,
median and
mode of two
log-normal distributions with the same medians and different skewnesses. Other measures of skewness have been used, including simpler calculations suggested by
Karl Pearson (not to be confused with Pearson's moment coefficient of skewness, see above). These other measures are:
Pearson's first skewness coefficient (mode skewness) The Pearson mode skewness, or first skewness coefficient, is defined as
Pearson's second skewness coefficient (median skewness) The Pearson median skewness, or second skewness coefficient, is defined as Which is a simple multiple of the
nonparametric skew.
Quantile-based measures Bowley's measure of skewness (from 1901), also called '''Yule's coefficient''' (from 1912) is defined as: \frac{\frac{Q(3/4) + Q(1/4)}{2} - Q(1/2)}{\frac{Q(3/4) - Q(1/4)}{2}} = \frac{Q(3/4) + Q(1/4) - 2 Q(1/2)}{Q(3/4) - Q(1/4)}, where
Q is the
quantile function (i.e., the inverse of the
cumulative distribution function). The numerator is difference between the average of the upper and lower quartiles (a measure of location) and the median (another measure of location), while the denominator is the
semi-interquartile range (Q(3/4)}-{Q(1/4))/2, which for symmetric distributions is equal to the
MAD measure of
dispersion. Other names for this measure are Galton's measure of skewness, the Yule–Kendall index and the quartile skewness, Similarly, Kelly's measure of skewness is defined as \frac{Q(9/10) + Q(1/10) - 2 Q(1/2)}{Q(9/10) - Q(1/10)}. A more general formulation of a skewness function was described by Groeneveld, R. A. and Meeden, G. (1984): \gamma( u ) = \frac{ Q( u ) +Q( 1 - u )-2Q( 1 / 2 ) }{Q( u ) -Q( 1 - u ) } The function satisfies and is well defined without requiring the existence of any moments of the distribution. defined as the
supremum of this over the range . Another measure can be obtained by integrating the numerator and denominator of this expression.
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 . If
X is a random variable taking values in the -dimensional Euclidean space, has finite expectation, is an independent identically distributed copy of , and \|\cdot\| denotes the norm in the Euclidean space, then a simple
measure of asymmetry with respect to location parameter is \operatorname{dSkew}(X) := 1 - \frac{\operatorname{E}\|X-X'\|}{\operatorname{E}\|X+X'-2 \theta\|} \text{ if } \Pr(X=\theta)\ne 1 and for (with probability 1). Distance skewness is always between 0 and 1, equals 0 if and only if
X is diagonally symmetric with respect to ( and have the same probability distribution) and equals 1 if and only if
X is a constant
c (c \neq \theta) with probability one. Thus there is a simple consistent
statistical test of diagonal symmetry based on the
sample distance skewness: \operatorname{dSkew}_n(X) := 1 - \frac{\sum_{i,j} \|x_i-x_j\| }{\sum_{i,j} \|x_i+x_j-2\theta \|}.
Medcouple The
medcouple is a scale-invariant robust measure of skewness, with a
breakdown point of 25%.{{cite journal h(x_i, x_j) = \frac{ (x_i - x_m) - (x_m - x_j)}{x_i - x_j} taken over all couples (x_i, x_j) such that x_i \geq x_m \geq x_j, where x_m is the median of the
sample \{x_1, x_2, \ldots, x_n\}. It can be seen as the median of all possible quantile skewness measures. == See also ==