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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
1.
2.
3.
4.
5.
6.
7.
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
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 x  3s
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
USL  LSL
Cp 
6
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
USL  LSL
Ĉ p 
6ˆ
Where the estimate ̂ can be calculated using the sample
standard deviation, S, or R / d 2
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
74.05  73.95
Ĉ p 
6(0.0099)
 1.68
Introduction to Statistical Quality Control,
4th Edition
7-3.1 Use and Interpretation of Cp
One-Sided Specifications
USL  
C pu 
3
  LSL
C pl 
3
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:
1. The quality characteristic has a normal distribution.
2. The process is in statistical control
3. 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
USL  LSL
C pc 

6
EXT
2
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
USL  LSL
C p (q ) 
x 0.99865  x 0.00135
•
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.
USL  LSL
C pm 
6
where  is the squre root of expected squared
deviation from target: T =½(USL+LSL),


  E x  T     (  T)
2
2
2
2
Introduction to Statistical Quality Control,
4th Edition
7-3.4 More About Process
Centering
•
Cpm can be rewritten another way:
USL  LSL
C pm 
6  2  (  T ) 2

Cp
1 2
where
T


Introduction to Statistical Quality Control,
4th Edition
7-3.4 More About Process
Centering
•
A logical estimate of Cpm is:
Ĉpm 
Ĉp
1  V2
where
Tx
V
S
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:
Cp
1.0
C pm 

 1.0
2
1 0
1 
•
since process A is centered.
For process B:
Cp
2.0
C pm 

 0.63
2
2
1 
1  (3)
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
C pkm 

•
C pk
T
1 

  
C pk
2
1 2
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
Ĉ p

2
1  / 2 , n 1
n 1
 C p  Ĉ p

Introduction to Statistical Quality Control,
4th Edition
2
 / 2 , n 1
n 1
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
62  38
Ĉ p 
 2.29
6(1.75)
95% confidence interval on Cp is
8.91
32.85
 C p  2.29
19
19
1.57  C p  3.01
2.29
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


1
1 
1
1 
  Cpk  Ĉpk 1  Z / 2

Ĉpk 1  Z / 2


9nĈpk 2(n  1) 
9nĈpk 2(n  1) 


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


1
1 
1
1 
1.331  1.96


  C pk  1.331  1.96

9
(
20
)
1
.
33
2
(
19
)
9
(
20
)
1
.
33
2
(
19
)




0.99  C pk  1.67
•
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
Ĉ pc
 sc 
, n 1  c n 


2
1 t 
where
 C pc 
Ĉ pc
 sc 
, n 1  c n 


2
1 t 
1 n
c   xi  T
n i 1
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


1
1 
1
1 
1.331  1.96


  C pk  1.331  1.96

9
(
20
)
1
.
33
2
(
19
)
9
(
20
)
1
.
33
2
(
19
)




0.99  C pk  1.67
•
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):
C p (High )
C p (Low )
•
•

1.66
 1.25
1.33
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:
C  Cp(Low )(1.10)
 1.33(1.10)
 1.46
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
2
2
Total
 2product  gage
Introduction to Statistical Quality Control,
4th Edition
7-6.1 Control Charts and Tabular
Methods
X and R Charts
• The variability seen on the X 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
R
ˆ gage 
d2
•
The P/T ratio is
P/T 
6ˆ gage
USL  LSL
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

2
Total

2
product

2
gage
2
2
ˆ
ˆ
S   product   gage
2
2
ˆ 2product  S2  ˆ gage
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:
ˆ gage

 100
ˆ product

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
2
2measuremen t error  gage
 2reproducibility  2repeatability
•
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
R
ˆ repeatability 

where
R
d2
R1  R 2    R p
p
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
Rx
ˆ reproducibility 
d2
R x  x max  x min
x max  max( x1 , x 2 , x p )
x min  min( x1 , x 2 , x p )
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
 i  1,2,...a

yijk    i   j  ()ij   ijk  j  1,2,..., b
k  1,2,..., n

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
V( yijk )        
2

2

2

2
2 , 2 , 2 , 2 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
ˆ 2  MS E
ˆ
MS AB  MS E

n
MS B  MS AB

an
MS A  MS AB

bn
2

ˆ 2
ˆ 2
Introduction to Statistical Quality Control,
4th Edition