Relation of variance and skewness for a skewed Gaussian variable? I understand that skewness and variance are not inherently related, but given the type of distribution (in this case a skewed Gaussian distribution), can a relationship between the variance and Pearson's skewness factor $$E\left(\frac{X-\mu}{\sigma}\right)^3$$ be found?

Consulting the side-bar of the wikipedia we see that they express it with three parameters: $\xi$, $\omega$ and $\alpha$. If the parameter $\alpha$ is set to zero, the distribution reduces to $N(\xi,\omega^2)$ which of course has zero skew.

The variance is in terms of $\omega$ and $\alpha$ and the skew is in terms of $\alpha$ alone. So when you change $\alpha$ to increase or decrease the skew it does change the variance for fixed $\omega.$ However, you can always adjust $\omega$ to compensate to keep the variance fixed. So you can independently vary the skew and the variance and there is no fixed relationship between them.