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Not being very familiar with statistics I have been trying to figure this out for some time:

Suppose that I am trying to determine the relative amounts of several animal species. To do this, I find as many animals as I can get (randomly) and compute the percent of each species to get an approximation of the true percent.

However, there is less error with the approximated ratios when the number of animals found is larger. How would I quantify this?

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Take a look at the Wikipedia page for Margin of error. Your margin of error is inversely proportional to the square root of the size of your sample. So if your sample includes $n$ observations, you'd divide the value of the error function at your desired percentile by $\sqrt{2n}$.

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