8
Process Capability Analysis
8.1
Introduction
• A process capability analysis relates the inherent variability in a process to specifications
or requirements for the product produced by that process.
• There are many ways of analyzing the capability of a process. The most common being:
(1) Histograms and probability plots
(2) The control chart
(3) Process capability ratios.
(4) Designed experiments.
• Process capability measures the uniformity of a process. Process variability (variance) and
systematic deviations from a target value (bias) are the primary sources of nonuniformity.
• We will study the two major components of process variability:
– Short-term variability which reflects the inherent random variability at a point in time.
– Long-term variability which reflects the variability over time.
• It is common to take a 6σ spread as a measure of process capability (where σ comes from the
distribution of the product quality characteristic of interest).
• When the distribution is assumed to be normal N (µ, σ), we define the natural tolerance
limits to be µ ± 3σ. In this case, 99.73% of process output will be within the tolerance limits.
• One way to estimate of process capability is to find a probability distribution that best describes data from that process (e.g. normal, weibull, gamma, lognormal, etc.). Once an
acceptable distribution has been found a process capability analysis is performed by comparing the properties of fitted distribution to specification limits.
• When the researcher observes the process directly and can control or monitor the datacollection procedure, the study is a true process capability study because by controlling data
collection and knowing the time sequence of the data, inferences can be made about the
stability of the process over time.
• Major applications of data from a process capability analysis are:
1. Predicting how well the process will meet tolerances.
2. Assisting, when necessary, in adjusting a process.
3. Reducing the variability in a manufacturing process.
4. Specifying performance requirements for new equipment.
5. Selecting between competing suppliers.
8.2
Using a Histogram or Probability Plots
• One advantage of using a histogram is the immediate visual impression of process performance
and that it could possibly indicate a reason for poor performance (off-target, outliers, skewness,
bimodality, etc.).
• For a histogram to be moderately stable so that it can reliably estimate process capability,
Montgomery recommends that at least 100 observations be taken from the process.
105
• The histogram along with the mean x and standard deviation s enable us to assess process
capability by looking first at the shape of the histogram. If it reasonably approximates a
normal distribution, then x ± 3s can be used when assessing process capability.
• A normal probability plot with a test for normality (such as a Kolmogorov-Smirnov test) are
commonly used as supplementary checks of normality.
Example 1: I used SAS to generate two data sets of 250 values from two distributions having
µ = 20.
• The first data set contains 250 random values from a normal N (20, 1) distribution. The
variable is denoted NORMAL.
• The second data set contains 250 random values from a gamma (.5,40) distribution. The
variable is denoted GAMMA.
• Suppose the lower and upper specification limits are LSL=17 and USL=23, respectively.
• Histograms (1) and (2) have a normal pdf superimposed on the normal and gamma data
histograms, respectively.
• Histograms (3) and (4) have a gamma pdf superimposed on the normal and true gamma data
histograms, respectively.
• The estimated parameters shown below each plot are the maximum likelihood estimates
(MLEs).
• The quality of the fitted distribution to the hypothesized distribution can be assessed with
goodness-of-fit tests.
• SAS can output the results for the (i) Anderson-Darling Test, (ii) Cramer Von-Mises Test, (3)
Kolmogorov-Smirnov Test, and (4) the (not-recommended) Chi-Square Goodness-of-Fit Test.
SAS Summary Statistics for the Normal(20,1) sample data:
-------------------------------------------------------The CAPABILITY Procedure
Variable: _normal
Moments
N
Mean
Std Deviation
Skewness
Uncorrected SS
Coeff Variation
250
20.0544549
0.98469082
0.12451955
100786.725
4.91008519
Sum Weights
Sum Observations
Variance
Kurtosis
Corrected SS
Std Error Mean
250
5013.61374
0.96961602
0.28268622
241.434388
0.06227732
Basic Statistical Measures
Location
Mean
Median
Mode
20.05445
19.98213
.
Variability
Std Deviation
Variance
Range
Interquartile Range
106
0.98469
0.96962
5.90262
1.36525
Tests for Normality
Test
--Statistic---
-----p Value------
Shapiro-Wilk
Kolmogorov-Smirnov
Cramer-von Mises
Anderson-Darling
W
D
W-Sq
A-Sq
Pr
Pr
Pr
Pr
0.994503
0.037547
0.059499
0.378384
Quantiles
W
D
W-Sq
A-Sq
0.5028
>0.1500
>0.2500
>0.2500
Extreme Observations
Quantile
100% Max
99%
95%
90%
75% Q3
50% Median
25% Q1
10%
5%
1%
0% Min
<
>
>
>
Estimate
23.3518731
22.4471940
21.7529221
21.2904965
20.7315978
19.9821281
19.3663452
18.9203281
18.4856329
17.8064801
17.4492496
-------Lowest-------
-------Highest------
Value
Obs
Value
Obs
17.4492496
17.5315695
17.8064801
17.8153951
17.8168153
239
29
234
214
5
22.0698526
22.1224929
22.4471940
22.9921756
23.3518731
33
213
70
56
61
Specification Limits
--------Limit--------
------Percent-------
Lower (LSL)
Target
Upper (USL)
% < LSL
% Between
% > USL
17.00000
20.00000
23.00000
0.00000
99.60000
0.40000
Process Capability Indices
Index
Cp
CPL
CPU
Cpk
Cpm
Value
1.015547
1.033981
0.997113
0.997113
1.013998
95% Confidence Limits
0.926361
0.934039
0.900105
0.900280
0.926976
1.104631
1.133481
1.093675
1.093946
1.104973
SAS Summary Statistics for the Gamma(.5,40) sample data:
-------------------------------------------------------The CAPABILITY Procedure
Variable: _gamma
Moments
N
Mean
Std Deviation
250
19.834919
3.20121968
Sum Weights
Sum Observations
Variance
107
250
4958.72976
10.2478074
Skewness
Uncorrected SS
Coeff Variation
0.300444
100907.707
16.1393131
Kurtosis
Corrected SS
Std Error Mean
-0.1739938
2551.70405
0.20246291
Basic Statistical Measures
Location
Mean
Median
Mode
Variability
19.83492
19.73126
.
Std Deviation
Variance
Range
Interquartile Range
3.20122
10.24781
18.15474
4.66168
Tests for Normality
Test
--Statistic---
-----p Value------
Shapiro-Wilk
Kolmogorov-Smirnov
Cramer-von Mises
Anderson-Darling
W
D
W-Sq
A-Sq
Pr
Pr
Pr
Pr
0.990459
0.048292
0.109171
0.650947
Quantiles
W
D
W-Sq
A-Sq
Estimate
-------Lowest-------
30.1185447
27.1717936
25.3220567
23.9723333
22.2090666
19.7312570
17.5473890
15.7160413
14.9488884
13.5740292
11.9638075
-------Highest------
Value
Obs
Value
Obs
11.9638075
13.2888758
13.5740292
13.6244628
13.6678656
218
174
234
1
17
26.9835419
27.1096947
27.1717936
27.2287022
30.1185447
173
28
96
176
5
Specification Limits
--------Limit--------
------Percent-------
Lower (LSL)
Target
Upper (USL)
% < LSL
% Between
% > USL
17.00000
20.00000
23.00000
18.40000
64.00000
17.60000
Process Capability Indices
Index
Cp
CPL
CPU
Cpk
Cpm
0.1010
>0.1500
0.0878
0.0909
Extreme Observations
Quantile
100% Max
99%
95%
90%
75% Q3
50% Median
25% Q1
10%
5%
1%
0% Min
<
>
>
>
Value
0.312381
0.295192
0.329570
0.295192
0.311966
95% Confidence Limits
0.284947
0.246202
0.278902
0.246412
0.285194
0.339783
0.343748
0.379784
0.343971
0.339956
108
PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA
Distribution of _normal
25
20
N
Cp
Cpk
Cpm
250
1.02
1.00
1.01
Summary Statistics
Mean
Std Dev
Skewness
Kurtosis
20.05
0.985
0.125
0.283
Percent
15
10
5
0
17.1
17.7
18.3
18.9
19.5
20.1
20.7
21.3
21.9
22.5
23.1
_normal
Specifications and Curve
Lower=17
Normal(Mu=20.054 Sigma=0.9847)
Target=20
Upper=23
PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA
Distribution of _gamma
25
20
N
Cp
Cpk
Cpm
250
0.31
0.30
0.31
Summary Statistics
Mean
Std Dev
Skewness
Kurtosis
19.83
3.201
0.300
-.174
Percent
15
10
5
0
10
12
14
16
18
20
22
24
26
28
_gamma
Specifications and Curve
Lower=17
Normal(Mu=19.835 Sigma=3.2012)
109
Target=20
Upper=23
30
PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA
Distribution of _normal
25
20
N
Cp
Cpk
Cpm
250
1.02
1.00
1.01
Summary Statistics
Mean
Std Dev
Skewness
Kurtosis
20.05
0.985
0.125
0.283
Percent
15
10
5
0
17.1
17.7
18.3
18.9
19.5
20.1
20.7
21.3
21.9
22.5
23.1
_normal
Specifications and Curve
Lower=17
Gamma(Theta=0 Alpha=417 Sigma=0.05)
Target=20
Upper=23
PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA
Distribution of _gamma
30
25
N
Cp
Cpk
Cpm
250
0.31
0.30
0.31
Summary Statistics
Mean
Std Dev
Skewness
Kurtosis
19.83
3.201
0.300
-.174
Percent
20
15
10
5
0
10
12
14
16
18
20
22
24
26
28
30
_gamma
Specifications and Curve
Lower=17
Gamma(Theta=0 Alpha=38.6 Sigma=0.51)
110
Target=20
Upper=23
SAS Code for Process Capability Example with Normal and Gamma Data
DM ’LOG; CLEAR; OUT; CLEAR;’;
* ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\cp1.pdf’;
ODS LISTING;
OPTIONS LS=78 PS=500 NONUMBER NODATE;
*******************************************************************;
*** NORMAL AND GAMMA VARIATES FROM DISTRIBUTIONS WITH MEAN = 20 ***;
*******************************************************************;
DATA in;
DO N = 1 TO 250;
_normal = 20 + RANNOR(5510);
** NORMAL(20,1)
**;
_gamma = .5*RANGAM(20921,40);
** GAMMA(.5,40)
**;
OUTPUT;
END;
SYMBOL1 VALUE=dot WIDTH=3 L=1;
TITLE ’PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA’;
PROC CAPABILITY DATA=in;
VAR _normal _gamma;
SPEC LSL=17 USL=23 TARGET=20 ;
** Specify responses ;
** Enter specifications;
*** Make histograms of the normal and gamma data ;
*** with the MLE normal pdf and statistics superimposed ;
HISTOGRAM _normal _gamma / NORMAL(INDICES);
INSET MEAN (5.3) STD=’Std Dev’ (5.3)
SKEWNESS (5.3) KURTOSIS (5.3) /
HEADER = ’Summary Statistics’
POS = NE;
INSET N CP (4.2) CPK (4.2) CPM (4.2) / POS = NW;
*** Make histograms of the normal and gamma data ;
*** with the MLE gamma pdf and statistics superimposed ;
HISTOGRAM _normal _gamma / GAMMA(THETA=0 INDICES);
INSET MEAN (5.3) STD=’Std Dev’ (5.3)
SKEWNESS (5.3) KURTOSIS (5.3) /
HEADER = ’Summary Statistics’
POS = NE;
INSET N CP (4.2) CPK (4.2) CPM (4.2) / POS = NW;
*** Make empirical CDF plots of the normal and gamma data ;
*** with the MLE normal CDF superimposed
;
CDFPLOT
_normal _gamma / NORMAL;
*** Make QQ and PP plots of the normal data ;
QQPLOT _normal / NORMAL;
PPPLOT _normal / NORMAL;
RUN;
• As an alternative to the histogram, we can use probability plots (such as CDF plots, percentilepercentile (PP) plots, and quantile-quantile (QQ) plots) to study process capability.
• Plot (5) has the fitted normal CDF plot superimposed on the empirical CDF plot for the
random normal data.
• Plot (6) has the fitted normal CDF plot superimposed on the empirical CDF plot for the
random gamma data.
• Plot (7) is quantile plot of the random normal data versus the quantiles assuming a normal
distribution.
• Plot (8) is a normal probability plot of the random normal data.
111
PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA
Cumulative Distribution Function for _normal
100
Cumulative Percent
80
60
40
20
0
18
20
22
24
_normal
Specifications and Normal Curve
Lower=17
Mu=20.054 Sigma=0.9847
Target=20
Upper=23
PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA
Cumulative Distribution Function for _gamma
100
Cumulative Percent
80
60
40
20
0
10
15
20
25
30
_gamma
Specifications and Normal Curve
Lower=17
Mu=19.835 Sigma=3.2012
112
Target=20
Upper=23
35
PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA
Q-Q Plot for _normal
24
_normal
22
20
18
16
-3
-2
-1
0
1
2
3
Normal Quantiles
Specifications
Lower=17
Target=20
Upper=23
PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA
P-P Plot for _normal
Cumulative Distribution of _normal
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
Normal(Mu=20.054 Sigma=0.9847)
113
0.8
1.0
• SAS output (9A) contains results for these tests fitting a (hypothesized) normal distribution
to the random normal data. All p-values are large so we fail to reject the null hypothesis of a
normal distribution.
• SAS output (9B) contains results for these tests fitting a (hypothesized) normal distribution
to the random gamma data. All p-values are relatively small so there is evidence to reject the
null hypothesis of a normal distribution.
PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA
OUTPUT 9A
The CAPABILITY Procedure
Fitted Normal Distribution for _normal
Parameters for Normal Distribution
Parameter
Symbol
Estimate
Mean
Std Dev
Mu
Sigma
20.05445
0.984691
Goodness-of-Fit Tests for Normal Distribution
Test
----Statistic-----
Kolmogorov-Smirnov
Cramer-von Mises
Anderson-Darling
Chi-Square
D
W-Sq
A-Sq
Chi-Sq
0.0375466
0.0594991
0.3783838
11.3270226
DF
7
------p Value-----Pr
Pr
Pr
Pr
>
>
>
>
D
W-Sq
A-Sq
Chi-Sq
>0.150
>0.250
>0.250
0.125
Percent Outside Specifications for Normal Distribution
Lower Limit
LSL
Obs Pct < LSL
Est Pct < LSL
Upper Limit
17.000000
0
0.096127
USL
Obs Pct > USL
Est Pct > USL
23.000000
0.400000
0.138878
Capability Indices Based on Normal Distribution
Cp
CPL
CPU
Cpk
Cpm
1.015547
1.033981
0.997113
0.997113
1.013998
Quantiles for Normal Distribution
Percent
1.0
5.0
10.0
25.0
50.0
75.0
90.0
95.0
99.0
------Quantile-----Observed
Estimated
17.8065
18.4856
18.9203
19.3663
19.9821
20.7316
21.2905
21.7529
22.4472
17.7637
18.4348
18.7925
19.3903
20.0545
20.7186
21.3164
21.6741
22.3452
114
OUTPUT 9B
The CAPABILITY Procedure
Fitted Normal Distribution for _gamma
Parameters for Normal Distribution
Parameter
Symbol
Estimate
Mean
Std Dev
Mu
Sigma
19.83492
3.20122
Goodness-of-Fit Tests for Normal Distribution
Test
----Statistic-----
Kolmogorov-Smirnov
Cramer-von Mises
Anderson-Darling
Chi-Square
D
W-Sq
A-Sq
Chi-Sq
0.04829201
0.10917094
0.65094719
6.39603200
DF
7
------p Value-----Pr
Pr
Pr
Pr
>
>
>
>
D
W-Sq
A-Sq
Chi-Sq
>0.150
0.088
0.091
0.494
Percent Outside Specifications for Normal Distribution
Lower Limit
LSL
Obs Pct < LSL
Est Pct < LSL
Upper Limit
17.000000
18.400000
18.792339
USL
Obs Pct > USL
Est Pct > USL
23.000000
17.600000
16.140229
Capability Indices Based on Normal Distribution
Cp
CPL
CPU
Cpk
Cpm
0.312381
0.295192
0.329570
0.295192
0.311966
Quantiles for Normal Distribution
Percent
1.0
5.0
10.0
25.0
50.0
75.0
90.0
95.0
99.0
------Quantile-----Observed
Estimated
13.5740
14.9489
15.7160
17.5474
19.7313
22.2091
23.9723
25.3221
27.1718
12.3878
14.5694
15.7324
17.6757
19.8349
21.9941
23.9374
25.1005
27.2821
• Output 10A and 10B contains the parameter estimates for fitting the normal data to a gamma
distribution (10A) and for fitting the gamma data to a gamma distribution (10B).
• They also contain tables of the observed versus estimated quantiles. If the fitted distribution
is a good choice, then the quantiles should be close.
• The output also contains process capability indices that we will discuss in the next section.
115
OUTPUT 10A
The CAPABILITY Procedure
Fitted Gamma Distribution for _normal
Parameters for Gamma Distribution
Parameter
Symbol
Estimate
Threshold
Scale
Shape
Mean
Std Dev
Theta
Sigma
Alpha
0
0.048128
416.6901
20.05445
0.982436
Goodness-of-Fit Tests for Gamma Distribution
Test
----Statistic-----
Kolmogorov-Smirnov
Cramer-von Mises
Anderson-Darling
Chi-Square
D
W-Sq
A-Sq
Chi-Sq
0.0313208
0.0488707
0.3381429
11.2562464
DF
7
------p Value-----Pr
Pr
Pr
Pr
>
>
>
>
D
W-Sq
A-Sq
Chi-Sq
>0.500
>0.500
>0.500
0.128
Percent Outside Specifications for Gamma Distribution
Lower Limit
LSL
17.000000
Obs Pct < LSL
0
Est Pct < LSL
0.054527
Upper Limit
USL
23.000000
Obs Pct > USL
0.400000
Est Pct > USL
0.199484
Capability Indices Based on Gamma Distribution
Cp
CPL
CPU
Cpk
Cpm
1.017742
1.083847
0.957809
0.957809
0.968746
Quantiles for Gamma Distribution
Percent
1.0
5.0
10.0
25.0
50.0
75.0
90.0
95.0
99.0
------Quantile-----Observed
Estimated
17.8065
18.4856
18.9203
19.3663
19.9821
20.7316
21.2905
21.7529
22.4472
17.8400
18.4663
18.8062
19.3834
20.0384
20.7081
21.3234
21.6973
22.4105
116
OUTPUT 10B
Fitted Gamma Distribution for _gamma
Parameters for Gamma Distribution
Parameter
Symbol
Estimate
Threshold
Scale
Shape
Mean
Std Dev
Theta
Sigma
Alpha
0
0.513866
38.59943
19.83492
3.192567
Goodness-of-Fit Tests for Gamma Distribution
Test
----Statistic-----
Kolmogorov-Smirnov
Cramer-von Mises
Anderson-Darling
Chi-Square
D
W-Sq
A-Sq
Chi-Sq
0.03369770
0.04294087
0.27089970
3.64833166
DF
7
------p Value-----Pr
Pr
Pr
Pr
>
>
>
>
D
W-Sq
A-Sq
Chi-Sq
>0.500
>0.500
>0.500
0.819
Percent Outside Specifications for Gamma Distribution
Lower Limit
LSL
17.000000
Obs Pct < LSL
18.400000
Est Pct < LSL
18.947093
Upper Limit
USL
23.000000
Obs Pct > USL
17.600000
Est Pct > USL
15.960379
Capability Indices Based on Gamma Distribution
Cp
CPL
CPU
Cpk
Cpm
0.312763
0.330698
0.299782
0.299782
0.269220
Quantiles for Gamma Distribution
Percent
1.0
5.0
10.0
25.0
50.0
75.0
90.0
95.0
99.0
------Quantile-----Observed
Estimated
13.5740
14.9489
15.7160
17.5474
19.7313
22.2091
23.9723
25.3221
27.1718
13.1701
14.8915
15.8692
17.5986
19.6639
21.8850
24.0206
25.3618
28.0075
• In general, we will relate the empirical distribution to a theoretical distribution. The parameters of the theoretical distribution can be specified or estimated from the data.
• We will choose a distribution that ‘best’ represents the data. This can be based on scientific or
engineering principles or by empirical modeling among competing distributions. The following
figure is a guide to the choice of a distribution by locating the measures of skewness (β1 ) and
kurtosis (β2 ) on the figure.
117
• From the data we get estimates:
q
βb1 =
Pn
where Mj =
i=1 (xi
n
− x)j
βb2 =
is the j th centered sample moment.
• The relations between the SAS measures of skewness and kurtosis and β1 and β2 are (i)
βb1 ≈ (SAS skewness)2 and βb2 ≈ (SAS kurtosis) + 3.
• If the point (βb1 , βb2 ) falls in a region where none of the distributions seem appropriate, you
will need to consider other families of distributions (e.g. Weibull).
118