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 $40 ± 3 \%$” ($±$ 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:
- It is approximated by another normal distribution, of mean 165 cm and a standard deviation of 7.0 cm
- 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.