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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?

7.2.5.3.

Tolerance intervals for a normal distribution

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 two-sided interval; questions (2) and (3) lead to one-sided intervals.
  1. What interval will contain p percent of the population measurements?
  2. What interval guarantees that p percent of population measurements will not fall below a lower limit?
  3. What interval guarantees that p percent of population measurements will not exceed an upper limit?
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 Y1, ..., YN :
  1. Y(L) = Ybar - k(2)*s; Y(U) = Ybar + k(2) 
  2. Y(L) = Ybar - k(1)*s 
  3. Y(U) = Ybar + k(1)*s
where the k factors are determined so that the intervals cover at least a proportion p of the population with confidence,gamma.
Calculation of k factor for a two-sided 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 andgamma for a two-sided tolerance interval (Howe, 1969) is

k(2) = SQRT{(N-1)*(1+(1/N))*Z((1-p)/2)**2)/Chi-Square(gamma,N-1)}

whereChoSquare(gamma,N-1) is the critical value of the chi-square distribution with degrees of freedom, N - 1, that is exceeded with probabilitygamma andZ((1-p)/2) is the critical value of the normal distribution which is exceeded with probability (1-p)/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 probabilitygamma = 0.99.
Use of tables in calculating two-sided tolerance intervals Values of the k factor as a function of p andgamma are tabulated in some textbooks, such as Dixon and Massey (1969). To use the tables in this handbook, follow the steps outlined below:
  1. Calculate alpha = (1 - p)/2 = 0.05
  2. Go to the table of upper critical values of the normal distribution and under the column labeled 0.05 find Z((1-p)/2) = 1.645.
  3. Go to the table of lower critical values of the chi-square distribution and under the column labeled 0.99 in the row labeled degrees of freedom = 42, find Chi-Square(gamma,N-1) = 23.650.
  4. Calculate
k(2) = SQRT{(N-1)*(1+(1/N))*Z((1-p)/2)**2)/Chi-Square(gamma,N-1)} = SQRT{42*(44/43)*(1.645)**2/23.650} = 2.217

The tolerance limits are then computed from the sample mean,Ybar, and standard deviation, s, according to case (1).

Important note The notation for the critical value of the chi-square 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 two-sided tolerance interval The Dataplot commands are:

let n = 43
let nu = n - 1
let p = .90
let g = .99
let g1=1-g
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)**.5
and 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 chi-square 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 twenty-five 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 y
Automatic output is given for several levels of coverage, and the tolerance interval for 90% coverage is shown below in bold:

 2-SIDED NORMAL TOLERANCE LIMITS: XBAR +- K*S

     NUMBER OF OBSERVATIONS    =     25
     SAMPLE MEAN               = 97.069832
     SAMPLE STANDARD DEVIATION = 0.26798090E-01

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 one-sided tolerance interval for a normal distribution The calculation of an approximate k factor for one-sided tolerance intervals comes directly from the following set of formulas (Natrella, 1963):

k(1) = [z(1-p) + SQRT((z(1-p))**2 - a*b]/a;
   a = 1 - (z(1-gamma))**2/(2*(N-1));  b = (z(1-p))**2 - (z(1-gamma))**2/N

where Z(1-p) is the critical value from the normal distribution that is exceeded with probability 1-p and Z(1-gamma) is the critical value from the normal distribution that is exceeded with probability 1-gamma.

Dataplot commands for calculating the k factor for a one-sided 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 one-sided tolerance interval are:

let n = 43
let p = .90
let g = .99
let nu = n-1
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)/a
and 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+01
The upper (one-sided) tolerance limit is therefore 97.07 + 1.8752*2.68 = 102.096.
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