The problems of approximating this (exact) population distribution using a parametric model become explicit if we attempt to fit either a normal or lognormal distribution.
For example,
- You might assume the location of this distribution would 1, in other words where _{T} = _{P}. However the mean of this population of statistics is 1.169, the geometric mean is 0.9543, and their median has to be interpolated between 0.72 and 1.19
Moreover, given the catches she observed, none of these values are possible under this model.
- Although the (population) standard deviation can be worked out readily enough (it is 10/9 × the sample standard deviation) there is no satisfactory way of rescaling the probability density to the second graph because there are no consistent class-intervals.
- Aside from the problems of approximating discrete distributions using continuous ones, the all-important tails of the exact and parametric distributions are radically different.