Standard Error
A standard error of some statistic is the standard deviation of that statistic's sampling distribution.
This concept is particularly common for describing the variance of a sample mean; standard error of the mean is sometimes abbreviated SEM.
Evaluation
For an independent sample, the standard error of a mean measurement is the standard deviation of the measurements divided by the root of the sample size: σX‾ = σX/(√n).
Given a random sample of n observations (xi) from a larger unknown population (X), the standard error can be estimated using the sample standard deviation (sX).
Bernoulli
For a Bernoulli-distributed mean (i.e., p), the true standard error is p(1-p).
When the mean is unknown or abstracted out, it can be appropriate to assume p=0.5. This maximizes the standard error at 0.25, and is generally considered 'close enough' for a mean between 0.2 and 0.8. As an example, when evaluating sampling plans, standard errors can be 'calculated' for subpopulations without considering any specific measurement by assuming the maximum possible error.
Finite Population Correction
The above evaluation assumes that the population (X) is unknown and/or infinitely large. If the population is in fact finite and the sampling rate is high (generally above 5%), that evaluation of standard error is inflated.
The finite population correction (FPC) is a correction factor, calculated as:
Intuitively, the FPC is 0 when n = N because there is no room for sampling error in a census. FPC approaches 1 when n approaches 0, demonstrating that the factor is meaningless for low sampling rates.