To check for overdispersion in GLM with a Poisson distribution one can compare the residual deviance with the residual degrees of freedom. If they are equal the Poisson error assumption is appropriate (The R Book, Crawley, 2007). If the residual deviance is bigger than the residual degrees of freedom overdispersion is indicated. Crawley indicates that overdispersion for Poisson distributions can be corrected with the quasi-Poisson distribution.
My question is how can I check for overdispersion with the Gaussian distribution and how can I correct for it?