Standard error of a proportion

If statistical theory does not provide the error distribution for the estimator of interest, a simulation can often be used to find properties of the error distribution numerically.

This methodology is illustrated with a simulation to find the standard error of a sample proportion. Since we already have a formula,

standard error  =  

a simulation is unnecessary, but it allows us to simply illustrate the method.

Example

A sample of n = 36 values are selected from a population with probability π of success, so the number of successes will have a binomial distribution,

X  ~  binomial (n = 36,  π)

If we knew the value of π, we could take repeated samples from this binomial distribution, find the estimation error, (p - π) for each sample, and build up the error distribution.

If x = 17 successes are observed, our best estimate of π is p = 17/36 = 0.472, so we could perform this simulation using p instead of π.

This simulation of 100 samples provides approximations to the bias (-0.004) and standard error (0.080) for this type of estimator. These are fairly close to the values from the formulae,

bias  =  0

standard error  =    =  0.0832

In practice, the formula would be used for the standard error of a proportion, but we can use simulations for other examples where a formula does not exist.