Cubic mean

The cubic mean (written as $\bar{x}_\mathrm{cubic}$ ) is a specific instance of the generalized mean with $p=3$ .

Definition

For $n$ real numbers $x_i \in \mathbb R$ the cubic mean is defined as:

: $\bar{x}_\mathrm{cubic} = \sqrt[3]{\frac{1}{n}\sum_{i=1}^n{x_i^3}} = \sqrt[3]{{x_1^3 + x_2^3 + \cdots + x_n^3} \over n}.$ {{cite web|title=calculation formulas|url=http://prmeh.ru/pub/catalogues/skf/data/skf_products/linear_motion/screws/calculation_formules.pdf}}{{cite book|last1=Svarovski|first1=Ladislav|title=Solid-Liquid Separation|date=31 October 2000|publisher=Elsevier |isbn=9780080541440|url=https://books.google.com/books?id=8dj5IJbEwu4C&q=cubic+mean+arithmetic+mean&pg=PA38|accessdate=2015-01-20}}{{cite web|title=Equivalent Load|url=http://www.creativemotioncontrol.com/home-copy/cmc-roller-screws/calculation-and-selection/equivalent-load/|website=Creative Motion Control|access-date=2015-01-19|archive-url=https://web.archive.org/web/20150120013504/http://www.creativemotioncontrol.com/home-copy/cmc-roller-screws/calculation-and-selection/equivalent-load/|archive-date=2015-01-20|url-status=dead}}

For example, the cubic mean of two numbers is:

: $\sqrt[3]{\frac{x_1^3+x_2^3}{2}}$ .

Applications

The cubic mean is used to predict the life expectancy of machine parts.{{cite web|title=ISO 4301-1-1986 Cranes and lifting appliances; Classification; Part 1 : General|year=1986|url=http://www.iso.org/iso/catalogue_detail.htm?csnumber=10151 |website=www.iso.org}} also available from {{cite web|url=http://www.freestd.us/soft/17753.htm|title=freestd.us|accessdate=2015-01-20}}{{cite book|last1=Babu & Sridhar|title=Design of Machine Elements|year=2010|publisher=McGraw-Hill Education (India) Pvt Limited |isbn=9780070672840|url=https://books.google.com/books?id=Px9kbvsG3WYC&q=cubic+mean+load&pg=PA354|accessdate=2015-01-20}}{{cite book|title=Essential Concepts of Bearing Technology, Fifth Edition|isbn = 9781420006599|url=https://books.google.com/books?id=0XYE2p0QyFwC&q=cubic+mean+load&pg=PA230|accessdate=2015-01-20|last1 = Harris|first1 = Tedric A.|last2 = Kotzalas|first2 = Michael N.|date = 9 October 2006| publisher=CRC Press }}

The cubic mean wind speed has been used a measure of local potential for wind energy.{{cite book|last=Da Rosa|first=Aldo Vieira|title=Fundamentals of renewable energy processes|page=696|year=2013|publisher=Academic Press |isbn=9780123972194}}

The cubic mean is also used in biology to measure the mean dimensions of spherical bacteria (cocci){{cite book|last=Rodina|first=Antonina Gavrilovna|title=Methods in aquatic microbiology|page=158|isbn=083910071X|year=1972|publisher=University Park Press }} and of larger animals that are approximately spheroidal in shape.{{cite book|last1=Rice|first1=Dale W.|last2=Wolman|first2=Allen A.|title=The life history and ecology of the gray whale (Eschrichtius robustus)|page=34|year=1971|publisher=American Society of Mammalogists|location=Stillwater, Oklahoma}} 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

Category:Means