If we do not know μ and σ, we must use early production test results to determine if the distribution can be regarded as Normal
and to estimate the mean and standard deviation. As was pointed out earlier, estimates of μ and σ calculated from samples
are subject to uncertainty due to sampling variability. This uncertainty requires us to adopt a probabilistic approach to
the tolerance limits such that we make statements like the following: "We are 99% confident that 99% of the measurements will
fall within the calculated tolerance limits." The uncertainty, or lack of confidence, is reflected in sigma multipliers that
are larger than would be the case if we knew μ and σ. The increase depends on the sample size. The comments attributed to
Shewhart suggest that the effect of sample variability will be close to zero for samples larger than 250.
Methods of calculating probabilistic tolerance limits can be found in the National Institute of Standards of Technology (NIST)/SEMATECH
e-Handbook of Statistical Methods.3
The method for a two-sided tolerance interval can be found in a 1969 paper attributed to Howe.4 This author calculates a variable MUL to provide us with a level of confidence of C% that the limits (x mean – MUL * s) and (x mean + MUL * s) will contain D% of the distribution.
The method for calculating probabilistic one-sided tolerance intervals that is set out in the NIST/SEMATECH e-Handbook of
Statistical Methods is attributed to Natrella.5 The upper limit providing a level of confidence of C% that D% of the values will be less than it is given by (x mean + MU * s). The lower limit is defined similarly as (x mean – ML * s).
Since the effect of sample variability will be close to zero for samples larger than 250, when constructing Table 1, the aim
was to produce values of the sigma multipliers (MUL, ML, and MU) that are around 3.0 when the sample size is 250. Thus C% was set to 99% and D% was set to 99.25%.
Table 1. Two-sided multipliers (MUL) and one-sided multipliers (MU or ML) for sample sizes up to N = 200
We can use the residual 1,3–diacetyl benzene results shown in Figure 1 as an example. Typically, there is an upper specification
limit for such residual compounds. The statistical software Minitab was used to test for Normality and to calculate estimates
of the mean and standard deviation.
The Anderson-Darling A-squared statistic was 0.24 and the P-value was 0.75. A P-value larger than 0.05 indicates that the
distribution is not significantly different than Normal. Thus, we can regard the distribution of the residual 1,3–diacetyl
benzene results as Normal.
The one-sided tolerance interval procedure was used because only an upper limit is needed. Table 1 shows the value of MU for a sample size of 62 to be 3.46. The calculation set out below resulted in an upper limit of 460 μg/g:
Mean = 245.7 μg/g
Standard deviation = 61.91 μg/g
Sample size = 62
Sigma multiplier = 3.46
Upper specification limit = 460 μg/g
What can be done when the distribution looks Normal but fails a test?
A similar upper specification limit was required for 1,4-diacetyl benzene, but this time the Anderson-Darling test indicated
that that the distribution of the sample was significantly different from a Normal.