Computational Approaches for Process Capability

Methods M1 through M6

Specifically, the different approaches are typically labeled with letter M followed by a number (for example, M2) to denote a specific approach. In the ISO specifications, greater granularity is possible by including 2 additional subscripts, to denote a specific approach to the estimation of location (such as mean, median, etc.), dispersion (such as  range, standard deviation), and another subscript to account for so-called additional variation.

The Statistica implementation follows the simpler notation proposed in DIN 55319; note that the computational methods in both standards are mostly identical (DIN 55319 being typically a subset of all possible methods described in ISO 21747).

However, there are important differences in the labeling of the main approaches, which can be mapped as follows:

Distribution Model and Recommended Computational Approach

The following table, which is adapted from DIN 55319 identifies the relationship between a particular time-dependent distribution model, and the recommended computational approach (note the differences in notation in DIN and ISO standards):

Computational method Time-distribution-model:

DIN/ISO M1₁ through M1₄

Here, the standard capability indices are computed as

Cp=(USL-LSL)/ 6σ

Cpk=min(USL- μ, μ-LSL)/ 3σ

The differences in M1₁ through M1₄ pertain to the estimation of sigma, the variability of the process, with:

M1₁ - sigma is estimated from the pooled within-sample variance

M1₂ - sigma is estimate from the average standard deviation, using the standard correction factor (e.g., as used in S charts in Quality Control)

M1₃ - sigma is estimated from ranges (see also the Computational Detailsfor R-charts, in Quality Control)

M1₄ - sigma is estimated from the total variance

DIN M2 (ISO M4)

Cp = not available

Cpk=min(z_(1-pL),z_(1-pU))/3

Here the actual faction non-conforming below the lower specification limits ( pL) and above the specification limits ( pU) are estimated from the data, and the respective normal distribution variate values (z_(1-pL),z_(1-pU)) are then used in the computations.

DIN M3

Cp = (USL-LSL)/R

Cpk= min((USL- μ)/( x_max- μ), ( μ-LSL)/ ( μ-x_min)

Here μ (the location parameter) can either be estimated as the mean or the median (see the options on the Options M3, M4 tab); R is the range, and x_max and x_min are the maximum and minimum values (respectively) over all observations. Note that the capability index computed in this manner will typically be affected by the total number of observations, as well as the numbers of samples.

DIN M4 (Note: ISO M4*)

Cp = (USL-LSL)/(Q_0.99865-Q_0.00135)

Cpk= min(USL- μ)/( Q_0.99865- μ),( μ-LSL) /( μ-Q_0.00135))

The Q_p values in these computations are the percentile values for the respective (non-normal) distribution fitted to the outcoming (resultant) distribution; μ can be estimated as Q_0.50 (the median) or the mean (see also the Options M3, M4 tab). Note that for the normal distribution, these indices will be identical to those computed under M1.

ISO 21747 provides no quantile-based method like DIN M4; because this method is generally applicable, in particular to "difficult" distribution models (e.g., B, D), STATISTICA provides this method for ISO-compliant computations, and labels it M4* (to distinguish it from the ISO M4 method, which is the equivalent of DIN M2).

DIN M5 (ISO M2)

Cp=(USL-LSL)/( 6σ+μ_add)

Cpk=min(USL-μ, μ-LSL)/(3σ+μ_add/2)

Here σ can be estimated in the same manner as described under M1 through M4. The additional variability μ_add is estimated via ANOVA (analysis of variance), or from the range of sample means (and options are provided on the Options M5, M6 tab)

DIN M6 (ISO M3)

Cp=(USL-LSL-μ_add)/6σ

Cpk=min(USL-μ-μ_add/2, μ-LSL-μ_add/2)/3σ

Here σ can be estimated in the same manner as described under M1 through M4. The additional variability μ_add is estimated via ANOVA (analysis of variance), or from the range of sample means (and options are provided on the Options M5, M6 tab).