7.
Product and Process Comparisons
7.2. Comparisons based on data from one process 7.2.5. What intervals contain a fixed percentage of the population values?


Definition of a tolerance interval  A confidence interval covers a population parameter with a stated confidence, that is, a certain proportion of the time. There is also a way to cover a fixed proportion of the population with a stated confidence. Such an interval is called a tolerance interval. The endpoints of a tolerance interval are called tolerance limits. An application of tolerance intervals to manufacturing involves comparing specification limits prescribed by the client with tolerance limits that cover a specified proportion of the population.  
Difference between confidence and tolerance intervals  Confidence limits are limits within which we expect a given population parameter, such as the mean, to lie. Statistical tolerance limits are limits within which we expect a stated proportion of the population to lie. Confidence intervals shrink towards zero as the sample size increases. Tolerance intervals tend towards a fixed value as the sample size increases.  
Not related to engineering tolerances  Statistical tolerance intervals have a probabilistic interpretation. Engineering tolerances are specified outer limits of acceptability which are usually prescribed by a design engineer and do not necessarily reflect a characteristic of the actual measurements.  
Three types of tolerance intervals  Three types of questions can be addressed by
tolerance intervals. Question (1) leads to a twosided interval; questions
(2) and (3) lead to onesided intervals.


Tolerance intervals for measurements from a normal distribution  For the questions above, the
corresponding tolerance intervals are defined by lower (L) and upper (U)
tolerance limits which are computed from a series of measurements Y_{1},
...,
Y_{N} :


Calculation of k factor for a twosided tolerance limit for a normal distribution  If the data are from a normally distributed
population, an approximate value for the factor as a function of p
and for a twosided
tolerance interval (Howe, 1969)
is
where is the critical value of the chisquare distribution with degrees of freedom, N  1, that is exceeded with probability and is the critical value of the normal distribution which is exceeded with probability (1p)/2. 

Example of calculation  For example, suppose that we take a sample of N = 43 silicon wafers from a lot and measure their thicknesses in order to find tolerance limits within which a proportion p = 0.90 of the wafers in the lot fall with probability = 0.99.  
Use of tables in calculating twosided tolerance intervals  Values of the k factor as a function
of p and
are tabulated in some textbooks, such as Dixon
and Massey (1969). To use the tables in this handbook, follow the steps
outlined below:
The tolerance limits are then computed from the sample mean,, and standard deviation, s, according to case (1). 

Important note  The notation for the critical value of the chisquare distribution can be confusing. Values as tabulated are, in a sense, already squared; whereas the critical value for the normal distribution must be squared in the formula above.  
Dataplot commands for calculating the k factor for a twosided tolerance interval 
The Dataplot commands are:
let n = 43 let nu = n  1 let p = .90 let g = .99 let g1=1g let p1=(1+p)/2 let cg=chsppf(g1,nu) let np=norppf(p1) let k = nu*(1+1/n)*np**2 let k2 = (k/cg)**.5and the output is: THE COMPUTED VALUE OF THE CONSTANT K2 = 0.2217316E+01 

Another note  The notation for tail probabilities in Dataplot is the converse of the notation used in this handbook. Therefore, in the example above it is necessary to specify the critical value for the chisquare distribution, say, as chsppf(1.99, 42) and similarly for the critical value for the normal distribution.  
Direct calculation of tolerance intervals using Dataplot 
Dataplot also has an option for calculating
tolerance intervals directly from the data. The commands for producing
tolerance intervals from twentyfive
measurements of resistivity from
a quality control study at a confidence level of 99% are:
read 100ohm.dat cr wafer mo day h min op hum ... probe temp y sw df tolerance yAutomatic output is given for several levels of coverage, and the tolerance interval for 90% coverage is shown below in bold: 2SIDED NORMAL TOLERANCE LIMITS: XBAR + K*S NUMBER OF OBSERVATIONS = 25 SAMPLE MEAN = 97.069832 SAMPLE STANDARD DEVIATION = 0.26798090E01 CONFIDENCE = 99.% COVERAGE (%) LOWER LIMIT UPPER LIMIT 50.0 97.04242 97.09724 75.0 97.02308 97.11658 90.0 97.00299 97.13667 95.0 96.99020 97.14946 99.0 96.96522 97.17445 99.9 96.93625 97.20341 

Calculation for a onesided tolerance interval for a normal distribution 
The calculation of an approximate k factor for onesided tolerance
intervals comes directly from the following set of formulas
(Natrella, 1963):
where is the critical value from the normal distribution that is exceeded with probability 1p and is the critical value from the normal distribution that is exceeded with probability 1. 

Dataplot commands for calculating the k factor for a onesided tolerance interval 
For the example above, it may also be of interest to guarantee with
0.99 probability (or 99% confidence) that 90% of the wafers have
thicknesses less than an upper tolerance limit. This problem falls under
case (3), and the Dataplot commands for
calculating the factor for the onesided tolerance interval are:
let n = 43 let p = .90 let g = .99 let nu = n1 let zp = norppf(p) let zg=norppf(g) let a = 1  ((zg**2)/(2*nu)) let b = zp**2  (zg**2)/n let k1 = (zp + (zp**2  a*b)**.5)/aand the output is: THE COMPUTED VALUE OF THE CONSTANT A = 0.9355727E+00 THE COMPUTED VALUE OF THE CONSTANT B = 0.1516516E+01 THE COMPUTED VALUE OF THE CONSTANT K1 = 0.1875189E+01The upper (onesided) tolerance limit is therefore 97.07 + 1.8752*2.68 = 102.096. 