As described in Confidence Intervals, a confidence interval provides a way of estimating a population parameter by a corresponding sample statistic to a given level of confidence. We show how to estimate the population mean (the parameter) by the sample mean (the statistic). In particular, if the experiment is repeated a sufficiently large number of times, then the true population parameter will lie in the 1–α confidence interval in 100(1–α) percent of the samples.

The tolerance interval, on the other hand, is an interval pertaining to the entire population and not just to a specific parameter. In particular, we expect that 100(1–α) percent of the entire population will lie in the 1–α tolerance interval.

We now show how to calculate a tolerance interval based on sample data taken from a normal distributed population. In particular, we want to find an interval of the form

that contains p% of the population with 100(1–α)% confidence, where x̄ is the sample mean and s is the sample standard deviation

We provide the following estimate for k (due to Howe):

where n is the sample size, χ²_crit is the critical value of the chi-square distribution at α with df = n–1 degrees of freedom and z_crit is the critical value of the standard normal distribution at the value where the cdf is (p+1)/2. Thus

Guenther recommends using wk instead of k, especially for smaller samples, where

There are also one sided tolerance intervals. In particular, we want to find the value of k such p% of the population falls in the interval (x̄–ks, ∞) with 100(1–α)% confidence. The same value of k will ensure that p% of the population falls in the interval (-∞, x̄+ks) with 100(1–α)% confidence.

We provide the following estimate for k (due to Natrella)

where

and

z_p = NORM.S.INV(p)            z_α = NORM.S.INV(α)

There is also an alternative estimate based on the noncentral t distribution, namely

where t_crit is the critical value at α of the noncentral t distribution T(n–1, z_p√n). This can be calculated via the Real Statistics formula

t_crit = NT_INV(α, n–1, NORM.S.INV(p)*SQRT(n))

You can also estimate the sample size required to obtain a particular tolerance interval by using Excel’s Goal Seek capability.

Real Statistics Function: The Real Statistics Resource Pack contains the following function:

TOLERANCE_NORM(n, p, α, type) = k value of the tolerance interval for a normal distribution (actually k′ for the two-sided interval)

n = sample size, p = tolerance (default .9), α = significance level (default .05)

type = 2 (default) for two-sided interval, type = 1 for a one-side interval using a non-central t distribution and type = 0 for a one-sided interval using the Natrella approach.