Over-dispersion can occur with one-parameter distributions, where mean and variance are tied together (Poisson, Binomial, Exponential). In real data, variance is usually much greater than would be allowed. Over-dispersion creates over-confidence (e.g. too narrow CIs), but usually does not introduce biases. In practical modelling, this problem can be resolved in one of three ways:
- quasi-likelihood or generalized equation estimation
- two-parameter distributions, such as negative-binomial or beta-binomial
- observation-level random effects
I am discussing the issue and solutions 2 & 3 in my book.