Tolerance interval

A tolerance interval (TI) is a statistical interval within which, with some confidence level, a specified sampled proportion of a population falls. "More specifically, a 100×p%/100×(1−α) tolerance interval provides limits within which at least a certain proportion (p) of the population falls with a given level of confidence (1−α)." "A (p, 1−α) tolerance interval (TI) based on a sample is constructed so that it would include at least a proportion p of the sampled population with confidence 1−α; such a TI is usually referred to as p-content − (1−α) coverage TI." "A (p, 1−α) upper tolerance limit (TL) is simply a 1−α upper confidence limit for the 100 p percentile of the population." One-sided normal tolerance intervals have an exact solution in terms of the sample mean and sample variance based on the noncentral t-distribution. Two-sided normal tolerance intervals can be obtained based on the noncentral chi-squared distribution. "In the parameters-known case, a 95% tolerance interval and a 95% prediction interval are the same." If we knew a population's exact parameters, we would be able to compute a range within which a certain proportion of the population falls. For example, if we know a population is normally distributed with mean and standard deviation , then the interval includes 95% of the population (1.96 is the z-score for 95% coverage of a normally distributed population). However, if we have only a sample from the population, we know only the sample mean and sample standard deviation , which are only estimates of the true parameters. In that case, will not necessarily include 95% of the population, due to variance in these estimates. A tolerance interval bounds this variance by introducing a confidence level , which is the confidence with which this interval actually includes the specified proportion of the population. For a normally distributed population, a z-score can be transformed into a "k factor" or tolerance factor for a given via lookup tables or several approximation formulas.