Confidence Interval - Confidence Intervals For Proportions and Related Quantities

Confidence Intervals For Proportions and Related Quantities

An approximate confidence interval for a population mean can be constructed for random variables that are not normally distributed in the population, relying on the central limit theorem, if the sample sizes and counts are big enough. The formulae are identical to the case above (where the sample mean is actually normally distributed about the population mean). The approximation will be quite good with only a few dozen observations in the sample if the probability distribution of the random variable is not too different from the normal distribution (e.g. its cumulative distribution function does not have any discontinuities and its skewness is moderate).

One type of sample mean is the mean of an indicator variable, which takes on the value 1 for true and the value 0 for false. The mean of such a variable is equal to the proportion that have the variable equal to one (both in the population and in any sample). This is a useful property of indicator variables, especially for hypothesis testing. To apply the central limit theorem, one must use a large enough sample. A rough rule of thumb is that one should see at least 5 cases in which the indicator is 1 and at least 5 in which it is 0. Confidence intervals constructed using the above formulae may include negative numbers or numbers greater than 1, but proportions obviously cannot be negative or exceed 1. Additionally, sample proportions can only take on a finite number of values, so the central limit theorem and the normal distribution are not the best tools for building a confidence interval. See "Binomial proportion confidence interval" for better methods which are specific to this case.