SMU
EMIS 7364
NTU
TO-570-N
Statistical Quality Control
Dr. Jerrell T. Stracener,
SAE Fellow
Measures of Process Capability
Process Capability Ratios
Updated: 2/4/04
1
Process Capability
Refers to the uniformity of the process.
Variability in the process is a measure of the
uniformity of the output.
- Instantaneous variability is the natural or
inherent variability at a specified time
- Variability over time
2
Process Capability
A critical performance measure that addresses
process results relative to process/product
specifications.
A capable process is one for which the process
outputs meet or exceed expectation.
3
Process Capability Measures or Indices
Process capability indices are used to measure the
process variability due to common causes present
in the process
• The Cp index
Inherent or potential measure of capability
Cp = specification spread
process spread
• The CpK index
Realized or actual measure of capability
• Other indices
CpM, CpMK
4
Measures of Process Capability
Customary to use the six sigma spread in the
distribution of the product quality characteristic
5
Key Points
The proportion of the process output that will fall
outside the natural tolerance limits.
• Is 0.27% (or 2700 nonconforming parts per million)
if the distribution is normal
• May differ considerably from 0.27% if the
distribution is not normal
6
7
Measure of Potential Process Capability, Cp
• Cp measures potential or inherent capability of the
process, given that the process is stable
• Cp is defined as
USL  LSL , for two-sided
Cp 
specifications
6σ
and
C pL 
C pU
  LSL
3σ
USL  

3σ
, for lower
specifications only
, for upper
specifications only
8
Interpretation of Cp
 1 
P    100%
C 
 p
is the percentage of the specification band used up
by the process
9
Measure of Potential Process Capability, CpK
• CpK measures realized process capability relative to
actual production, given a stable process
• CpK is defined as
C pK
   LSL USL   
 min 
,
3σ 
 3σ
10
Interpretation of CpK
< 1, then conclude that the
process is stable
If CpK
= 1, then conclude that the
process is marginally capable
> 1, then conclude that the
process is capable
11
Recommended Minimum Values of the Process
Capability Ratio
Existing process
Two-sided
specifications
1.33
One-sided
specifications
1.25
New processes
1.50
1.45
Safety, strength,
or critical parameter
existing process
1.50
1.45
Safety, strength,
or critical parameter
new process
1.67
1.60
12
Process Fallout (in defective ppm)
PCR
0.25
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
2.00
One-sided specs
226628
66807
35931
17865
8198
3467
1350
1484
159
48
14
4
1
0.17
0.03
0.0009
Two-sided specs
453255
133614
71861
35729
16395
6934
2700
967
318
96
27
7
2
0.34
0.06
0.0018
13
Estimation of Process Capability Ratios
14
Estimation of Cp
A point estimate of Cp is:
^
Cp 
where
USL  LSL
^
6σ
n 1
σs
n
^


2
1 n
s
xi  x

n - 1 i 1
15
Estimation of Cp - example
If the specification limits are
LSL = 73.95 and USL = 74.05 and
^
Cp 
^
  0.0099
USL  LSL
^
6σ
74.05  73.95

60.0099
 1.68
16
Estimation of CP - example
and the process uses
 1 
P
 100%
 1.68 
^
 59.5%
of the specification band.
17
Example
To illustrate the use of the one sided process
capability ratios, suppose that the lower specification
limit on bursting strength is 200 psi. We will use
x = 264 and S = 32 as estimates of  and ,
respectively, and the resulting estimate of the one
sided lower process capability ratio is
^
C pL 
  LSL
^
3σ
264  200

3  32
64

 0.67
96
18
Example
The fraction of defective bottles produced by this
process is estimated by finding the area to the left
^ = (200 - 264)/32 = -2 under
of Z = (LSL - ^
)/
the standard normal distribution. The estimated
fallout is about 2.28% defective, or about 22,800
nonconforming bottles per million. Note that if the
normal distribution were an inappropriate model
for strength, then this last calculation would have
to be performed using the appropriate probability
distribution.
19
Estimation of Cp
A (1 - )  100% confidence interval for Cp is
C
PL
, CPU
where

12 / 2,n 1 ^
USL

LSL
C pL =
Cp
6S
n 1
C pU = USL  LSL  / 2,n 1
Cp
6S
n 1
2
^
12 / 2,n 1
n 1
 / 2,n 1
2
n 1
where χ1α/ 2 ,n 1 and χ α/ 2 ,n 1 are the lower /2 and upper
/2 percentage points of the chi-squared distribution
with n - 1 degrees of freedom.
2
2
20
The Chi-Squared Distribution
Definition - A random variable X is said to have the Chi-Squared
distribution with parameter , called degrees of freedom, if the
probability density function of X is:
1
f (x)   / 2
x
2  / 2
0

1
x / 2
2
e
,
for x > 0
,
elsewhere
where  is a positive integer.
21
The Chi-Squared Model
Remarks:
The Chi-Squared distribution plays a vital role in statistical
inference. It has considerable application in both
methodology and theory. It is an important component of
statistical hypothesis testing and estimation.
The Chi-Squared distribution is a special case of the
Gamma distribution, i.e., when  = /2 and  = 2.
22
Properties of the Chi-Squared Model
• Mean or Expected Value

• Standard Deviation
  2
23
Properties of the Chi-Squared Model
It is customary to let 2 represent the value above
which we find an area of p. This is illustrated by the
shaded region below.
f(x)
1  F ( 2, )

0
2 ,
2
x
For tabulated values of the Chi-Squared distribution see the
Chi-Squared table, which gives values of 2 for various values
of p and . The areas, p, are the column headings; the degrees
of freedom, , are given in the left column, and the table entries
are the 2 values.
24
Estimation of CpK
An approximate (1 - )  100% confidence interval
for CpK is C , C

where
PK L
PKU



^
1
1 

C pKL = C pK 1  Z / 2

2
^

2n  1 
9n C pK




^
1
1 

C pKU = C pK 1  Z / 2

2
^

2n  1 
9n C pK


25
Example
A sample of size n = 20 is taken from a stable process
is used to estimate CpK, with the result that
^
C pK = 1.33. An approximate 95% confidence
interval on CpK is


^
1
1 


C pKL = C pK 1  Z / 2
2
^

2n  1 
9n C pK



1
1 
= 1.331  1.96


2
9201.33 219 

26
Example


1
1 

= C pK 1  Z / 2

2
^

2n  1 
9n C pK



1
1 

= 1.331  1.96

2
9201.33 219 

^
C pKU
0.99  CpK  1.67
This is an extremely wide confidence interval. Based on the
sample data, the ratio CpK could be less than one (a very bad
situation), or it could be as large as 1.67 (a very good situation).
Thus, we have learned very little about the actual process
capability, because CpK is very imprecisely estimated. The reason
for this is that a very small sample (n = 20) has been used.
27
Testing Hypotheses About PCR’s
A practice that is becoming increasingly common
in industry is to require a supplier to demonstrate
process capability as part of the contractual
agreement. Thus, it is frequently necessary to
demonstrate that the process capability ratio Cp
meets or exceeds some particular target value say, Cp0. This problem may be formulated as a
hypothesis testing problem.
H0: Cp = Cp0 (or the process is not capable)
H1: Cp > Cp0 (or the process is capable)
28
Testing Hypotheses About PCR’s
 =  = 0.10
Sample
Size
10
20
30
40
50
60
70
80
90
100
Cp(high)/
Cp(low)
1.88
1.53
1.41
1.34
1.30
1.27
1.25
1.23
1.21
1.20
C/Cp(low)
1.27
1.20
1.16
1.14
1.13
1.11
1.10
1.10
1.10
1.09
 =  = 0.05
Cp(high)/
Cp(low)
2.26
1.73
1.55
1.46
1.40
1.36
1.33
1.30
1.28
1.26
C/Cp(low)
1.37
1.26
1.21
1.18
1.16
1.15
1.14
1.13
1.12
1.11
29
Testing Hypotheses About PCR’s
We would like to reject H0, thereby demonstrating
that the process is capable. We can formulate the
^
statistical test in terms of Cp, so that we will reject
^ exceeds a critical value C.
H0, if C
p
30
Example
A customer has told his supplier that, in order to
qualify for business with his company, the supplier
must demonstrate that his process capability
exceeds Cp = 1.33. Thus his supplier is interested
in establishing a procedure to test the hypothesis.
H0: Cp = 1.33
H1: Cp > 1.33
The supplier wants to be sure that if the process
capability is below 1.33 there will be a high
probability of detecting this (say, 0.90), whereas if
the process capability exceeds 1.66 there will be a
high probability of judging the process capable
31
Example
again (say, 0.90). This would imply that Cp(low) =
1.33, Cp(high) = 1.66 and  =  = 0.10. To find
the sample size and critical value C from the table,
compute
C p (high)
1.66

 1.25
C p (low) 1.33
and enter the table value where  =  = 0.10. This
yields
n = 70
and
C
 1.10
C p (low)
32
Example
from which we calculate
C  C p (low) 1.10
 1.331.10
 1.46
Thus, to demonstrate capability, the supplier must
take a sample of n = 70 parts and the sample
^
process ratio Cp must exceed C = 1.46.
33
Testing Hypotheses
In many situations the reason for gathering and
analyzing data is to provide a basis for deciding
on a course of action. Let us assume that either
of two courses of action is possible: A1 or A2, and
that we would be clear whether one or the other
is the better action, if only we knew the nature of
a certain population - that is, if we knew the
probability distribution of a certain random
variable.
34
Testing Hypotheses
The whole population or the distribution of
probability is usually unattainable, therefore, we
are forced to settle for such information as
can be gleaned from a sample and to make our
choice between the two actions on the basis of
the sample.
1. Obtain random sample of size n
2. Apply decision rule to data
35
Testing Hypotheses
Statistical Hypothesis - is a statement about a
probability distribution and is usually a statement
about the values of one or more parameters of
the distribution. For example, a company may
want to test the hypothesis that the true average
lifetime of a certain type of TV is at least 500
hours, i.e., that   500.
36
Testing Hypotheses
The hypothesis to be tested is called the null
hypothesis and is denoted by H0. To construct a
criterion for testing a given null hypothesis, an
alternate hypothesis must be formulated,
denoted by H1.
Remark: To test the validity of a statistical
hypothesis the test is conducted, and according
to the test plan the hypothesis is rejected if the
results are improbable under the hypothesis. If
not, the hypothesis is accepted. The test leads
to one of two possible actions: accept H0 or
reject H0 (accept H1)
37
Testing Hypotheses
Test Statistic - The statistic upon which a test of
a statistical hypothesis is based.
Critical Region - The range of values of a test
statistic which, for a given test, requires the
rejection of H0.
Remark: Acceptance or rejection of a statistical
hypothesis does not prove nor disprove the
hypothesis! Whenever a statistical hypothesis is
accepted or rejected on the basis of a sample,
there is always the risk of making a wrong
decision. The uncertainty with which a decision
is made is measured in terms of probability.
38
Testing Hypotheses
There are two possible decision errors associated
with testing a statistical hypothesis:
A Type I error is made when a true hypothesis is rejected.
A Type II error is made when a false hypothesis is accepted.
Decision
Accept H0
Reject H0
(Accept H1)
True Situation
H0 true
H0 false
correct
Type II error
Type I errorcorrect
39
Uses of Results 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 on
processes or tolerances.
7. Reducing the variability in a manufacturing
process.
40