Section A

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Figure IV.1: Within – and Between-Subgroup Variation

CHAPTER IV – Section A Definition of Process Terms

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CHAPTER IV - Section A

Definitions of Process Terms

Process variation has various aspects:

• Inherent Process Variation – That portion of process variation due to common (systematic) causes only.

• Within-subgroup Variation ( ) This is the variation due only to the variation within the subgroups. If the process is in statistical control this variation is a good estimate of the inherent process variation. It can be estimated from control charts by or

• Between-subgroup Variation – This is the variation due to the variation between subgroups. If the process is in statistical control this variation should be zero.

• Total Process Variation ( ) – This is the variation due to both within-subgroup and between-subgroup variation. If the process is not in statistical control the total process variation will reading, is the average of the individual readings, and n is the total number of individual readings.

• Process Capability — The 6 range of inherent process variation, for statistically stable processes only, where is usually estimated

by or

• Process Performance — The 6 range of total process variation, where is usually estimated by s , the total process standard deviation.

If the process is in statistical control the process capability will be very close to the process performance. A large difference between the capability and performance indicates the presence of a special cause(s).

CHAPTER IV – Section A Definition of Process Terms

39As 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.

132

Process Measures for Predictable Processes

Indices —

Bilateral Tolerances

This section discusses commonly used indices where the specification has both an upper and lower limit. ³⁹

CAUTION: The indices discussed below are valid only when the process is stable (in statistical control). If the process is not in statistical control then these indices can be very misleading, as can be seen by Figure IV.4.

C p : This is a capability index. It compares the process capability to the maximum allowable variation as indicated by the tolerance. This index provides a measure of how well the process will satisfy the variability requirements. Cp is calculated by Cp =

Cp is not impacted by the process location. This index can be calculated only for two-sided (bilateral) tolerances.

C^pk: This is a capability index. It takes the process location as well as the capability into account. For bilateral tolerances C_pk will always be less than or equal to C p .

Cpk ≤ Cp

C_pk will be equal to C_p only if the process is centered.

C_pk is calculated as the as minimum of CPU or CPL where:

a n d

C_pkand Cp should always be evaluated and analyzed together. A C_p value significantly greater than the corresponding C_{p k} indicates an opportunity for improvement by centering the process.

C

C

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Pp: This is a performance index. It compares the process performance to the maximum allowable variation as indicated by the tolerance. This index provides a measure of how well the process will satisfy the variability requirements. Pp is calculated by

P_p is not impacted by the process location.

P_pk: This is a performance index. It takes the process location as well as the performance into account. For bilateral tolerances Ppk will always be less than or equal to P_P. P_pk will be equal to P_p only if the process is centered.

Ppk ≤ Pp

P_pk is calculated as the as the minimum of PPU or PPL where:

P

P

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134

P_pk and P_p should always be evaluated and analyzed together. A P_p value significantly greater than the corresponding Ppk indicates an opportunity for improvement by centering the process.

If the process is in statistical control the process capability will be very close to the process performance. A large difference between the C and P indices indicate the presence of a special cause(s). See Figure IV.3 and IV.4.

CR: This is the capability ratio and is simply the reciprocal of Cp:

PR: This is the performance ratio and is simply the reciprocal of PP;

NOTE: Example calculations for all of these measures are shown in Appendix F.

A parts-per-million (ppm) nonconformance rate is sometimes used as a supplemental measure of process capability. To estimate the nonconformance rate using capability index information, a probability distribution of the data must be defined. While the normal distribution often is used for this purpose, this is an assumption that should be validated using a goodness-of-fit test before proceeding further. The nonlinear relationship between the capability index and the proportion nonconforming should be understood in order to make correct inferences (see Wheeler (1999) for a detailed discussion of this subject).

CR PR

PPM

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135

Note that the Range charts are identical since the within subgroup variation is the same for both processes

Figure IV.3: Comparison between a Predictable and Immature Process

CHAPTER IV – Section A Definition of Process Terms

136

Process below has same within subgroup variation as above but no between subgroup variation Ppk = 0.71

Cpk = 1.80

Ppk = 1.71 Cpk = 1.80

Figure IV.4: Cpk and Ppk Values produced by a Predictable and Immature Process

CHAPTER IV – Section A Definition of Process Terms

137

Indices —

Unilateral Tolerances

This section discusses commonly used indices where the specification has either an upper or lower limit but not both.

CP:This is a capability index. It compares the process capability to the maximum allowable variation as indicated by the tolerance. This index has no meaning for unilateral tolerances.

If the product characteristic has a physical limit (e.g., flatness cannot be less than zero), a C_p could be calculated using the physical limit (0.0) as a surrogate lower limit. But this number will not have the same relationship to Cpk as it does in the bilateral case.

C_pk: This is a capability index. It takes the process location as well as the capability into account. With unilateral tolerances with a physical limit, Cpk can be less than, equal to or greater than C_p.

Cpk is directly related to the proportion nonconforming produced by the process. It is equal to CPU or CPL depending whether the tolerance is an U S L or a L S L where:

PP:This is a performance index. It compares the process performance to the maximum allowable variation as indicated by the tolerance.

This index has no meaning for unilateral tolerances.

If the product characteristic has a physical limit (e.g., flatness cannot be less than zero), a Pp could be calculated using the physical limit (0.0) as a surrogate lower limit. But this number will not have the same relationship to Ppk as it does in the bilateral case.

C _p

C _pk

C _pk

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Ppk is directly related to the proportion nonconforming produced by the process. It is equal to PPU or PPL depending whether the tolerance is an USL or a LSL where:

An alternate notation for P_pk in the case of unilateral tolerances is P_pku or P_pkl depending on whether the limit is an USL or LSL.

CR: This is the capability ratio and is simply the reciprocal of C_p. As such, this index has no meaning for unilateral tolerances.

PR: This is the performance ratio and is simply the reciprocal of P_p. As such, this index has no meaning for unilateral tolerances.

NOTE: Example calculations for all of these measures are shown in Appendix F.

P _pk

CR

PR

CHAPTER IV – Section B