Section B

40For non-normal distributions, see the pages that follow.

139

CHAPTER IV - Section B Description of Conditions

It is appropriate to point out that process variation and process centering are two separate process characteristics. Each needs to be understood separately from the other. To assist in this analysis it has become convenient to combine the two characteristics into indices, such as Cp, Cpk or Pp, Ppk.These indices can be useful for:

• Measuring continual improvement using trends over time.

• Prioritizing the order in which processes will be improved.

The capability index, Cpk , is additionally useful for determining whether or not a process is capable of meeting customer requirements. This was the original intent of the capability index.

The performance index, P_pk, shows whether the process performance is actually meeting the customer requirements. For these indices (as well as all of the other process measures described in Chapter IV, Section A) to be effectively used, the CONDITIONS which surround them must be understood. If these conditions are not met, the measures will have little or no meaning and can be misleading in understanding the processes from which they were generated. The following three conditions are the minimum that must be satisfied for all of the capability measures described in Section A:

• The process from which the data come is statistically stable, that is, the normally accepted SPC rules must not be violated.

• The individual measurements from the process data form an approximately normal distribution. ⁴⁰

• The specifications are based on customer requirements.

Commonly, the computed index (or ratio) value is accepted as the

"true" index (or ratio) value; i.e., the influence of sampling variation on the computed number is discounted. For example, computed indices C_pkof 1.30 and 1.39 can be from the same stable process simply due to sampling variation.

See Bissell, B.A.F. (1990), Boyles, R. A. (1991) and Dovich, R. A.

(1991) for more on this subject.

CHAPTER IV – Section B Description of Conditions

41 As discussed in Chapter II, Section A, process analysis requires that the data have been collected using measurement system(s) that are consistent with the process and have acceptable measurement system characteristics.

140

Handling Non-Normal and Multivariate Distributions

^{4 1}

Although the normal distribution is useful in describing and analyzing a wide variety of processes, it cannot be used for all processes. Some processes are inherently non-normal, and their deviations from normality are such that using the normal distribution as an approximation can lead to erroneous decisions. Other processes have multiple characteristics that are interrelated and should be modeled as a multivariate distribution.

Of the indices described above, C^p, P^p, CR, and PR are robust with respect to non-normality. This is not true for C^pk,and P^pk.

Relationship of Indices and Proportion Nonconforming

Although many individuals use the C^pk,and P^pk indices as scalar-less (unit-less) metrics, there is a direct relationship between each index and the related process parameter of proportion nonconforming (or ppm). Assuming that Cp > 1, the capability index relationship is given by:

proportion nonconforming = 1 where zc = 3Cpk and

Cpk = min {CPU, CPL}

Similarly, Ppk is related to the performance proportion nonconformance through:

zp = 3Pk

With this understanding of Cpk, and Ppk, indices for non-normal distributions can be developed with the same relationships between the index and the process proportion nonconforming.

The determination of these indices for non-normal distributions requires extensive tables or the use of iterative approximation techniques. They are rarely calculated without the assistance of a computer program.

Non-Normal Distributions Using Transformations

One approach is to transform the non-normal form to one that is (near) normal. The specifications are also transformed using the same parameters.

CHAPTER IV – Section B Description of Conditions

42Box, G. E. P., Hunter, W. G., and Hunter, J. S.,

Statistics for Experimenters,

John Wiley and Sons, New York, 1978, pg.239.

43See Johnson (1949).

141

The Cpk, and PPk indices are then determined in the transformed space using standard calculations based on the normal distribution.

Two general transformation approaches which have gained support are:

• Box-Cox Transformations

The methods of analysis of designed experiments are "appropriate and efficient when the models are

(a) structurally adequate,

^and

the (supposedly independent) errors

(b)

^have

constant variance

^and

(c) are

normally distributed.

⁴²

"

Box and Cox (1964) discussed a transformation which reasonably satisfies all three of these requirements. This transformation is given by:

W=

Where -5 ≤ ≤ 5

and = 0 for the natural log transformation = 0.5 for the square root transformation

Although this transformation was developed with the focus of the analysis of designed experiments, it has found an application in the transformation of process data to normality.

• Johnson Transformations

In 1949, Norman L. Johnson developed a system of transformations which yields approximate normality. ⁴³ This system is given by:

S_B Bounded

SL Log Normal

S_U Unbounded

As in the case of the Pearson Family of distributions (see below), this system of curves encompasses all the possible unimodal distributional forms; i.e., it covers the entire feasible

skewness-kurtosis

plane. It also contains as a boundary form the familiar lognormal distribution. However, in the general case, the Johnson curves are four parameter functions.

CHAPTER IV – Section B

Non-normal forms model the process distribution and then determine the proportion nonconforming, i.e., the area of the non-normal distribution outside the specifications.

A common approach to the modeling of the non-normal distribution is to use the Pearson Family of Curves. The most appropriate member of this family is determined by the method of matching moments; i.e., the curve with skewness (SK) and kurtosis (KU) that match that of the sampled distribution is used as a model for the underlying form. As in the case of the Johnson Transformation System (see above), this family of curves encompasses all the possible unimodal distributional forms; i.e., it covers the entire feasible SK-KU plane.

To calculate the non-normal equivalent to the P_pk index, the non-normal form ( f (x)) is used to determine the proportion nonconforming, i.e., the area of the non-normal distribution outside the upper and lower specifications:

and

These values are converted to a z value using the inverse standard normal distribution. That is, the z_L and zU values in the following equations are determined such that:

and

Then

Although the standard calculation of Pp is a robust estimate, a more exact estimate can be found using the convention that the process spread is defined as the range that includes 99.73% of the distribution (representing the equivalence of a ±3 normal distribution spread). The limits of this range are called the "0.135% quantile" (Q0.00135) and the "99.865%

quantile" (Q0.99865). That is, 0.135% of the values of the population are to be found both below Q0.00135 and above Q0.99865 . ⁴⁴

CHAPTER IV – Section B Description of Conditions

143

0.00135 = f (x) dx and

0.99865 = f (x) dx.

The calculation for Ppthen is:

P_p= =

where the non-normal form is used to calculate the quantiles.

The capability index C_p is calculated as above replacing S with

Because this approach uses the total variation to calculate the proportion nonconforming, there is no analogue of a non-normal C_pk available.

An alternate approach to calculating Ppk using quantiles is given in some documents by:

Ppk = min

This approach does not tie the Ppk index to the proportion nonconforming. That is, different non-normal forms will have the same index for different proportion nonconforming. To properly interpret and compare these indices, the non-normal form as well as the index value should be considered.

CHAPTER IV – Section B Description of Conditions

144

Multivariate Distributions

When multiple characteristics are interrelated, the process distribution should be modeled using a multivariate form. The process performance index Ppk can be evaluated by first determining the proportion nonconforming, i.e., the area of the multivariate distribution outside the specifications.

For many geometrically dimensioned (GD&T) characteristics, the bivariate normal form is useful in describing the process.

A pair of random variables X and Y have a bivariate normal distribution if and only if their joint probability density is given by

f (x,y) =

where z =

= cov (x,y) =

For ; where

To calculate the multivariate equivalent to the P_pk index, the multivariate form (e.g., f (x y)) used to determine the proportion nonconforming, i.e., the volume of the multivariate distribution outside the specification (tolerance) zone. In the bivariate case this would be:

Pz = f (x, y) dx dy and

This value is converted to a z value using the inverse standard normal distribution. That is, the z value such that:

tolerance zone

CHAPTER IV – Section B Description of Conditions

45 See also Bothe (2001) and Wheeler (1995).

145 Then Ppk =

An estimate Pp can be found using:

Specification area Pp =

Est 99.73% area

where the multivariate form is used to calculate the estimated 99.73% area.

Because this approach uses the total variation to calculate the proportion nonconforming, there is no analogue of a multivariate Cpk

available. ⁴⁵

CHAPTER IV – Section B Description of Conditions

146

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CHAPTER IV – Section C

In document SPC 2nd Edition 2005 (Reference) (Page 149-157)