introduction to statistical quality control.ppt

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

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

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.

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 tolerances
7. Reducing the variability in a manufacturing process.

4th Edition
7-1. Introduction
Techniques used in process capability analysis
1. Histograms or probability plots
2. Control Charts
3. Designed Experiments

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.
s
3
x 

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

̂

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).

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
LSL
USL
Cp

4th Edition
The estimate of Cp is given by
Where the estimate can be calculated using the sample
standard deviation, S, or



ˆ
6
LSL
USL
Ĉp
̂
2
d
/
R

4th Edition
Piston ring diameter in Example 5-1
• The estimate of Cp is
68
.
1
)
0099
.
0
(
6
95
.
73
05
.
74
Ĉp




4th Edition
One-Sided Specifications
These indices are used for upper specification and
lower specification limits, respectively








3
LSL
C
3
USL
C
pl
pu

4th Edition
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.

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)

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
T
X
E
2
6
LSL
USL
Cpc





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
.
0
99865
.
0
p
x
x
LSL
USL
)
q
(
C




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)

4th Edition
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
LSL
USL
Cpm
 
  2
2
2
2
)
T
(
T
x
E 








4th Edition
Centering
• Cpm can be rewritten another way:
where
2
p
2
2
pm
1
C
)
T
(
6
LSL
USL
C














T

4th Edition
Centering
• A logical estimate of Cpm is:
where
2
p
pm
V
1
Ĉ
Ĉ


S
x
T
V



4th Edition
Centering
Example 7-3. Consider two processes A and B.
• For process A:
since process A is centered.
• For process B:
0
.
1
0
1
0
.
1
1
C
C
2
p
pm 





63
.
0
)
3
(
1
0
.
2
1
C
C
2
2
p
pm 







4th Edition
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
pk
pkm
1
C
T
1
C
C















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
1
n
Ĉ
C
1
n
Ĉ
2
1
n
,
2
/
p
p
2
1
n
,
2
/
1
p





 





4th Edition
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
.
3
C
57
.
1
19
85
.
32
29
.
2
C
19
91
.
8
29
.
2
p
p




29
.
2
)
75
.
1
(
6
38
62
Ĉp 



4th Edition
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
n
(
2
1
Ĉ
n
9
1
Z
1
Ĉ
C
)
1
n
(
2
1
Ĉ
n
9
1
Z
1
Ĉ
pk
2
/
pk
pk
pk
2
/
pk

4th Edition
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
1
96
.
1
1
33
.
1
C
)
19
(
2
1
33
.
1
)
20
(
9
1
96
.
1
1
33
.
1 pk
67.1 C 99.0
pk
 

4th Edition
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




















n
c
s
t
1
Ĉ
C
n
c
s
t
1
Ĉ
c
1
n
,
2
pc
pc
c
1
n
,
2
pc
 


n
1
i
i T
x
n
1
c

4th Edition
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
1
96
.
1
1
33
.
1
C
)
19
(
2
1
33
.
1
)
20
(
9
1
96
.
1
1
33
.
1 pk
67.1 C 99.0
pk
 

4th Edition
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

4th Edition
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

4th Edition
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.

4th Edition
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
.
1
33
.
1
66
.
1
)
Low
(
C
)
High
(
C
p
p


46
.
1
)
10
.
1
(
33
.
1
)
10
.
1
)(
Low
(
Cp
C




4th Edition
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.

4th Edition
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.

4th Edition
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.

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
gage
2
product
2
Total 





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

4th Edition
Methods
Precision to Tolerance (P/T) Ratio
• An estimate of the standard deviation for
measurement error is
• The P/T ratio is
2
gage
d
R
ˆ 

LSL
USL
ˆ
6
T
/
P
gage




4th Edition
Methods
• Total variability can be estimated using
the sample variance. An estimate of
product variability can be found using
2
gage
2
2
product
2
gage
2
product
2
2
gage
2
product
2
Total
ˆ
S
ˆ
ˆ
ˆ
S














4th Edition
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:
100
ˆ
ˆ
product
gage




4th Edition
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.

4th Edition
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.
2
ity
repeatabil
2
ility
reproducib
2
gage
2
error
t
measuremen 







4th Edition
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.
2
ity
repeatabil
d
R
ˆ 

p
R
R
R
R
p
2
1 





4th Edition
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
x
ility
reproducib
d
R
ˆ 

)
x
,
x
,
x
min(
x
)
x
,
x
,
x
max(
x
x
x
R
p
2
1
min
p
2
1
max
min
max
x







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.

4th Edition
of Variance
• The measurements, yijk, could be represented by
the model
where i = part, j = operator, k = measurement.


















n
,...,
2
,
1
k
b
,...,
2
,
1
j
a
,...
2
,
1
i
)
(
y ijk
ij
j
i
ijk

4th Edition
of Variance
• The variance of any observation can be given by
are the variance
components.
2
2
2
2
ijk )
y
(
V 






 


2
2
2
2
,
,
, 


 



4th Edition
of Variance
• Estimating the variance components can be
accomplished using the following formulas
bn
MS
MS
ˆ
an
MS
MS
ˆ
n
MS
MS
ˆ
MS
ˆ
AB
A
2
AB
B
2
E
AB
2
E
2















introduction to statistical quality control.ppt

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