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let f(x) be the probability density function of X. We can define the right and the left and right hand sided moments of X with respect to m as follows:

Left hand one sided k-th moment of x with respect to m = int_[-inf m] (x-m)^k m-x)^k f(x) dx

Right hand one sided k-th moment of x with respect to m = int[m inf] (x-m)^k = f(x) dx

One observes the following analogies

1. The median is the statistic for which the zeroth left and right hand one sided moments are equal (the zeroth moments are just probabilities)

2. For the mean, the first left and right hand one sided moments are equal.

3. For the statistic defined in the question, the second left and right hand one sided moments are equal.

1

let f(x) be the probability density function of X. We can define the right and the left and right hand sided moments of X with respect to m as follows:

Left hand one sided k-th moment of x with respect to m = int_[-inf m] (x-m)^k f(x) dx

Right hand one sided k-th moment of x with respect to m = int[m inf] (x-m)^k = f(x) dx

One observes the following analogies

1. The median is the statistic for which the zeroth left and right hand one sided moments are equal (the zeroth moments are just probabilities)

2. For the mean, the first left and right hand one sided moments are equal.

3. For the statistic defined in the question, the second left and right hand one sided moments are equal.