Skewness Measures for the Weibull Distribution

The skewness of a statistical distribution is often used to determine whether that distribution is symmetric or not. Such a determination is misleading. To show this we have analyzed a broad range of (classes of) skewness measures – complying with the requirements of a general skewness measure – and applied the measures to the Weibull distribution. This distribution is known to be asymmetric for any value of its parameters. Nevertheless skewness measures might be equal to zero for a specific set of parameter values of the Weibull distribution, leading to the incorrect conclusion that the distribution is symmetric. Further, different general skewness measures might be zero for different values of the shape parameter (a) of the Weibull distribution, which makes it not possible to classify those distributions as being left- or right-skew. Consequently a Weibull distribution can be classified as being right-skew if a<1/(1-ln2), but cannot be classified (nor as being left-skew, nor as being symmetric) otherwise.