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Cubic root of the mean of the cubes
The cubic mean (written as
x
¯
c
u
b
i
c
{\displaystyle {\bar {x}}_{\mathrm {cubic} }}
generalized mean with
p
=
3
{\displaystyle p=3}
Definition For
n
{\displaystyle n}
real numbers
x
i
∈
R
{\displaystyle x_{i}\in \mathbb {R} }
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}}.}
[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}}}}
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
The cubic mean is also used in
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
![{\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b779561c8e64a5bfda052e9a4847cb7683e25b7c)
![{\displaystyle {\sqrt[{3}]{\frac {x_{1}^{3}+x_{2}^{3}}{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e600e24ebb4cc85ef2dc73bafdab5366ff343693)
