Basic Analysis and Graphing - SAS

Chapter 2 Performing Univariate Analysis 75
Statistical Details for the Distribution Platform
Table 2.19 Descriptions of Capability Indices and Computational Formulas (Continued)
Index Index Name Formula
CPL
CPU
process capability ratio
of one-sided lower
spec
process capability ratio
of one-sided upper
spec
(mean - LSL)/3s
(USL - mean)/3s
• A capability index of 1.33 is considered to be the minimum acceptable. For a normal distribution, this
gives an expected number of nonconforming units of about 6 per 100,000.
• Exact 100(1 - α)% lower and upper confidence limits for CPL are computed using a generalization of
the method of Chou et al. (1990), who point out that the 100(1 - α) lower confidence limit for CPL
(denoted by CPLLCL) satisfies the following equation:
Pr{ T n – 1
( δ = 3 n)CPLLCL ≤ 3CPL n} = 1 – α
where T n-1 (δ) has a non-central t-distribution with n - 1 degrees of freedom and noncentrality
parameter δ.
• Exact 100(1 - α)% lower and upper confidence limits for CPU are also computed using a generalization
of the method of Chou et al. (1990), who point out that the 100(1 - α) lower confidence limit for CPU
(denoted CPULCL) satisfies the following equation:
Pr{ T n – 1
( δ = 3 n)CPULCL ≥ 3CPU n} = 1 – α
where T n-1 (δ) has a non-central t-distribution with n - 1 degrees of freedom and noncentrality
parameter δ.
Note: Because of a lack of supporting research at the time of this writing, computing confidence intervals
for capability indices is not recommended, except for cases when the capability indices are based on the
standard deviation.
• Sigma Quality is defined as the following
% outside
Sigma Quality = Normal Quantile1
– ---------------------- 100
+ 1.5
% above
Sigma Quality Above = Normal Quantile1
– ------------------- + 1.5
100
% below
Sigma Quality Below = Normal Quantile1
– ------------------- + 1.5
100
For example, if there are 3 defects in n=1,000,000 observations, the formula yields 6.03, or a 6.03 sigma
process. The results of the computations of the Sigma Quality Above USL and Sigma Quality Below
LSL column values do not sum to the Sigma Quality Total Outside column value because calculating
Sigma Quality involves finding normal distribution quantiles, and is therefore not additive.