# standard error for difference in proportions between a population and a subset

I have a reference population and a non-random subset of that population. I am interested in the difference in proportions between the subset and the population.

If my subset were a random sample of the population, then the standard error of this sample proportion would be simply $\sqrt{\frac{p(1-p)}{n}}$. But the subset was not selected randomly.

I believe that a pooled estimate of standard error of this difference is not appropriate since (a) the groups intersect and (b) the population proportion is known. (Also, in my particular case, since the subset is fairly small, typically less than 1% of the population, the pooled estimate would approach that of the subset.)

What is the typical practice in this case?

Are there any other pitfalls when comparing proportions of a population and a subset?

• Am I overthinking this? If the population proportion is a known constant, then the standard error of the difference between the subset proportion and the constant is the same as the standard error of the subset proportion? – C8H10N4O2 Jan 14 '16 at 15:56