This seems to be a pretty basic question for this forum, please bear with me.
Consider that I have 10,000 measurements, say of the height of 10,000 people. These height measurements vary from 140 to 190 cm. I now define three height-groups: short (<150 cm), medium (150 to 170 cm) and tall (>170 cm). I can now calculate proportions of those groups in my set of 10,000 people (eg: “40% of the people are short, 30% are medium, 30% are tall”).
Now, consider that there is a random error associated with each of the height measurements. By a separate experiment, I concluded that this error distribution is well-approximated by a normal distribution, with a mean of zero and a standard deviation of 10 cm. That is, the measurers are unbiased, but do make some random errors.
Now, I would like to propagate this estimated error in height measurement to the proportions. That is, I would like to say something like “the percent of short men in ” ( could be standard error). Is there a theoretical way to go about this problem, rather than resorting to a Monte Carlo simulation?
The original data of 10,000 measurements could be described in two ways:
1. It is approximated by another normal distribution, of mean 165 cm and a standard deviation of 7.0 cm
2. It is described in a programming language data structure context; it is in a R vector, say "origData". Here, I am expecting R code that will take this vector and other inputs (from the question) and give me the standard errors.