Home Original page

Function Centroid

FunctionCentroid

By analogy with the geometric centroid, the centroid of an arbitrary function f(x) is defined as

<x>=(intxf(x)dx)/(intf(x)dx),

(1)

where the integrals are taken over the domain of f(x). For example, for the Gaussian function f(x)=e^(-(x-x_0)^2/(2sigma^2)), the centroid is

<x>=(int_(-infty)^inftyxe^(-(x-x_0)^2/(2sigma^2))dx)/(int_(-infty)^inftye^(-(x-x_0)^2/(2sigma^2))dx)=(sigmasqrt(2pi)x_0)/(sigmasqrt(2pi))=x_0.

(2)

If f(x) is normalized so that

intf(x)dx=1,

(3)

then its centroid is equivalent to its mean.

See also

Geometric Centroid, Mean, Triangle Centroid

Explore with Wolfram|Alpha

References

Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 139-140 and 156, 1999.

Referenced on Wolfram|Alpha

Function Centroid

Cite this as:

Weisstein, Eric W. "Function Centroid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FunctionCentroid.html

Subject classifications