Why a Bivariate Distribution Has No Unique Standard Median

Summary

The document asks how to extend the familiar univariate definition of a median, obtained from the inverse cumulative distribution function at one half, to a bivariate distribution. Its answer is that there is no standard, unique median for multiple dimensions; generalized notions of multivariate median exist.

The response points to an external explanation but does not describe a particular definition, derive a method, or provide examples or evidence. The useful takeaway is conceptual: a bivariate median requires choosing a generalized definition, rather than directly applying the one-dimensional inverse-CDF rule. The note is brief and leaves the available definitions and their properties unexplained.

Key ideas

• The inverse-CDF definition gives a familiar median for a univariate distribution.
• There is no single standard median definition for a multivariate distribution.
• A bivariate median requires selecting a generalized notion suited to the intended use.

Tags

# How to define the median for bivariate function?


# How to define the median for bivariate function?












I know if we define a function f(x) and its cdf is F(x). The inverse function of cdf is inverseF.

I can define its median as follows: median = inverseF(0.5).

But if I want to get the median for a bivariate function f(x,y). How to define it? Thanks very much!

## Answer by Bob Jansen (score 1)

https://quant.stackexchange.com/a/11427

The standard median is not defined in multiple dimensions but a generalized notion exists as explained in this CrossValidated answer.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.