You can get an immediate idea of how critical PQn is by examining the distribution where PQn is equal to 5. The graph below shows the distribution of the means of 5000 equal-sized random samples from populations containing P successes. We have repeated these simulation for range of sample sizes, but arranged things such that in each case PQn is close to 5.

{Fig. 1}

There is surprisingly little change in the shape of this distribution with sample size. About the most we can say is, when compared to a normal distribution, it tends to be slightly skewed to the right.

The illustration below shows how the distribution changes where PQn is much below 5.

{Fig. 2}

There seems to be limited scope for transforming most of these distributions to normal. Below, we have re-plotted these results using a logarithmic scale.

{Fig. 3}

Now we can see why log transformations are not used to normalize data where PQn is less than 5. Unfortunately, whilst the log transformation have given these distributions approximately the correct shape, they are truncated. The normal distribution cannot be sensibly fitted nor defended in these circumstances.