This simplified variance formula can be justified by several routes.
- Where P is close to zero, then P2 approaches zero even more closely, and 1 - P approaches one. Therefore P[1 - P] becomes almost the same as P × 1, and P - P2 become indistinguishable from P - 0. So the population variance of a frequency, PQn or Pn - P2, reduces to Pn, and our sample estimate of it becomes pn.
- Or, where n is very large compared to f, then n - f is very similar to n, so [n - f]/n approaches one - and f[n-f]/n is the same as f. In this case f is our sample estimate of λ, in other words Pn.