# Generalizing the wilson score confidence interval to other distributions

This article describes the 'Wilson score confidence interval', and describes how to use it to derive the lower bound on the nth percentile confidence interval for determining sorting criteria for thumbs-up/thumbs-down type ratings in a ratings system.

How can this be generalized to a ratings system that doesn't form a binomial distribution? Specifically, how can this be determined when each rating is a real number between 0 and 1, or when each rating is one of a set of discrete ratings (eg, 1, 2, 3, 4 or 5)?

With a normal distribution, it appears that this could be simply the average minus some number of standard deviations - but to the best of my knowledge, it's not a true normal distribution, since it's limited to the range (0, 1).

• See also Brown, Cai & DasGupta, "Interval estimation for a binomial proportion" which compares Wilson to other alternatives. – Charles Jun 3 '10 at 18:58
• Anyone interested in this question or related questions, please have a look at the new proposed statistics stack-exchange site. area51.stackexchange.com/proposals/33/statistical-analysis – Noah Snyder Jun 17 '10 at 20:43

## 1 Answer

The rating is performed according to the sample mean. If the sample size is big enough >30, the sample mean is approximately Gaussian distributed and the confidence interval can be computed from the Gaussian formula using the sample standard deviation. When the sample size is small, one can treat the random variable (x-mu)/(sigma/sqrt(n)) (mu is the sample mean, sigma is the samole standard deviation and n is the sample size) as t-distributed and calculate the confidence interval from the t-distribution table. Here is a tutorial explaining this computation.