. 696. 2013. 9780123972194.
8. Book: Rodina, Antonina Gavrilovna. Methods in aquatic microbiology. 158. 083910071X. 1972.
9. Book: Rice. Dale W.. Wolman. Allen A.. The life history and ecology of the gray whale (Eschrichtius robustus). 34. 1971. American Society of Mammalogists. Stillwater, Oklahoma.