process capability analysis8 process capability analysis 8.1 introduction a process capability...

8 Process Capability Analysis

8.1 Introduction

• A process capability analysis relates the inherent variability in a process to specificationsor 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 (3) Process capability ratios.(2) The control chart (4) Designed experiments.

• Process capability measures the uniformity of a process. Process variability (variance) andsystematic 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 thedistribution of the product quality characteristic of interest).

• When the distribution is assumed to be normal N(µ, σ), we define the natural tolerancelimits 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 de-scribes data from that process (e.g. normal, weibull, gamma, lognormal, etc.). Once anacceptable distribution has been found a process capability analysis is performed by compar-ing the properties of fitted distribution to specification limits.

• When the researcher observes the process directly and can control or monitor the data-collection procedure, the study is a true process capability study because by controlling datacollection and knowing the time sequence of the data, inferences can be made about thestability 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 performanceand 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.

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• The histogram along with the mean x and standard deviation s enable us to assess processcapability by looking first at the shape of the histogram. If it reasonably approximates anormal 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) arecommonly used as supplementary checks of normality.

Example 1: I used SAS to generate two data sets of 250 values from two distributions havingµ = 20.

• The first data set contains 250 random values from a normal N(20, 1) distribution. Thevariable is denoted NORMAL.

• The second data set contains 250 random values from a gamma (.5,40) distribution. Thevariable 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 datahistograms, respectively.

• Histograms (3) and (4) have a gamma pdf superimposed on the normal and true gamma datahistograms, 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 withgoodness-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 ProcedureVariable: _normal

Moments

N 250 Sum Weights 250Mean 20.0544549 Sum Observations 5013.61374Std Deviation 0.98469082 Variance 0.96961602Skewness 0.12451955 Kurtosis 0.28268622Uncorrected SS 100786.725 Corrected SS 241.434388Coeff Variation 4.91008519 Std Error Mean 0.06227732

Basic Statistical Measures

Location Variability

Mean 20.05445 Std Deviation 0.98469Median 19.98213 Variance 0.96962Mode . Range 5.90262

Interquartile Range 1.36525

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Tests for Normality

Test --Statistic--- -----p Value------

Shapiro-Wilk W 0.994503 Pr < W 0.5028Kolmogorov-Smirnov D 0.037547 Pr > D >0.1500Cramer-von Mises W-Sq 0.059499 Pr > W-Sq >0.2500Anderson-Darling A-Sq 0.378384 Pr > A-Sq >0.2500

Quantiles Extreme Observations

Quantile Estimate -------Lowest------- -------Highest------

100% Max 23.3518731 Value Obs Value Obs99% 22.447194095% 21.7529221 17.4492496 239 22.0698526 3390% 21.2904965 17.5315695 29 22.1224929 21375% Q3 20.7315978 17.8064801 234 22.4471940 7050% Median 19.9821281 17.8153951 214 22.9921756 5625% Q1 19.3663452 17.8168153 5 23.3518731 6110% 18.92032815% 18.48563291% 17.80648010% Min 17.4492496

Specification Limits

--------Limit-------- ------Percent-------

Lower (LSL) 17.00000 % < LSL 0.00000Target 20.00000 % Between 99.60000Upper (USL) 23.00000 % > USL 0.40000

Process Capability Indices

Index Value 95% Confidence Limits

Cp 1.015547 0.926361 1.104631CPL 1.033981 0.934039 1.133481CPU 0.997113 0.900105 1.093675Cpk 0.997113 0.900280 1.093946Cpm 1.013998 0.926976 1.104973

SAS Summary Statistics for the Gamma(.5,40) sample data:--------------------------------------------------------

The CAPABILITY ProcedureVariable: _gamma

Moments

N 250 Sum Weights 250Mean 19.834919 Sum Observations 4958.72976Std Deviation 3.20121968 Variance 10.2478074

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Skewness 0.300444 Kurtosis -0.1739938Uncorrected SS 100907.707 Corrected SS 2551.70405Coeff Variation 16.1393131 Std Error Mean 0.20246291

Basic Statistical Measures

Location Variability

Mean 19.83492 Std Deviation 3.20122Median 19.73126 Variance 10.24781Mode . Range 18.15474

Interquartile Range 4.66168

Tests for Normality

Test --Statistic--- -----p Value------

Shapiro-Wilk W 0.990459 Pr < W 0.1010Kolmogorov-Smirnov D 0.048292 Pr > D >0.1500Cramer-von Mises W-Sq 0.109171 Pr > W-Sq 0.0878Anderson-Darling A-Sq 0.650947 Pr > A-Sq 0.0909

Quantiles Extreme Observations

Quantile Estimate -------Lowest------- -------Highest------

100% Max 30.1185447 Value Obs Value Obs99% 27.171793695% 25.3220567 11.9638075 218 26.9835419 17390% 23.9723333 13.2888758 174 27.1096947 2875% Q3 22.2090666 13.5740292 234 27.1717936 9650% Median 19.7312570 13.6244628 1 27.2287022 17625% Q1 17.5473890 13.6678656 17 30.1185447 510% 15.71604135% 14.94888841% 13.57402920% Min 11.9638075

Specification Limits

--------Limit-------- ------Percent-------

Lower (LSL) 17.00000 % < LSL 18.40000Target 20.00000 % Between 64.00000Upper (USL) 23.00000 % > USL 17.60000

Process Capability Indices

Index Value 95% Confidence Limits

Cp 0.312381 0.284947 0.339783CPL 0.295192 0.246202 0.343748CPU 0.329570 0.278902 0.379784Cpk 0.295192 0.246412 0.343971Cpm 0.311966 0.285194 0.339956

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PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATAPROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA

17.1 17.7 18.3 18.9 19.5 20.1 20.7 21.3 21.9 22.5 23.1

_normal

0

5

10

15

20

25P

erce

nt

0.283Kurtosis0.125Skewness0.985Std Dev20.05Mean

Summary Statistics

1.01Cpm1.00Cpk1.02Cp250N

Normal(Mu=20.054 Sigma=0.9847)Upper=23Target=20Lower=17

Specifications and Curve

Distribution of _normal


10 12 14 16 18 20 22 24 26 28 30

_gamma

0

5

10

15

20

25

Per

cent

-.174Kurtosis0.300Skewness3.201Std Dev19.83Mean

Summary Statistics

0.31Cpm0.30Cpk0.31Cp250N

Normal(Mu=19.835 Sigma=3.2012)Upper=23Target=20Lower=17


Distribution of _gamma

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17.1 17.7 18.3 18.9 19.5 20.1 20.7 21.3 21.9 22.5 23.1

_normal

0

5

10

15

20

25P

erce

nt

0.283Kurtosis0.125Skewness0.985Std Dev20.05Mean

Summary Statistics

1.01Cpm1.00Cpk1.02Cp250N

Gamma(Theta=0 Alpha=417 Sigma=0.05)Upper=23Target=20Lower=17


Distribution of _normal


10 12 14 16 18 20 22 24 26 28 30

_gamma

0

5

10

15

20

25

30

Per

cent

-.174Kurtosis0.300Skewness3.201Std Dev19.83Mean

Summary Statistics

0.31Cpm0.30Cpk0.31Cp250N

Gamma(Theta=0 Alpha=38.6 Sigma=0.51)Upper=23Target=20Lower=17


Distribution of _gamma

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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; ** Specify responses ;SPEC LSL=17 USL=23 TARGET=20 ; ** 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, percentile-percentile (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 therandom normal data.

• Plot (6) has the fitted normal CDF plot superimposed on the empirical CDF plot for therandom gamma data.

• Plot (7) is quantile plot of the random normal data versus the quantiles assuming a normaldistribution.

• Plot (8) is a normal probability plot of the random normal data.

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18 20 22 24

_normal

0

20

40

60

80

100C

umul

ativ

e P

erce

nt

Mu=20.054 Sigma=0.9847Upper=23Target=20Lower=17

Specifications and Normal Curve

Cumulative Distribution Function for _normal


10 15 20 25 30 35

_gamma

0

20

40

60

80

100

Cum

ulat

ive

Per

cent

Mu=19.835 Sigma=3.2012Upper=23Target=20Lower=17

Specifications and Normal Curve

Cumulative Distribution Function for _gamma

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-3 -2 -1 0 1 2 3

Normal Quantiles

16

18

20

22

24_n

orm

al

Upper=23Target=20Lower=17Specifications

Q-Q Plot for _normal


0.0 0.2 0.4 0.6 0.8 1.0

Normal(Mu=20.054 Sigma=0.9847)

0.0

0.2

0.4

0.6

0.8

1.0

Cum

ulat

ive

Dis

trib

utio

n of

_no

rmal

P-P Plot for _normal

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• SAS output (9A) contains results for these tests fitting a (hypothesized) normal distributionto the random normal data. All p-values are large so we fail to reject the null hypothesis of anormal distribution.

• SAS output (9B) contains results for these tests fitting a (hypothesized) normal distributionto the random gamma data. All p-values are relatively small so there is evidence to reject thenull hypothesis of a normal distribution.

PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA

OUTPUT 9A The CAPABILITY ProcedureFitted Normal Distribution for _normal

Parameters for Normal Distribution

Parameter Symbol Estimate

Mean Mu 20.05445Std Dev Sigma 0.984691

Goodness-of-Fit Tests for Normal Distribution

Test ----Statistic----- DF ------p Value------

Kolmogorov-Smirnov D 0.0375466 Pr > D >0.150Cramer-von Mises W-Sq 0.0594991 Pr > W-Sq >0.250Anderson-Darling A-Sq 0.3783838 Pr > A-Sq >0.250Chi-Square Chi-Sq 11.3270226 7 Pr > Chi-Sq 0.125

Percent Outside Specifications for Normal Distribution

Lower Limit Upper Limit

LSL 17.000000 USL 23.000000Obs Pct < LSL 0 Obs Pct > USL 0.400000Est Pct < LSL 0.096127 Est Pct > USL 0.138878

Capability Indices Based on Normal Distribution

Cp 1.015547CPL 1.033981CPU 0.997113Cpk 0.997113Cpm 1.013998

Quantiles for Normal Distribution

------Quantile------Percent Observed Estimated

1.0 17.8065 17.76375.0 18.4856 18.4348

10.0 18.9203 18.792525.0 19.3663 19.390350.0 19.9821 20.054575.0 20.7316 20.718690.0 21.2905 21.316495.0 21.7529 21.674199.0 22.4472 22.3452

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OUTPUT 9B The CAPABILITY ProcedureFitted Normal Distribution for _gamma

Parameters for Normal Distribution


Mean Mu 19.83492Std Dev Sigma 3.20122

Goodness-of-Fit Tests for Normal Distribution


Kolmogorov-Smirnov D 0.04829201 Pr > D >0.150Cramer-von Mises W-Sq 0.10917094 Pr > W-Sq 0.088Anderson-Darling A-Sq 0.65094719 Pr > A-Sq 0.091Chi-Square Chi-Sq 6.39603200 7 Pr > Chi-Sq 0.494

Percent Outside Specifications for Normal Distribution

Lower Limit Upper Limit

LSL 17.000000 USL 23.000000Obs Pct < LSL 18.400000 Obs Pct > USL 17.600000Est Pct < LSL 18.792339 Est Pct > USL 16.140229

Capability Indices Based on Normal Distribution

Cp 0.312381CPL 0.295192CPU 0.329570Cpk 0.295192Cpm 0.311966

Quantiles for Normal Distribution


1.0 13.5740 12.38785.0 14.9489 14.5694

10.0 15.7160 15.732425.0 17.5474 17.675750.0 19.7313 19.834975.0 22.2091 21.994190.0 23.9723 23.937495.0 25.3221 25.100599.0 27.1718 27.2821

• Output 10A and 10B contains the parameter estimates for fitting the normal data to a gammadistribution (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 distributionis a good choice, then the quantiles should be close.

• The output also contains process capability indices that we will discuss in the next section.

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OUTPUT 10A The CAPABILITY ProcedureFitted Gamma Distribution for _normal

Parameters for Gamma Distribution


Threshold Theta 0Scale Sigma 0.048128Shape Alpha 416.6901Mean 20.05445Std Dev 0.982436

Goodness-of-Fit Tests for Gamma Distribution



Percent Outside Specifications for Gamma Distribution

Lower Limit Upper LimitLSL 17.000000 USL 23.000000Obs Pct < LSL 0 Obs Pct > USL 0.400000Est Pct < LSL 0.054527 Est Pct > USL 0.199484

Capability Indices Based on Gamma Distribution

Cp 1.017742CPL 1.083847CPU 0.957809Cpk 0.957809Cpm 0.968746

Quantiles for Gamma Distribution


1.0 17.8065 17.84005.0 18.4856 18.4663

10.0 18.9203 18.806225.0 19.3663 19.383450.0 19.9821 20.038475.0 20.7316 20.708190.0 21.2905 21.323495.0 21.7529 21.697399.0 22.4472 22.4105

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OUTPUT 10B Fitted Gamma Distribution for _gamma

Parameters for Gamma Distribution


Threshold Theta 0Scale Sigma 0.513866Shape Alpha 38.59943Mean 19.83492Std Dev 3.192567

Goodness-of-Fit Tests for Gamma Distribution



Percent Outside Specifications for Gamma Distribution

Lower Limit Upper LimitLSL 17.000000 USL 23.000000Obs Pct < LSL 18.400000 Obs Pct > USL 17.600000Est Pct < LSL 18.947093 Est Pct > USL 15.960379

Capability Indices Based on Gamma Distribution

Cp 0.312763CPL 0.330698CPU 0.299782Cpk 0.299782Cpm 0.269220

Quantiles for Gamma Distribution


1.0 13.5740 13.17015.0 14.9489 14.8915

10.0 15.7160 15.869225.0 17.5474 17.598650.0 19.7313 19.663975.0 22.2091 21.885090.0 23.9723 24.020695.0 25.3221 25.361899.0 27.1718 28.0075

• In general, we will relate the empirical distribution to a theoretical distribution. The param-eters 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 orengineering principles or by empirical modeling among competing distributions. The followingfigure is a guide to the choice of a distribution by locating the measures of skewness (β1) andkurtosis (β2) on the figure.

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• From the data we get estimates:√β̂1 = β̂2 =

where Mj =

∑ni=1(xi − x)j

nis the jth centered sample moment.

• The relations between the SAS measures of skewness and kurtosis and β1 and β2 are (i)

β̂1 ≈ (SAS skewness)2 and β̂2 ≈ (SAS kurtosis) + 3.

• If the point (β̂1, β̂2) falls in a region where none of the distributions seem appropriate, youwill need to consider other families of distributions (e.g. Weibull).

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process capability analysis8 process capability analysis 8.1 introduction a process capability...

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