The cubic mean (written as

\bar_\mathrm

) is a specific instance of the

generalized mean In mathematics, generalised means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic mean, arithmetic, geometric mean, ge ...

with

p=3

Definition

For

n

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

x_i \in \mathbb R

the cubic mean is defined as: :

\bar_\mathrm = \sqrt = \sqrt

For example, the cubic mean of two numbers is: :

\sqrt /math>.

Applications

The cubic mean is used to predict the life expectancy of machine parts. The cubic mean wind speed has been used a measure of local potential for

wind energy Wind power is the use of wind energy to generate useful work. Historically, wind power was used by sails, windmills and windpumps, but today it is mostly used to generate electricity. This article deals only with wind power for electricity ...

. The cubic mean is also used in

biology Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...

to measure the mean dimensions of spherical

bacteria Bacteria (; : bacterium) are ubiquitous, mostly free-living organisms often consisting of one Cell (biology), biological cell. They constitute a large domain (biology), domain of Prokaryote, prokaryotic microorganisms. Typically a few micr ...

(

cocci Bacterial cellular morphologies are the shapes that are characteristic of various types of bacteria and often key to their identification. Their direct examination under a light microscope enables the classification of these bacteria (and archaea ...

) and of larger animals that are approximately spheroidal in shape. In this case using the conventional

arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...

will not give an accurate result because the size of a spherical bacterium increases as the ''cube'' of the radius.

References

{{Statistics, descriptive Means