Skewness
Distance skewness

A value of skewness equal to zero does not imply that the probability distribution is symmetric. Thus there is a need for another measure of asymmetry that has this property: such a measure was introduced in 2000. It is called distance skewness and denoted by dSkew. If X is a random variable taking values in the d -dimensional Euclidean space, X has finite expectation, X ' is an independent identically distributed copy of X, and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} denotes the norm in the Euclidean space, then a simple measure of asymmetry with respect to location parameter θ is

and dSkew(X) := 0 for X = θ (with probability 1). Distance skewness is always between 0 and 1, equals 0 if and only if X is diagonally symmetric with respect to θ (X and 2θ− X have the same probability distribution) and equals 1 if and only if X is a constant c (c ≠ θ {\displaystyle c\neq \theta }) with probability one. Thus there is a simple consistent statistical test of diagonal symmetry based on the sample distance skewness:
combination:two_responses