## (Basic Statistics for Citizen Data Scientist)

# Tolerance Interval

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, *χ*^{2}_{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)

and

*z _{p}* = NORM.S.INV(

*p*)

*z*= NORM.S.INV(

_{α}*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.

Statistics for Beginners in Excel – Confidence Intervals for Sampling Distributions

## Statistics for Beginners in Excel – Tolerance Interval using Real Statistics

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