TOLERANCE INTERVALS 
Bibliographic References:  
Gerald J. Hahn, and William W. Meeker. Statistical Intervals  A Guide for Practitioners. Wiley:&Sons, Inc., 1991.  
Robert E. Odeh, and Donald B. Owen. Tables for Normal Tolerance Limits, Sampling Plans, and Screening. Marcel Dekker, Inc. 1980.  
Introduction:  
Tolerance intervals method is a statistical method to estimate the overall proportion of a population which is between specifications with a certain level of confidence.  
Engineers, managers and scientists may need to draw conclusions from a limited sample of variables data and apply it to the overall population. Statistical intervals, also called tolerance intervals quantify the "margin of error" from the collected data to extract the maximum of information and reach a larger conclusion with a certain level of confidence.  
Assumption: Variable data must come from a population which followed a normal distribution. If it is not the case, a mathematical transformation can be used to transform the data and departure from normality can be reassessed.  
The method can be onesided for the calculation of a lower or upper confidence limit or twosided for the calculation of a lower and upper confidence limit at the level of confidence g and for the proportion of the overall population p. We will take an example of a lower tolerance limit which could be applied to testing properties which need to meet a minimum requirement.  
The method uses the sample mean and standard deviation of a measured output to determine the lower tolerance limit (1sided) above which a specified proportion p (for medical devices, p=99.9% for example) of the overall population can be claimed to reside with a high degree of confidence g (g =0.95 for example). The calculated lower tolerance limit could then be compared to a minimum specification.  
Perform the following:  
Compare L to the minimum specification (Spec.):
Figure 1 presents a schematic representation of the calculation of the lower tolerance limit for a population and its comparison to a minimum specification. 

Figure 1: Lower tolerance limit for a normal distribution. Comparison to specification. 

Conclusion:  
Tolerance intervals methods are for variable data. They cannot be used for attribute data: pass/fail, go/nogo (other methods are available such as reliability analysis). Tolerance interval techniques can be used to check if a given percentage of the population is within specifications, or to evaluate which percentage of the population is within specifications. They are very useful methods which are easy to apply and require a relatively small sample size for high levels of confidence. 
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