What property of a distribution is measured by its skewness?
One measure of skewness, \(\gamma\), is given by
\[
\displaystyle
\gamma=
\frac{ \E\big((X-\mu)^3\big)}{\sigma^3}\,,
\]
where \(\mu\) and \(\sigma^2\) are the mean and variance of the random variable \(X\).
Show that
\[
\gamma = \frac{ \E(X^3) -3\mu \sigma^2 - \mu^3}{\sigma^3}\,.
\]
The continuous random variable \(X\) has probability density function \(\f\) where
\[
\f(x)
= \begin{cases}
2x & \text{for } 0\le x\le 1\,, \\[2mm]
0 & \text{otherwise}\,.
\end{cases}
\]
Show that for this distribution \(\gamma= -\dfrac{2\sqrt2}{5}\).
The decile skewness, \(D\), of a distribution is defined by
\[D=
\frac { {\rm F}^{-1}(\frac9{10}) - 2{\rm F} ^{-1}(\frac12) + {\rm F}^{-1} (\frac1{10}) }
{{\rm F}^{-1}(\frac9{10}) - {\rm F} ^{-1} (\frac1{10})}\,,
\]
where \({\rm F}^{-1}\) is the inverse of the cumulative distribution function.
Show that, for the above distribution, \( D= 2 -\sqrt5\,.\)
The Pearson skewness, \(P\), of a distribution is defined by
\[
P = \frac{3(\mu-M)}{\sigma}
\,,\]
where \(M\) is the median.
Find \(P\) for the above distribution and show that \(D > P > \gamma\,\).
Solution: Skewness is a measure of the symmetry (specifically the lack-thereof) in the distribution. How much mass is there on one side rather than another.