What is the sampling distribution of the difference between two proportions when they are parts of the same whole. It's relatively easy when one proportion is binomial, the one of the common ways to approximate the standard error is to use a normal approximation = $\sqrt{\frac{p(1-p)}{n}}$.
However, say I have a survey where the respondents have 3 options for an answer: positive, negative, neutral; and I am interested in the sampling distribution of the difference between the proportion of positive responses and the proportion of negative responses, a sort-of a net-positve score. I can't add the variances of the sampling distributions of each proportion (treating them as binomial, e.g. positive responses vs. all other responses), because they are not independent.
