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### Chapter 7

Process and Measurement System Capability Analysis

Introduction to Statistical Quality Control, 4th Edition

7-1. Introduction

- 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

7-1. Introduction

- 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

7-1. Introduction

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

7-1. Introduction

Techniques used in process capability analysis

- Histograms or probability plots
- Control Charts
- Designed Experiments

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

7-2.2 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

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

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

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

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

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

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

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

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

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

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

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

7-3.4 More About Process Centering

- Cpm can be rewritten another way:

where

Introduction to Statistical Quality Control, 4th Edition

7-3.4 More About Process Centering

- A logical estimate of Cpm is:

where

Introduction to Statistical Quality Control, 4th Edition

7-3.4 More About Process Centering

Example 7-3. Consider two processes A and B.

- For process A:

since process A is centered.

- For process B:

Introduction to Statistical Quality Control, 4th Edition

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

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

Cp

- 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

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

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

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

Example 7-5. n = 20, Ĉpk = 1.33. 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

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

7-3.5 Confidence Intervals and Tests on Process Capability Ratios

Example 7-5. n = 20, Ĉpk = 1.33. 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

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

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

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 0.90. Giving Cp(Low) = 1.33
- High probability of detecting if process capability exceeds 1.66, say 0.90. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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