Meaning of "Overdispersion" in Statistics I am trying to understand what "overdispersion" means in statistics.
Based on the Wikipedia page, "overdispersion" is defined as follows : "In statistics, overdispersion is the presence of greater variability (statistical dispersion) in a data set than would be expected based on a given statistical model."
However, I have heard other interpretations of "overdispersion" which suggest that "overdispersion refers to situations where the variance within the data is a function of the mean" - in other words, there is a non-constant relationship between the mean and variance within the data.
My Question: Can someone please tell how to mathematically measure and define "overdispersion"? For instance, I have heard that the Normal Distribution and the Poisson Distribution can be defined as "Dispersion Models". I have also heard that many models can be considered as "Dispersion Models" so long as a "Dispersion Parameter" can be inserted into the model. Using these definitions - is the Normal Distribution an example of "Overdispersion"? For example, there is a lot more variation in a Normal Distribution around the peak - and relatively less variation in a Normal Distribution around the tails. Is all this correct?
References:

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*https://en.wikipedia.org/wiki/Overdispersion

The only place I run into over-dispersion issues is when e.g. fitting a GLM model based on Poisson count data (Poisson regression).
As you know, for Poisson, the variance is equal to the mean, so
\begin{equation}
\mathrm{Var}(Y_i) = \mathrm{E}(Y_i).
\end{equation}
But commonly the variance exceeds the mean, so an attempt is made to recover the relationship above by introducing an over-dispersion parameter, $\phi$ in the functional form
\begin{equation}
\mathrm{Var}(Y_i) = \phi\mathrm{E}(Y_i),
\end{equation}
which is fitted along with the parameters during maximization of the log-likelihood.  For Poisson, $\phi=1$, and if you allow $\phi>1$, you no longer have a distribution from the exponential family.  The end result of implementing $\phi$ is that $V(\beta)$ is purposely inflated, leading to an over-estimation of standard errors -- essentially an under-statement of significance of the parameters.  By not taking $\phi$ into account, it leads to underestimated standard errors, or over-statement about significance.
Another work-around is to use quantile regression, which is hinged to ranks (non-parametrics).
