Introduction to Statistical Quality Control, 4th Edition
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 = + 3LNTL = - 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
1. Predicting how well the process will hold the tolerances.2. Assisting product developers/designers in selecting or
modifying a process.3. Assisting in Establishing an interval between sampling
for process monitoring.4. Specifying performance requirements for new
equipment.5. Selecting between competing vendors.6. Planning the sequence of production processes when
there is an interactive effect of processes on tolerances7. Reducing the variability in a manufacturing process.
Introduction to Statistical Quality Control, 4th Edition
7-1. Introduction
Techniques used in process capability analysis
1. Histograms or probability plots
2. Control Charts
3. 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.
s3x
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.
6
LSLUSLCp
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
ˆ6
LSLUSLCp
2d/R
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
68.1
)0099.0(6
95.7305.74Cp
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
3
LSLC
3
USLC
pl
pu
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:
1. The quality characteristic has a normal distribution.2. The process is in statistical control3. 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
TXE2
6
LSLUSLCpc
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
00135.099865.0p xx
LSLUSL)q(C
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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),
6
LSLUSLCpm
2222 )T(TxE
Introduction to Statistical Quality Control, 4th Edition
7-3.4 More About Process Centering
• Cpm can be rewritten another way:
where
2
p
22pm
1
C
)T(6
LSLUSLC
T
Introduction to Statistical Quality Control, 4th Edition
7-3.4 More About Process Centering
• A logical estimate of Cpm is:
where
2
ppm
V1
CC
S
xTV
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:
0.101
0.1
1
CC
2
ppm
63.0)3(1
0.2
1
CC
22
ppm
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.
2
pk
2
pkpkm
1
C
T1
CC
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
1nCC
1nC
21n,2/
pp
21n,2/1
p
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
01.3C57.119
85.3229.2C
19
91.829.2
p
p
29.2)75.1(6
3862Cp
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
)1n(2
1
Cn9
1Z1CC
)1n(2
1
Cn9
1Z1C
pk
2/pkpk
pk
2/pk
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.)
)19(2
1
33.1)20(9
196.1133.1C
)19(2
1
33.1)20(9
196.1133.1 pk
67 . 1 C 99 . 0pk
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
nc
st1
CC
nc
st1
C
c
1n,2
pcpc
c
1n,2
pc
n
1ii Tx
n
1c
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.)
)19(2
1
33.1)20(9
196.1133.1C
)19(2
1
33.1)20(9
196.1133.1 pk
67 . 1 C 99 . 0pk
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:
25.133.1
66.1
)Low(C
)High(C
p
p
46.1
)10.1(33.1
)10.1)(Low(CpC
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
2gage
2product
2Total
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.
X
X
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 is2
gage d
Rˆ
LSLUSL
ˆ6T/P gage
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
2gage
22product
2gage
2product
2
2gage
2product
2Total
ˆSˆ
ˆˆS
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:
100ˆ
ˆ
product
gage
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.
2ityrepeatabil
2ilityreproducib
2gage
2errortmeasuremen
Introduction to Statistical Quality Control, 4th Edition
7-6.1 Control Charts and Tabular MethodsStatistics 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.
2ityrepeatabil d
Rˆ
p
RRRR p21
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
2
xilityreproducib d
Rˆ
)x,x,xmin(x
)x,x,xmax(x
xxR
p21min
p21max
minmaxx
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.
n,...,2,1k
b,...,2,1j
a,...2,1i
)(y ijkijjiijk
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.
2222ijk )y(V
2222 ,,,
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
bn
MSMSˆ
an
MSMSˆ
n
MSMSˆ
MSˆ
ABA2
ABB2
EAB2
E2
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