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."

Assume observations or random variates ${\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})}$ as realization of independent random variables ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})}$ which have a common distribution ${\displaystyle F_{\theta }}$, with unknown parameter ${\displaystyle \theta }$. Then, a tolerance interval with endpoints ${\displaystyle (L(\mathbf {x} ),U(\mathbf {x} )]}$ which has the defining property:

where ${\displaystyle \inf\{\}}$ denotes the infimum function.

This is in contrast to a prediction interval with endpoints ${\displaystyle [l(\mathbf {x} ),u(\mathbf {x} )]}$ which has the defining property:

Here, ${\displaystyle X_{0}}$ is a random variable from the same distribution ${\displaystyle F_{\theta }}$ but independent of the first ${\displaystyle n}$ variables.

Notice ${\displaystyle X_{0}}$ is not involved in the definition of tolerance interval, which deals only with the first sample, of size n.

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 estimated using the 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 ${\displaystyle \mu }$ and standard deviation ${\displaystyle \sigma }$, then the interval ${\displaystyle \mu \pm 1.96\sigma }$ includes 95% of the population (1.96 is the z-score for 95% coverage of a normally distributed population).