Is binomial dispersion the same as binomial distribution? I have been asked by a client to write a program in R which calculates the "binomial dispersion test." I have not found anything for this, but I have found that you can calculate the binomial distribution. 
Are they the same thing? I suspect they are.
 A: Based on the discussion in this link, it would appear that the binomial dispersion test assesses whether the dispersion of the data is substantially divergent from what we would expect of a binomial distribution. The binomial distribution and binomial dispersion test are not the same, but are clearly related, since the former assesses a feature of data which may have been drawn from the latter.
The binomial distribution is a probability distribution describing the results of a series Bernoulli trials with a specified size and probability of success. The classic example is describing the probability that a dice roll will return 6 if I roll the dice some number of times (say, 10). Knowing $Pr(6)$ allows one to describe the range of outcomes and the probability of each.
Here, dispersion is the concept underlying the term-of-art variance. This is important because the binomial distribution specifies a particular variance given the parameters (specifically, $np(1-p)$ for size $n$ and probability of success $p$). A sample with dispersion sufficiently far from its theoretical value under the assumption that it is drawn from a binomial distribution may be evidence that the distributional assumption is not met.
