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I have data fitting a binomial distribution with $n$ total observations and m positive observations. My estimate of $p$ is $\frac{m}{n}$, but is there a way I can estimate the error or variance in $p$? That is to say, can I quantify the error and determine how close $\frac{m}{n}$ is to the true $p$?

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You can calculate confidence intervals for the success probability estimate in a binomial distribution. Literature goes back at least to Clopper & Pearson (Biometrika, 1934), whose method is used in R's binom.test() function. The Wikipedia page is informative.

Note that the confidence interval will usually be asymmetric, especially if your estimate for $p$ is far from 0.5. Symmetric confidence intervals are possible but have disadvantages as per Rob Hyndman's answer to the linked question.

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