1. Process and Measurement System Capability Analysis
Chapter 7
Process and Measurement System Capability Analysis
Introduction to Statistical Quality Control, 4th Edition

2. Introduction to Statistical Quality Control, 4th Edition
Process capability refers to the uniformity of the process.
Variability in the process is a measure of the uniformity of output.
Two types of variability:
Natural or inherent variability (instantaneous)
Variability over time
Assume that a process involves a quality characteristic that follows a normal distribution with mean , and standard deviation, . The upper and lower natural tolerance limits of the process are
UNTL =  + 3
LNTL =  - 3
Introduction to Statistical Quality Control, 4th Edition

3. Introduction to Statistical Quality Control, 4th Edition
Process capability analysis is an engineering study to estimate process capability.
In a product characterization study, the distribution of the quality characteristic is estimated.
Introduction to Statistical Quality Control, 4th Edition

4. Introduction to Statistical Quality Control, 4th Edition
Major uses of data from a process capability analysis
Predicting how well the process will hold the tolerances.
Assisting product developers/designers in selecting or modifying a process.
Assisting in Establishing an interval between sampling for process monitoring.
Specifying performance requirements for new equipment.
Selecting between competing vendors.
Planning the sequence of production processes when there is an interactive effect of processes on tolerances
Reducing the variability in a manufacturing process.
Introduction to Statistical Quality Control, 4th Edition

5. Introduction to Statistical Quality Control, 4th Edition
Techniques used in process capability analysis
Histograms or probability plots
Control Charts
Designed Experiments
Introduction to Statistical Quality Control, 4th Edition

6. Introduction to Statistical Quality Control, 4th Edition
7-2. Process Capability Analysis Using a Histogram or a Probability Plot
7-2.1 Using a Histogram
The histogram along with the sample mean and sample standard deviation provides information about process capability.
The process capability can be estimated as
The shape of the histogram can be determined (such as if it follows a normal distribution)
Histograms provide immediate, visual impression of process performance.
Introduction to Statistical Quality Control, 4th Edition

7. 7-2.2 Probability Plotting
Probability plotting is useful for
Determining the shape of the distribution
Determining the center of the distribution
Determining the spread of the distribution.
Recall normal probability plots (Chapter 2)
The mean of the distribution is given by the 50th percentile
The standard deviation is estimated by
84th percentile – 50th percentile
Introduction to Statistical Quality Control, 4th Edition

8. 7-2.2 Probability Plotting
Cautions in the use of normal probability plots
If the data do not come from the assumed distribution, inferences about process capability drawn from the plot may be in error.
Probability plotting is not an objective procedure (two analysts may arrive at different conclusions).
Introduction to Statistical Quality Control, 4th Edition

9. 7-3. Process Capability Ratios
7-3.1 Use and Interpretation of Cp
Recall
where LSL and USL are the lower and upper specification limits, respectively.
Introduction to Statistical Quality Control, 4th Edition

10. 7-3.1 Use and Interpretation of Cp
The estimate of Cp is given by
Where the estimate can be calculated using the sample standard deviation, S, or
Introduction to Statistical Quality Control, 4th Edition

11. 7-3.1 Use and Interpretation of Cp
Piston ring diameter in Example 5-1
The estimate of Cp is
Introduction to Statistical Quality Control, 4th Edition

12. 7-3.1 Use and Interpretation of Cp
One-Sided Specifications
These indices are used for upper specification and lower specification limits, respectively
Introduction to Statistical Quality Control, 4th Edition

13. 7-3.1 Use and Interpretation of Cp
Assumptions
The quantities presented here (Cp, Cpu, Clu) have some very critical assumptions:
The quality characteristic has a normal distribution.
The process is in statistical control
In the case of two-sided specifications, the process mean is centered between the lower and upper specification limits.
If any of these assumptions are violated, the resulting quantities may be in error.
Introduction to Statistical Quality Control, 4th Edition

14. 7-3.2 Process Capability Ratio an Off-Center Process
Cp does not take into account where the process mean is located relative to the specifications.
A process capability ratio that does take into account centering is Cpk defined as
Cpk = min(Cpu, Cpl)
Introduction to Statistical Quality Control, 4th Edition

15. 7-3.3 Normality and the Process Capability Ratio
The normal distribution of the process output is an important assumption.
If the distribution is nonnormal, Luceno (1996) introduced the index, Cpc, defined as
Introduction to Statistical Quality Control, 4th Edition

16. 7-3.3 Normality and the Process Capability Ratio
A capability ratio involving quartiles of the process distribution is given by
In the case of the normal distribution Cp(q) reduces to Cp
Introduction to Statistical Quality Control, 4th Edition

17. 7-3.4 More About Process Centering
Cpk should not be used alone as an measure of process centering.
Cpk depends inversely on  and becomes large as  approaches zero. (That is, a large value of Cpk does not necessarily reveal anything about the location of the mean in the interval (LSL, USL)
Introduction to Statistical Quality Control, 4th Edition

18. 7-3.4 More About Process Centering
An improved capability ratio to measure process centering is Cpm.
where  is the squre root of expected squared deviation from target: T =½(USL+LSL),
Introduction to Statistical Quality Control, 4th Edition

19. 7-3.4 More About Process Centering
Cpm can be rewritten another way:
where
Introduction to Statistical Quality Control, 4th Edition

20. 7-3.4 More About Process Centering
A logical estimate of Cpm is:
where
Introduction to Statistical Quality Control, 4th Edition

21. 7-3.4 More About Process Centering
Example Consider two processes A and B.
For process A:
since process A is centered.
For process B:
Introduction to Statistical Quality Control, 4th Edition

22. 7-3.4 More About Process Centering
A third generation process capability ratio, proposed by Pearn et. al. (1992) is
Cpkm has increased sensitivity to departures of the process mean from the desired target.
Introduction to Statistical Quality Control, 4th Edition

23. 7-3.5 Confidence Intervals and Tests on Process Capability RatiosCp
Ĉp is a point estimate for the true Cp, and subject to variability. A 100(1-) percent confidence interval on Cp is
Introduction to Statistical Quality Control, 4th Edition

24. 7-3.5 Confidence Intervals and Tests on Process Capability Ratios
Example 7-4. USL = 62, LSL = 38, n = 20,
S = 1.75, The process mean is centered. The point estimate of Cp is
95% confidence interval on Cp is
Introduction to Statistical Quality Control, 4th Edition

25. 7-3.5 Confidence Intervals and Tests on Process Capability Ratios
Cpk
Ĉpk is a point estimate for the true Cpk, and subject to variability. An approximate 100(1-) percent confidence interval on Cpk is
Introduction to Statistical Quality Control, 4th Edition

26. 7-3.5 Confidence Intervals and Tests on Process Capability Ratios
Example n = 20, Ĉpk = An approximate 95%
confidence interval on Cpk is
The result is a very wide confidence interval ranging from below unity (bad) up to 1.67 (good). Very little has really been learned about actual process capability (small sample, n = 20.)
Introduction to Statistical Quality Control, 4th Edition

27. 7-3.5 Confidence Intervals and Tests on Process Capability Ratios
Cpc
Ĉpc is a point estimate for the true Cpc, and subject to variability. An approximate 100(1-) percent confidence interval on Cpc is
where
Introduction to Statistical Quality Control, 4th Edition

28. 7-3.5 Confidence Intervals and Tests on Process Capability Ratios
Example n = 20, Ĉpk = An approximate 95%
confidence interval on Cpk is
The result is a very wide confidence interval ranging from below unity (bad) up to 1.67 (good). Very little has really been learned from this result, (small sample, n = 20.)
Introduction to Statistical Quality Control, 4th Edition

29. 7-3.5 Confidence Intervals and Tests on Process Capability Ratios
Testing Hypotheses about PCRs
May be common practice in industry to require a supplier to demonstrate process capability.
Demonstrate Cp meets or exceeds some particular target value, Cp0.
This problem can be formulated using hypothesis testing procedures
Introduction to Statistical Quality Control, 4th Edition

30. 7-3.5 Confidence Intervals and Tests on Process Capability Ratios
Testing Hypotheses about PCRs
The hypotheses may be stated as
H0: Cp  Cp0 (process is not capable)
H0: Cp  Cp0 (process is capable)
We would like to reject Ho
Table 7-5 provides sample sizes and critical values for testing H0: Cp = Cp0
Introduction to Statistical Quality Control, 4th Edition

31. 7-3.5 Confidence Intervals and Tests on Process Capability Ratios
Example 7-6
H0: Cp = 1.33
H1: Cp > 1.33
High probability of detecting if process capability is below 1.33, say Giving Cp(Low) = 1.33
High probability of detecting if process capability exceeds 1.66, say Giving Cp(High) = 1.66
 =  = 0.10.
Determine the sample size and critical value, C, from Table 7-5.
Introduction to Statistical Quality Control, 4th Edition

32. 7-3.5 Confidence Intervals and Tests on Process Capability Ratios
Example 7-6
Compute the ratio Cp(High)/Cp(Low):
Enter Table 7-5, panel (a) (since  =  = 0.10). The sample size is found to be n = 70 and C/Cp(Low) = 1.10
Calculate C:
Introduction to Statistical Quality Control, 4th Edition

33. 7-3.5 Confidence Intervals and Tests on Process Capability Ratios
Example 7-6
Interpretation:
To demonstrate capability, the supplier must take a sample of n = 70 parts, and the sample process capability ratio must exceed 1.46.
Introduction to Statistical Quality Control, 4th Edition

34. 7-4. Process Capability Analysis Using a Control Chart
If a process exhibits statistical control, then the process capability analysis can be conducted.
A process can exhibit statistical control, but may not be capable.
PCRs can be calculated using the process mean and process standard deviation estimates.
Introduction to Statistical Quality Control, 4th Edition

35. 7-5. Process Capability Analysis Designed Experiments
Systematic approach to varying the variables believed to be influential on the process. (Factors that are necessary for the development of a product).
Designed experiments can determine the sources of variability in the process.
Introduction to Statistical Quality Control, 4th Edition

36. 7-6. Gage and Measurement System Capability Studies
7-6.1 Control Charts and Tabular Methods
Two portions of total variability:
product variability which is that variability that is inherent to the product itself
gage variability or measurement variability which is the variability due to measurement error
Introduction to Statistical Quality Control, 4th Edition

37. 7-6.1 Control Charts and Tabular Methods
and R Charts
The variability seen on the chart can be interpreted as that due to the ability of the gage to distinguish between units of the product
The variability seen on the R chart can be interpreted as the variability due to operator.
Introduction to Statistical Quality Control, 4th Edition

38. 7-6.1 Control Charts and Tabular Methods
Precision to Tolerance (P/T) Ratio
An estimate of the standard deviation for measurement error is
The P/T ratio is
Introduction to Statistical Quality Control, 4th Edition

39. 7-6.1 Control Charts and Tabular Methods
Total variability can be estimated using the sample variance. An estimate of product variability can be found using
Introduction to Statistical Quality Control, 4th Edition

40. 7-6.1 Control Charts and Tabular Methods
Percentage of Product Characteristic Variability
A statistic for process variability that does not depend on the specifications limits is the percentage of product characteristic variability:
Introduction to Statistical Quality Control, 4th Edition

41. 7-6.1 Control Charts and Tabular Methods
Gage R&R Studies
Gage repeatability and reproducibility (R&R) studies involve breaking the total gage variability into two portions:
repeatability which is the basic inherent precision of the gage
reproducibility is the variability due to different operators using the gage.
Introduction to Statistical Quality Control, 4th Edition

42. 7-6.1 Control Charts and Tabular Methods
Gage R&R Studies
Gage variability can be broken down as
More than one operator (or different conditions) would be needed to conduct the gage R&R study.
Introduction to Statistical Quality Control, 4th Edition

43. 7-6.1 Control Charts and Tabular Methods
Statistics for Gage R&R Studies (The Tabular Method)
Say there are p operators in the study
The standard deviation due to repeatability can be found as
where
and d2 is based on the # of observations per part per operator.
Introduction to Statistical Quality Control, 4th Edition

44. 7-6.1 Control Charts and Tabular Methods
Statistics for Gage R&R Studies (the Tabular Method)
The standard deviation for reproducibility is given as
where
d2 is based on the number of operators, p
Introduction to Statistical Quality Control, 4th Edition

45. 7-6.2 Methods Based on Analysis of Variance
The analysis of variance (Chapter 3) can be extended to analyze the data from an experiment and to estimate the appropriate components of gage variability.
For illustration, assume there are a parts and b operators, each operator measures every part n times.
Introduction to Statistical Quality Control, 4th Edition

46. 7-6.2 Methods Based on Analysis of Variance
The measurements, yijk, could be represented by the model
where i = part, j = operator, k = measurement.
Introduction to Statistical Quality Control, 4th Edition

47. 7-6.2 Methods Based on Analysis of Variance
The variance of any observation can be given by
are the variance components.
Introduction to Statistical Quality Control, 4th Edition

48. 7-6.2 Methods Based on Analysis of Variance
Estimating the variance components can be accomplished using the following formulas
Introduction to Statistical Quality Control, 4th Edition