The discrete random variable X is a count variable of 45 Bernoulli trials and can take on values from 0 to 45. The mean is 2.34 and the variance is 18.39. The average probability of scoring 1 in a single Bernoulli trial is $p=2.34/45=0.05$. According to a rule of thumb, I could consider this binomial distribution approximating a Poisson distribution since p is small.
However, it seems that the assumption for a Poisson distribution, that the single Bernoulli trials are independent, does not hold, resulting in overdispersion (the variance is greater than the mean). For this reason I assume my variable approximates either a negative binomial distribution or an extra binomial (=beta binomial) distribution.
edit: it seems that the assumption is not that the 45 trials within one individual are independent, but that the 45 trials of one person is independent of the 45 trials in another person (?)
My question now is, how could I determine which of these two theoretical distributions (extraBin vs NegBin) best approximates my empirical distribution?
Proportion (count/45) distribution:
If any more information is needed to answer this question I gladly supply this.