Average cubic

Cubic average (also average cubic ^[1] ) - number $x$ ${\ displaystyle x}$ $x$ equal to the cubic root of the arithmetic mean cubes of given numbers $a_{1},a_{2},...,a_{n}$ ${\ displaystyle a_ {1}, a_ {2}, ..., a_ {n}}$ $a_ {1}, a_ {2}, ..., a_ {n}$ :

x={\sqrt[{3}]{\frac {a_{1}^{3}+a_{2}^{3}+\ldots +a_{n}^{3}}{n}}}

{\ displaystyle x = {\ sqrt [{3}] {\ frac {a_ {1} ^ {3} + a_ {2} ^ {3} + \ ldots + a_ {n} ^ {3}} {n} }}}

{\ displaystyle x = {\ sqrt [{3}] {\ frac {a_ {1} ^ {3} + a_ {2} ^ {3} + \ ldots + a_ {n} ^ {3}} {n} }}}

Properties

The cubic average is a special case of the average power and therefore obeys the inequality about the means . In particular, for any numbers it is not less than the arithmetic mean :

{\frac {a_{1}+a_{2}+\ldots +a_{n}}{n}}\leqslant {\sqrt[{3}]{\frac {a_{1}^{3}+a_{2}^{3}+\ldots +a_{n}^{3}}{n}}}

{\ displaystyle {\ frac {a_ {1} + a_ {2} + \ ldots + a_ {n}} {n}} \ leqslant {\ sqrt [{3}] {\ frac {a_ {1} ^ {3 } + a_ {2} ^ {3} + \ ldots + a_ {n} ^ {3}} {n}}}}

Application

The cubic average is a characteristic of volumetric features. It can be used, for example, to calculate the average volume of objects by their diameters. So, if the diameters of the eggs are known, then their average volume can be calculated using the average cubic ^[1] . The cubic average is used in statistics ^[2] .

Notes

↑ ¹ ² Frolov K.V.,. Encyclopedia of Mechanical Engineering XXL . - Engineering. - 1994-2013. - S. 42.
↑ Geometric mean, harmonic mean, quadratic mean and cubic mean (Russian) . Date of treatment May 20, 2018.

[xxl-1] ¹ ² Frolov K.V.,. Encyclopedia of Mechanical Engineering XXL . - Engineering. - 1994-2013. - S. 42.

[2] Geometric mean, harmonic mean, quadratic mean and cubic mean (Russian) . Date of treatment May 20, 2018.

Average cubic

Properties

Application

Notes

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