ASYMMETRICAL CONTROL LIMITS FOR
INDIVIDUAL MEASUREMENT X
AND MOVING RANGE (n = 2)
mR CONTROL CHARTS
BY
MICHAEL LEE ANKNEY
Industrial Engineering and Management
Oklahoma State University
1994
Submitted to the Faculty of the
Graduate College of
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
December, 1997
ASYMMETRICAL CONTROL LIMITS FOR
INDIVIDUAL MEASUREMENT X
AND MOVING RANGE (n = 2)
mR CONTROL CHARTS
Thesis Approved:
II
ACKNOWLEDGMENTS
I wish to extend my deepest appreciation to my major advisor and Chairman of
my Master committee, Dr. Kenneth Case for his understanding, support, and fri endship
during my studies at Oklahoma State University. Your constructive guidance and
inspiration were instrumental in completing my Master degree. I would al so like to
extend my sincere appreciation to my other committee members Dr. David Pratt and Dr.
Wayne Turner for their assistance during this research. I wish to express a special thanks
to Dr. Turner for allowing me to pursue opportunities in my other areas of interest.
A very special recognition goes to my wife, Cindy. Your love, support, patience
and understanding provided me with the strength and encouragement to complete my
studies. This would not have been possible without you.
Finally, I would like to thank the Department of Industrial Engineeri ng and
Management for the support over these two years of study.
iii
T ABLE OF CONTENTS
CHAPTER 1: THE PROBLEM AND ITS SETTING ..................................................... .. . 1
Introduction .............................................................................................................. .. ... 1
General Statement Of The Research Problem ...... ....... ......................................... ......... 4
The Delimitations .......................................................................................................... 5
Definition Of Terms ................................ ...... ........... .. ................................................... 6
Abbreviations And Notations .................................... .... ................................................ 8
The Assumptions ......................................................................................................... II
The Importance Of The Study ............................................................ ......... ................ 11
CHAPTER 2: REVIEW OF RELATED LITERATURE .............................. .. ................. 12
History ....................................................................................... ... ............................... 12
Shewhart Control Charts .............................................................................. ............... 13
X And R Control Charts .... ... ...... ..... ........... .. ....... .. ............................... .......... .......... .. 14
Individual Measurement X And mR Control Charts .................................................. . 17
Application Of X & mR Control Charts ................................................. .. ... .. ......... .. . 20
Current Research ................................................................ .. ........... ... ........... .. ............ 21
CHAPTER 3: THE RESEARCH METHOD ...... .... .......................................................... 23
1. Introduction .......................................................................................................... 23
2. General Data .................... .. ............................................................... .. ........ ....... .. 29
2.1 The Data .............. ........................................................................................... 29
2.2 The Criteria for Admissibility of the Data ..................................................... 30
2.3 The Research Methodology ....................................... .. .................................. 30
3. Specific Treatment Of Each Subobjective ............. .... .................... .. .......... ......... 30
3.1 Subobjective One ......................................................................................... 30
3.1.1 Individual measurement X control chart limits ...... ... ............................. 32
3.1.1.1 The upper control limit ............................................ .. .............. . 32
3.1.1.2 The lower control limit.. ........................................................... 36
3.1.2 Moving range n = 2 mR upper control chart limit ................................. 37
3.2 Subobjective Two ....................................................................... ........... ... .... 39
3.1 Subobjective Three ....................................................................................... 48
iv
CHAPTER 4: RESULTS AND ANALYSIS ......................... " ....... " ................. " ............. S4
1. Subobjective One ................................................................................................ 54
1.1 Individual Measurement X Control Chart Limits .......................... ................ 54
1.1.1 The upper control limit ........................................................................ 56
1.1.2 The lower control limit ........................................................................ 57
1.2 Moving Range n=2 Upper Control Chart Limit ............................................ 58
2. Subobjective Two ............................................................................................... 61
2.1 Norma] (40, 102) Algorithm .......................................................................... 62
2.2 Lognormal (0,12) Algorithm ......................................................................... 62
2.3 Gamma (a = 1.5, ~ = 1) Algorithm .................................................... .... ....... 63
2.4 Chisquare (df = 2) Algorithm ..................................................... .. ................ 63
2.5 Exponential (~ = I) Algorithm ...................................................................... 64
2.6 Analysis of the Normal Distribution; No Mean ShifL .................................. 74
2.7 Analysis of NonNormal Distributions; No Mean Shift.." ........ ... ................. 76
3. Subobjective Three ........................................................................................... 77
3.1 Analysis of Negative Shifts in the Mean ........................................................ 79
3.2 Analysis of Positive Shifts in the Mean ......................................................... 79
CHAPTER 5: CONCLUSIONS AND RECOMMENDATIONS ................................. .. . 80
I. Conclusions ....................................................................................... .. ................. 80
2. Research Contributions ..................................................... " ................................. 82
3. Future Research ..................................................................... .. ............................. 83
REFERENCES .................................................................................................................. 89
APPENDICES
APPENDIX A: The Scale Parameter and the Range ........................................................ 92
APPENDIX B: Sample Size Estimation ........................................................................... 95
APPENDIX C: T Statistic Table .............................................................. ......................... 98
APPENDIX 0: Regression Output for Control Limits ................................................... 100
APPENDIX E: Chisquare Distribution Turbo Pascal (version 6.0) Program ................ 108
v
APPENDIX F: Marse and Roberts Random Number Generator .................................. .. 120
APPENDIX G: Normal Distribution Program OUlpUt.. .... .......................................... .... 123
APPENDIX H: R vs. mR ............................................................................................... 127
APPENDIX I: Gamma Distribution At Different Alphas ............................................... 130
APPENDIX J: SubObjectives Two & Three Program Logic ............. .. ................... .. .... 133
vi
LIST OF TABLES
Table Page
31: crx Units From the Mean ........................... ... .............................. .............................. .. 34
32: Units From the Mean for the UCL of the Range Chart ............. ....... ... ........ .............. 40
33: Control Chart ARLs and VRLs ............. ..... ...... .................................. ..... ........ ..... ...... 49
34: ARLsNRLs for Shifts in the Process Mean ............................ ..... .. ... ....... .... .... ......... 53
41: Gamma Distribution Upper & Lower Control Limits ...... ... ..... .. .. ....... .... ,., ... , ... , ... .. .. 55
42: Gamma Distribution Upper Control Limits For Moving Range ..... ... , .. ... , .. ..... .. .. ..... 59
43: Control Chart ARLs and VRLs for No Mean Shift ..... ....... ....... ................................ 65
44: ARLsNRLs for Shifts in the Process Mean .... ....... ..... .. ... .................................. , ..... . 78
51: Theoretical Run Lengths for Exponential Distribution .. ............. ....... .. ... ..... ....... .. .... 85
52: ARLs For Control Limits Set On Different Number Of Observations ... ................... 88
VII
LIST OF FIGURES
Figure Page
31: Research Methodology Flow Chart ._ .... ... ............. .. ....... ...... .. ... ....... .. .. .............. .. ...... 25
41: Normal Distribution Run Lengths  Shewhart Control Limits ....... .... .. ..... ....... .... .. ... 66
42: Normal Distribution Run Lengths  Oyon Control Limits ......................... .. ...... ........ 66
43: Normal Distribution Run Lengths  Asymmetrical Control Limits ...... .. ... .. ...... ........ 67
44: Lognormal Distribution Run Lengths  Shewhart Control Limits ................... .. ....... 67
45: Lognormal Distribution Run Lengths  Oyon Control Limits .. ........... ...... .. ........... .. 68
46: Lognormal Distribution Run Lengths  Asymmetrical Control Limits ............ .. ...... 68
47: Gamma Distribution Run Lengths  Shewhart Control Limits ...... .................... .. ...... 69
48: Gamma Distribution Run Lengths  Oyon Control Limits ...... .. ... .. ..... .. .......... .... .. .. .. 69
49: Gamma Distribution Run Lengths  Asymmetrical Control Limits ........... ............... 70
4 J 0: Chisquare Distribution Run Lengths  Shewhart Control Limits .. .... .. ... .... .. .. ........ 70
411: Chisquare Di. tribution Run Lengths  Oyon Control Limits ..... ............................ 71
412: Chisquare Distribution Run Lengths  Asymmetrical Control Limits .................... 71
413: Exponential Distribution Run Lengths  Shewhart Control Limits ......... .. .............. 72
414: Exponential Distribution Run Lengths  Oyon Control Limits ................................ 72
415: Exponential Distribution Run Lengths  Asymmetrical Control Limits ......... .. ....... 73
VIII
CHAPTER!
THE PROBLEM AND ITS SETTING
INTRODUCTION
Dr. Walter A. Shewhart introduced the concept of control charts in the 1920' s.
The control charts were developed as tools for generating a picture of a process. The
basis of these charts was that there are two types of variation: controlled (common
cause) variation that is stable and consistent over time and uncontrolled (speci al cause)
variation which changes over time. Dr. Shewhart made the following conclusions based
on process variations; limits can be set, based on the natural vari ations of a process
(common cause), so that as long as there are fluctuations between these limits only controlled
(common cause) variation is present and fluctuations outside these limits indicate
uncontrolled (special cause) variation. If the process is influenced by only common
cause variation then it is in a state of statistical control (SOSC) and can be used as a
predictor of future occurrences; if influenced by speci al cause vari ati ons th en it is not in
a state of statistical control. Dr. Shewhart stated the following as concerned with stati stical
control: "A phenomenon will be said to be controlled when, through the use of past
experience, we can predict, at least within limits, how the phenomenon may be expected
to behave in the future (4, p. 6)."
There are many different types of control charts used to study processes. The
most commonly used control charts are X and R chalts which require meas urable quality
characteristics. The data used in these charts are made up of subgroups (typically
consisting of about four or five pieces of data) collected from the process in a rat ional
manner. The X and R values are plotted in series on their respective graph to build
control charts. These charts are utilized to monitor the process for changes in both location
and dispersion. The X control chart monitors the location of the process by plotting
the process average between subgroups. The R control chart monitors the dispersion
of the data within the subgroups by plotting the range of data poi nts within each
subgroup.
The X chart is a very robust tool although its statistical foundation is based all
the normal distribution. The robustness of this control charts is best explained by the
central limit theorem which states that for a random sample of size n, if n is significantly
large, the sample averages have approximately a nonna] di stribution. The assumption
of normality can be made even when the process 's underlying distribution is 110nnormal.
The R chart can also be used to monitor variations in process spread when the
underlying distribution is nonnormal. The robustness of the R chart cannot, however,
be explained by the central limit theorem. In fact, as sample sizes increa e, the di stribu tion
of the subgroup ranges become more di ssimilar from normal. Although the probabilities
of type I errors for nonnormal distributions fall short of those for the normal
distribution, " ... both the Average Chart and the Range Chart can be said to be robust to
those departures from normality which are likely to be encountered in practice. They can
be used with confidence. They will work and they will work well even when 'the measurements
are not normally distributed' (4, p.76)."
2
The X and R control charts are not suitable for all situations. Sometimes there
are special circumstances in a process that make subgroups impractical. Natural ubgroups
may not be feasible if there are long periods of time between measurement, a
single measurement represents one batch, measurements are too time consuming to obtain,
or measurements are too expensive to obtain. In cases such as these, where n= I,
X and R control charts are not applicable. Individual measurement X and moving
range n=2 mR control charts are commonly applied when only a single measurement is
taken at a time. An individual measurement X control chart is generated by plotting the
individual measurements on a graph to evaluate the process's location. The moving
range n=2 mR chart is generated by plotting the successive differences between the individual
values.
The individual measurement X and moving range n=2 mR control charts do not
possess the robustness of the X and R control charts. The underlying assumption of
normality is much more critical when there are no subgroups. Since the central limit
theorem does not apply to individual measurements, the quality characteri stic measurements
must be approximately normally distributed to easily and accurately generate existing
individual measurement X and moving range n=2 mR control charts.
In practice, all events cannot be explained by the normal distribution. There are
many instances where processes represent asymmetrical di stributions. According to
Irving Burr (1953), " .. . causes of nonnormality is that the di stribution may be unable to
go beyond a certain point, such as zero .... measurement has a physical limitation at zero
3
(5, p.80)." When the underlying distribution is asymmetrical the Pearson type ill family
of distributions can be used to approximate the data (5, p.67).
Despite the limitations, individual measurement X and moving range n=2 mR
control charts are used in applications with nonnormal distributions. As stated by
Schilling and Nelson (1976), "In many applications the chart is applied without knowledge
of the shape of the underlying di stribution of indi viduals (3, p.1 83 )." Conversely,
Duncan (1986) states, "Control charts for individuals must be very carefully interpreted
if the process shows evidence of marked departure from normality. In such cases, the
multiples of cr used to set control limits might be better derived from other distributions
for which the percentage points have been computed (6, pAOO). " There is only limited
research concerned with the use of individual measurement X and moving range n=2
mR control charts in industry when the underlying process distribution is nonnormal.
GENERAL STATEMENT OF THE RESEARCH PROBLEM
The problem of this research is to create and validate a mathematical model for
determining the location of upper and lower control limits on individual measurement X
and moving range n=2 mR control charts for asymmetrical distributions.
The subobjectives of this study are as follows:
(I) Develop mathematical models representative of the upper and lower control limits
for asymmetrical distributions based on the shape parameter (ex.) and the scale parameter
(~) from the Pearson Type ill family of distributions with location parameter
c=O (gamma distributions).
4
(2) Evaluate the performance of the individual measurement X and moving range n=2
mR contra} charts, based on average run lengths (ARL) and vari ation of run length
(VRL) using the Pearson type ill family of distributions with location parameter c=O
(gamma distribution) control limits determined from objective I. The performance
will be evaluated against a level that is acceptable for practical application in industry
and compared with methods having symmetrical control limits. A level that is
acceptable for practical application in industry means that the average run length
CARL) for both the individual measurement X and moving range n=2 mR control
charts is a minimum of 100, which is equivalent to a 1 % chance of a type I error,
when the process is in a state of statistical control.
(3) Compare the power of the individual measurement X and moving range n=2 mR
control charts using the Pearson Type ill family of distributions with location parameter
c=O (gamma distribution) asymmetrical control limits with those methods
having symmetrical control limits. The power in this case refers to the ability of the
control charts to detect shifts in process location of 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 Ox
units.
THE DELIMIT A TIONS
The following limitations pertain to this research:
• This study is limited to the evaluation of control limits for individual measurement
X and moving range n=2 mR control charts which apply to nonnormal distributions
5
generated by the Pearson type ill family of distribution where c=O (gamma di stribution).
• The Pearson type ill family of distributions with location parameter c=O have a
range of values from (0, +(0); therefore. values of X (quality characteristic) cannot
take on negative values.
• The Pearson type ill family of distributions with location parameter c=O will only be
evaluated where the shape parameter alpha (a) is greater than or equal to the value
of 1.
• Type I (a') errors are evaluated for points outside the upper and lower control limits.
Runs rules are not used in the evaluation of these errors. The notation (a') is used to
distinguish type I error fom the gamma distribution shape parameter (a).
• The evaluation of average run lengths CARL) do not consider shifts in the process
standard deviation. Only shifts in the mean are considered.
DEFINITION OF TERMS
Average run length(ARL)  The average number of subgroups taken before an outofcontrol
condition is given on the control chart.
Central limit theorem  Let X 1,X2, ••• ,Xn be a random sample from a distribution with
mean "J.1" and standard deviation "cr" . Then, if "n" is sufficiently large, the
sample average has approximately a normal di stribution with mean "J.1" and
standard deviation "cr/">lo". The larger the value of "n" the better the approx imation.
6
Control chart  A graphical chart with control limits and plotted values of some statistical
measure for a series of samples or indi vidual values. Control charts are tools
u ed to detect the presence of uncontrolled variation in a proce s in order to in dicate
when predictions regarding the future can be made.
Control limits  Limits on a control chart based on the data or standards given which are
used as criteria for action or for judging the significance of variations between
samples or individual values.
Individual measurement X control chart  A control chart used to evaluate the process
level in terms of a single observation per sample. These charts are usually used
when rational subgrouping is not appropriate.
Moving range  The successive absolute differences between individual values.
Moving range n=2 mR control chart  A control chart for evaluating the variability
within a process in terms of the range of the latest two observations in which the
current observation has replaced the oldest of the previous two observations.
Pearson Type III family of distributions  A family of distributions that, according to
Burr (5, p. 67), may be used as a second approximation of the curve shape of the
distribution if much asymmetry is present. The Pearson Type III family of di stributions
with location parameter c=O are gamma distributions which go from bell
shaped curves with range (0, +00) to Jshaped curves with range (0, +co).
Process  The set of individuals, items, or data from which a statistical sample is taken,
usually in time order.
7
Random sample  A sample that contains independent observation selected from the
same population or universe.
Range  The distance between the largest and smallest values in a subgroup. The range
is used as a measure of dispersion.
Run length  The number of subgroups taken before an outofcontrol condition is given
on the control chart.
Type I error (ex')  The probability of demonstrating that a process IS outoFcontrol
when it is in control. It is the probability of getting a false alarm.
ABREVIATIONS AND NOTATIONS
Symbol Term Definition
ARL Average run length lIP or liP'
ex' Type I error
ex. Shape parameter for the gamma di stribution
P' Type II error
P Scale parameter for the gamma di stribution
c Location parameter for the Pearson Type
III distribution
CLmR Center line for moving range control mR
charts
8

CLx
d2
d3
D4
E2
k
k'
LCLx
mR
mR
mR chart
n
N
P
P'
Center line for individual control charts
Bias correction factor
Bias correction factor
Control chart constant
Control chart constant
Number of subgroups
Number of subgroups used to set control
chart limits.
Lower control limit for individual measurement
X control charts
Mean of theoretical probability distribution
Moving range
A verage moving range
Moving range n=2 control chart
Number of items in a subgroup
Number samples or subgroups
Probability of detection on an X chart
probability of detection on an mR chart
9
X
RIO' or mR /a
aRIa
1+3 d3 /d2
3/d2
X  3 mR /d2
IXi+IXd
I mR I(N I)
Probability(UCLx<X or
X<LCLx)
Probability(mR>UCLrnR)
R Range of a set of data XmaxXlllil1
R A verage range LR,IN
s Sample standard deviation for a set of I(x,  X)2 I(nI)
data
aR Standard deviation of the theoretical dis d30"
tribution of ranges
ax Process standard deviation
SOSC State of statistical control
t Multiple of a units the control chart limits
are from the center line.
UCLmR Upper control limit for moving range D4mR
control charts
UCLx Upper control limits for individuals C011 X + 3 mR/d2
trol charts
VRL Variance of run length expressed as multiples
of standard deviations.
X An individual measurement
X A verage of a set of data Ix, In
X chart Individual measurement control chart
X(u, P) Gamma distribution with parameters 0:
and p
10

THE ASSUMPTIONS
The following assumptions pertain to this research:
• The use of individual measurement X and moving range n=2 mR control chart will
continue to have widespread use in industry in the future.
• The individual measurements are not correlated.
• The acceptable minimum average run length (ARL) in industry for the combination
of control charts (X and mR) is 100 when the process is in a state of statistical control
(SOSC). An ARL of 100 is equivalent to a 1 % risk of having a type I (a') error.
THE IMPORTANCE OF THE STUDY
The purpose of this research is to create a method of determining control limits
for nonnormal distributions which will support the widespread use of indi vidu al measurement
X and moving range n=2 mR control charts in industry.
11
CHAPTER 2
REVIEW OF RELATED LITERATURE
HISTORY
Throughout history, quality has been built into products. The early colonists and
immigrants in the United States fan owed the concepts of craftsmanship that were practiced
in their countries of origin. At an early age, a boy would become an apprentice and
learn a skilled trade from a master. One of the lessons learned from the master was to
control the quality of the product through inspection before sale. The quality of the
products was essential because the craftsman had a large stake in meeting customer
needs. Product quality was a reflection of the craftsman's skill.
The industrial revolution, which began in Europe, brought changes to controlling
the quality of products. The factory system of manufacturing products was becoming
increasingly popular. The trades that the craftsman practiced were divided into many
specialized tasks that could be performed by semiskilled or unskilled workers. The
skilled craftsman were no longer needed and the ability of a person to selfinspect a
product's quality throughout its entire ma.nufacture was lost. To maintain quali ty under
the factory system, fun time inspectors would report to departmental production sllpervisors.
Product was either "good" or "bad" based on specification limits.
In the 1920' s, Dr. Walter A. Shewhart introduced the concept of stati st ical quality
control to American industry. According to Dr. Shewhart, statistical tools could be
applied in a manufacturing setting to control the quality of manufactured product. One
of the tools of statistical quality control was the Shewhart control chart. The purpose of
12
Shewhart's control charts was to determine if a sequence of data may be u ed for predictions
of what will occur in the future and to warn of instability. These control charts
develop a picture of the process which aids in the evaluation of the process's performance.
The history of quality can be found in part or in full in numerous texts such as
Burr 1953 (5), Duncan 1986 (6), Joiner 1994 (11), and Juran 1995 ( I I).
SHEWHART CONTROL CHARTS
The basis of the Shewhart control charts is variation. There are two types of
variation that can affect a process; chance cause (common cause) variation and assignable
cause (special cause) variation. Chance cause variation, also referred to as controlled
or common cause, is present in the process all the time. It is characterized by a
stable and consistent pattern of variation over time. Assignable cause variation, also referred
to as uncontrolled or special cause, is not always present in the process . This
variation changes over time and comes from outside the process . References for process
variation and the basis of Dr. Shewhart's control charts can be fOllnd in many texts il1
eluding Burr 1953 (5), Duncan 1986 (6), Wheeler 1992 (4), Deming 1993 (9), and
Joiner 1994 (11 ).
Dr. Shewhart made the following conclusion based on process variations:
"Limits can be set, based on the natural variations of a process, so that as long as there
are fluctuations between these limits only controlled variation is present, and tluctu ations
outside these limits indicate uncontrolled (special cause) variation. If the process is
influenced by only common cause variation then it is in a state of statistical control
13

(SOSC) and can be used as a predictor of future occurrences, if influenced by special
cause variations then it is not in a state of statistical control. Dr. Shewhart stated the
following as concerned with statistical control: "A phenomenon will be said to be controlled
when, through the use of past experience, we can predict, at least within limi ts ,
how the phenomenon may be expected to behave in the future (4, p. 6)."
X AND R CONTROL CHARTS
There are many different types of control charts used in industry. The most
commonly used control charts are the X ancl R control charts. According to Juran.
"Where the characteristic under study can be measured along a scale of meas urement,
the X and R charts have proved to be of great value and should be ll sed in place of p
and c charts (7, p.389)." There are two requirements for using X and R charts. First, the
quality characteristic must be measurable, and second, these control charls require that
data be collected in subgroups. The subgroups should be collected in a rational manner.
In other words, the subgroups should be such that if special causes are present they will
show up in the differences between subgroups instead of within the subgroups.
The X control charts are used to monitor variation between subgroups. This is
accomplished by monitoring the differences between subgroup averages. According to
ANSVASQC Standard A 1 1978, "Averages are generally used for the purpose of determining
whether there are differences between subgroup levels ( 12, p.3)."
The X chart has a center line and control limits. The center line of the X control
chart is set at:
14
i o

CL x = x
where X is the average Xof all the data (or the average of the subgroup averages).
The control1imits DCLx and LCLx are set +/ 3 cr x units away from X.
UCL x = X +36 x
LCL x = X 36 x
where 6 x is an estimate of cr x derived from the data.
The estimate, 6 x ,depends on the subgroup size, n, and is calculated as follows:
6 x = dx /Vn
where d x is an estimate of the process standard deviation crx derived from the data.
The range (R) control charts monitor the variation within subgroups. This is accomplished
by monitoring the range of data points that are collected for each subgroup.
According to ANSI/ASQC Standard AII978. "Ranges of the individual observation
within the subgroup or sample are used to estimate the vari ability from chance cau se
within short time intervals and ordinarily should not include assignable causes. These
ranges serve to estimate the inherent variability within all essenti ally unchanging process
(12, p.3)." Although standard deviation is a more common measure of variability in
most applications, ranges are used because they are easier to compute. The range should
not be used, however, for subgroup sizes greater than 10 (n> 1 0).
The range control chart has a center line and control limits based solely on subgroup
ranges. The center line of the R control chart is set at:
15

CL R = R
where R is the average of all the subgroup ranges.
The control limits UCLRand LCLRare set +/ 3 aR units away from R .
UCLlF R + 3dI{
LCLIF R  36R
where dRis an estimate of the range standard deviation aRderived from the data.
The X and R control charts are considered very robust tools although their statistical
foundation is based on the normal distribution. The robustness of the X control
chart is best explained by the central limit theorem. The central limit theorem states:
"Let X"X2, ... ,Xn be a random sample from a distribution with mean "11" ancl standard
deviation "a". Then, if "n" is sufficiently large, the sample average has approximately a
normal distribution with mean "Jl" and standard deviation "a/Vn". The larger the value
of "n" the better the approximation." From the above definition of the central limit theorem,
the X chart can be used without having concern about lhe underlying distribution
of the process as long as the subgroup size is sufficiently large. According to Dr. Shewhart,
"Such evidence ... Ieads us to believe that in almost all cases in practice we may
establish sampling limits for averages of samples of four or more on the basis of normal
law theory (13),"
The R chart can also be used to monitor vari ations in process spread when the
underlying distribution is nonnormal. The robustness of the R chart cannot, howeve r,
be explained by the central limit theorem. In fact, as sample sizes increase, the di stribution
of the subgroup ranges may become more dissimilar to the parent distribution. In a
16

study perfonned by Wheeler and Chambers (4, 1992), the subgroup ranges of five nOI1
normal distributions were evaluated for type r errors using the common limit of 3aR
units from the center line. The evaluation was performed for sample sizes of n= 2, 4,
and 10. The resulting probabilities of a type r error for a highly skewed distribution were
0.026, 0.026, and 0.04, respectively. Although the probabilities fall short of the 0.0 I,
0.005, and 0.005 probability of a type I error for the normal di stribution, " ... both the
A verage Chart and the Range Chart can be said to be robust to those departures from
nonnality which are likely to be encountered in practice. They can be used witb confidence.
They will work and they will work well, even when 'the measurements are not
normally distributed' (4, p.76)."
INDIVIDUAL MEASUREMENT X AND MOVING RANGE n=2 mR CONTROL
CHARTS
The X and R control charts are not suitable for all industrial situations. Sometimes
there are special circumstances in a process that make subgroups impractical.
Natural subgroups may not be feasible if th ere are long periods of time between measurements,
a single measurement represents one batch, measurements are too time consuming
to obtain, or measurements are too expensive to obtain. In cases such as these,
where n=l, X and R control charts are not applicable. Individual measurement X and
moving range n=2 mR control charts are commonly applied when only a single measurement
is taken at a time. According to Wadsworth, et aI., "Their use is generally reserved
for process and product characteristics for which it is impractical or unreasonable
17

to rep licate observations and to form subgroups of observations to aid the study of process
variation (14, p.143)."
The individual measurement X control chart monitors the process level. This
control chart has a center line and control limits based on the indi vidual value X. The
center line on this control chart is set at
CLx= X
where X is the average of all the individual metl,>ures.
The control limits UCLx and LCLx are set at +/ tax from X (22, p. 2757)
UCLx = X + tdx
LCLx = X  td"x
where d x is an estimate of the process standard deviation ax derived from the individual
measurements X.
The common form of the individual measurement X control chart has an unde rlying
process distribution that is normal. In common form, the multiple of standard deviations,
t, the limits are from the mean is equal to 3 (4, p. 60). The control limits UCLx
and LCLx become
UCLx = X + 3dx
LCLx = X  3dx
The moving range n=2 mR control chart monitors the variation within the process.
This control chart has a center line and control I imits based on the range between
the two latest individual measurements X. The center line on this control chart is set at
18

CLR = mR
where mR is the average of kl moving ranges formed from consecutive n=2 observations.
The upper control limit UCLmR is set at t<JmR above the center line mR (22. p. 2757)
UCLmR = mR + td"rnR
The common form of the moving range n=2 mR control chart has an unde rlying
process distribution that is normal. In the common form, the multiple of standard devi ations,
t, the limit is away from the mean is equal to 3 (4, p. 60). The control limit
UCLrnR becomes
UCLmR = mR + 3dJIll{
The individual measurement X and moving range n=2 mR control charts do not
possess the robustness of the X and R control charts. The underlying assumption of
normality is much more critical when there are no subgroups (5, p. 2667). Since the
central limit theorem does not apply to individual measurements, because n= I. the
quality characteristic measurements must be approximately normally distributed to easily
and accurately generate existing individual measurement X and moving range n=2
mR control charts. When the process distribution is not approximately normally di stributed,
the value of t=3 may not produce control limits that are acceptable for use in
industry.
19

APPLICATION OF X & mR CONTROL CHARTS
In practice, all events cannot be expl ained by the normal di stribu tion. Many of
the distributions encountered in every day experiences are nonnormal. Economical,
physical, chemical, and biological factors typically have distributions that are skewed.
According to Irving Burr" ... cause of nonnormality is that the distribution may be unable
to go beyond a certain point, such as zero .... measurement has a physical limitation
at zero (5, p.80)." When the underlying distribution is asymmetrical, the Pearson type
III family of distributions can be used to approximate the data (5, p.67).
Despite the limitations, individual measurement X and movi ng range n=2 mR
control charts based on the normal distribution are used in applications with nonnormal
distributions. As stated by Schilling and Nelson, "In many application s the ch art is applied
without knowledge of the shape of the underlying distribution of ind ividu als (3,
p.183)." Conversely, Duncan states, "Control charts for individuals mllst be very carefully
interpreted if the process shows evidence of marked departure from normality_ In
such cases, the multiples of (j' used to set control limits might be better derived from
other distributions for which the percentage points have been computed (6, pAOO). " By
Duncan's statement above, the value of "t" used in setting control limits on individual
measurement X and moving range n=2 mR control charts for skewed distributions
should be ba<;ed on a distribution more accurately representing the process. As stated in
the previous paragraph, the Pearson type III family of distributions with location parameter
c=O can be used to approximate asymmetrical distributions.
20

Individual measurement X and moving range n=2 mR control charts are commonly
used in industry. Unfortunately, they may produce inaccurate representations of
the process if the underlying process distribution is nonnormal. To use indivi iual
measurement X and moving range n=2 mR control charts appropriately there must be a
method for setting control limits that more accurately predict th e stability of the process .
Research concerned with these control limits has been limited. although the need is justified.
CURRENT RESEARCH
The only research found that addresses nonnormal individual measurement X
and moving range n=2 mR control chart limits was performed by Jose Oyon, 1995.
Oyon (8), in an unpublished master of science thesis, studied the effect of nonnormality
on individual measurement X and moving range n=2 mR control charts. In this thes is,
Oyon did the following:
I. Evaluated the performance of the individual measurement X and moving range n=2
mR control charts using t.he constants d2, d3, and D4 under the assu mption of' normality
when the underlying distribution was Pearson type III family of di stri butions
with location parameter c=O.
2. Determined empirical functions for the control chart constants d2, d:;, and D4 when
the process distribution was approximated by the above distribution.
21

3. Compared the performance of the individual measurement X and moving range n=2
mR control charts with control limits based on the normal di stribution to those
based on the Pearson type III family of distributions with location paramete r c=O.
Oyon made the following conclusion from his research:
1. The individual measurement X and moving range n=2 mR control charts based on
the normal distribution do not work well when the underlying process distribution
shows a marked departure from normality.
2. Control chart constants based on the gamma distribution perform better than those
based on the normal distribution when the process distribution is nOIlnoTmal and
perform approximately the same when the process distribution is normal.
Although the gamma control chart constants perform better than the normal
control chart constants, the false alarm rate produced from the gamma control ChaJ1 constants
does not meet industry standard of I % when the process is in SOSc. One possible
reason for the high false alarm rates is that the gamma control cha rt constants are used
to produce symmetrical control limits for process distributions that are asymmetrical
(skewed). It may be possible to improve the performance of indiv idual measurement X
and moving range n=2 mR control charts for skewed distributions if asymme trical COI1
trollimits are developed.
No other work was found that addresses the effects of nonnormality on individual
measurement X and moving range n=2 mR control charts.
22

CHAPTER 3
THE RESEARCH METHOD
Section 1: INTRODUCTION
The following sections of Chapter Three explain the methodology for
performing this research. The sections of this chapter are outlined below:
1. Introduction
2. General Data
2.1 The Data
2.2 Criteria for Admissibility
2.3 The Research Methodology
3. Specific Treatment of the Data for Each Subobjective
3.1 Subobjective One
3.1.1 Individual Measurement X Control Chart Limits
3.1.1 .1 The Upper Control Limit
3.1.1.2 The Lower Control Limit
3.1.2 Moving Range n=2 mR Control Chart Limits
3.2 Subobjective Two
3.3 Subobjective Three
Section one of Chapter Three is intended to clarify the methodology of this
research. Section two is intended to characterize the data that is used to devclop the
asymmetrical control limits. The Data describes the primary source of the data used to
develop the control limits. The Criteria for Admissibility defines the established limits
23

and standards that the data must meet to be admitted into this research. The Research
Methodology classifies the methodology of this research.
Section three of this chapter explains (he specific steps for each subobjective of
this research. The flowchart on the following page (Figure 31) is included as a guide
for the research methodology. Section three is broken into three main subsections; subobjective
one, subobjective two, and subobjective three. The statement of the subobjectives
is found in their respective subsections. The following is an overvi ew of the
main subsections:
Subsection 3.1 overview
Subsection 3.1 develops mathematical models representative of the upper and
lower control limits for asymmetrical distributions based on the shape parameter (a) and
the scale parameter (~) from the Pearson Type III family of di stri butions with locati on
parameter c=O (gamma distribution s). These mathematical mode ls are for the multi ple
of standard deviations the control limits are from the mean (t values). The mathematical
models are generated in two different sections. One section is for the generation of the
mathematical models for the individual measurement X control chart (tl and t2) and the
other for the moving range n=2 mR control chart (t3 ).
Section 3 .t. I develops the mathematical models for the upper and lower control
limits of the individual measurement X chart. To develop th ese mathematical models an
upper control limit is foulld which leaves 0.00135 of the area under the Pearson type III
(c=O) distribution beyond the upper control limit and a lower control I imit is found
which
24

Determine lt1e UCL for the xchart
based on the Gamma
• dI stni bul'lo n I or d'If f erent va Iu es
1'01 a and 13· (Section 3.1 .1.1
step 1)
~
Create a mathematical model for
the UCL < t1> based on the
predictors a and 13 (Section
3.1.1 .1 steps 2 & 3)
~
Determine the LCL lor the X·
chart based on the Gamma
distribution for different values of
ex and 13. (Section 3.1.1 .2 step 1)
~
Create a mathematical model for
the LCL < 12 > based on the
predictors a and 13. (Section
3.1.1.2 steps 2 & 3)
~
Locate the UCllor the mR ehart
based on tile Gamma distribution for
diNerent values 01 a , Parameter ~ is
shown to have no effect, (Section
3.1,2 steDs 16)
~
Create a mathematical model lor
the UCL <b> based on the
predictors (l and ~, (Section 3,1,2
steps 7 & 8)
1
Produce 50 random variates
from a distribution. (Section
3.2 step 6)
~
Calculate the Normal
conlrollimits based on
the 50 random variates.
(Section 3.2 step 7)
~
Fit the 50 random variates to
the gamma distribution's a
parameter. (Section 3.2 step
7)
Calculate the d'2, d'l, and D'4
values from Oyan (1995). ~
(Section 3.2 step 8)
!
Calculate control limits based on
Oyan's "d" values. (Section 3.2 step
8)
l
Calculate the tl, \2, and values
from the mathematical models found
in previous steps of this research.
(Section 3.2 step 10)
~
Calculate the asymmetrical control
limits based upon the "r values,
(Section 3,2 step 11)
~
Adjust the previous contro llimils
(Normal, Oyon, and Asymmelrical) to
simulate a shift in the mean 01 0.5,
1.0, 1.5, 2.0, 2.5, and 3.0 standard
deviations. (Seelion 3.3 steps 1 & 2)
~
For each control chart, produce
random variates until a value lalls
Repeat
outside of the control limits, (Section
1000
3,2 step 12 & Section 3.3 step 3)
times ~
Calculate the ARL & VRl lor the
control charts based on the 1000 runs
and build Rl histograms.(Section 3.2
slep 13 & Section 33 step 4)
1
Place ARLs in Tables for analysis.
(Section 3.2 step 14 & Section 3.3
step 5)
!
( END )
Figure 31: Research Methodology Flow Chart
25
leaves 0.00135 of the area below the lower control limit. In this section the area is found
by integration. Control limits are located for diffe rent combinations of the shape
parameter (a) and the scale parameter (~) of the Pearson type III (c=O) di stribut ion. The
control limits are expressed as multiples of the standard deviation from the mean. The t l
value represents the multiple of standard deviations for the upper control limit and the t2
value represents the multiple of standard deviations for the lower control limit.
The next step in developing the mathematical models for tl and t2 is to use the
"t" values (from the different a's and ~'s) to develop the actual mathematical
expressions. Multiple regression models are developed which predict the "t" values
using the (a) and (~) parameters as the predictors. There are different mathematical
models which can represent the behavior of the "t" values, so, by trial and error, mode ls
are found which do a good job of predicting tl and t2 but may not be the only mode ls
that can be used. A global F test is used to test the va,lidity of the multiple regression
models selected.
Section 3. 1.2 develops the mathematical model for the upper control Ii mil of" the
moving range n=2 mR control chart. In this section of the research, two streams of
random numbers are generated from the Pearson type III (c=O) distribution and the range
for the corresponding values of those streams are found. The ranges for subgroups n=2
are used instead of moving range values for two reasons:
1. There is correlation between the moving range values.
2. Current methods for setting control I imits on the moving range charts are
based on the range of n=2.
26

The next step is to find an upper control limit which leaves 0.0027 of the ranges
beyond the limit. The area of 0.0027 is used because it is consistent with the
probabilities of the individual measurement X chart when the subgroup size is Ie than
seven (since only an upper control limit exists on the range chart). Control limits are
found for different shape parameters (a) of the Pearson type ill (c=O~ di stribution.
Previous analysis of the individual measurement X control charts indicate that ~ does
not have an effect on the control limit of the moving range chart. Appendix A
demonstrates that ~ has little or no effect on the control limits ; therefore, ~ is 110t
included in the development of the moving range n=2 mR control limit. The control
limit is stated as a multiple (t3) of the standard deviation of the i ndi vi dual ranges . A
mathematical model for t3 is found in the same manner as for the individual
measurement X control1imits.
Subsection 3.2 overview
Subsection 3.2 evaluates the performance of the individual rneasurement X and
moving range n=2 mR control charts based on average run lengths (ARL). Des pite
limitations, individual measurement X and moving range n=2 mR control charts based
on the normal distribution are used in applications with underlying process di stribu tions
that are nonnormal. As stated by Schilling and Nelson, "In many applications the chart
is applied without knowledge of the shape of the underlying distribution of individuals
(3, p.l83)." The idea of this subsection is to evaluate the performance of individual
measurement X and moving range n=2 mR control charts having asymmetrical co ntrol
limits based on the Pearson Type III family of distributions with location parameter c=O
27

(gamma distribution) to those having symmetrical control limits based on the ame
distribution. The asymmetrical control limits are also compared to those based on the
normal distribution since control chart limi ts based on normality are commonly used in
industry.
The evaluation is performed by generating random variates from a parent
distribution which is assumed to be unknown. Control limits are calculated for Normal
Shewhart limits, symmetrical control limits based on the Pearson type III (c=O)
distribution (Oyon 1995), and the asymmetrical control limits based on the Pearson type
ill (c=O) distribution. In order to calculate the latter two sets of control limits, the
randomly generated variates are fit to the Pearson type ill (c=O) distribution. The
method of fit used in this research generates (ex) and (~) values which are used in the
mathematical models for calculating the "d" values (needed for Oyon's limits) and the
"t' values from this research.
Next, random variates are generated from the same parent di stribution until an
outofcontrol signal is detected on each of the three individual meas urement X and
three moving range control charts. A run length (RL) is recorded for each control chart
(both X and mR) and the above steps (setting control limits and determining RL' s) are
repeated 1000 times. An average run length CARL) and variance of run length (YRL) is
found for each control chart (6 total) and recorded for analysis ill Chapter Four. The run
lengths are stored and presented as a histogram. The chart in Appendix J demonstrates
the logic used in the evaluation of subobjectives 3.2 and 3.3.
28

Subsection 3.3 overview
Subsection 3.3 compares the power of the individual measurement X and
moving range n=2 mR control charts using the Pearson Type III c=O asymmetrical
control limits with those methods having symmetrical control limits (Normal and
Oyon). The power, in this case, refers to the ability of the control charts to detect shifts
in process location of 0.5, 1.0, 1.5. 2.0, 2.5, and 3.0crx units. A type II error (~') is the
probability of concluding a process is incontrol when it is actually outofcontrol. The
power of a control chart is a function of a type II error. The power is equal to 1  ~ and
is the probability of detecting an outofcontrol condition. Generally, type I and type 11
errors are negatively correlated. As the type I errors are reduced, the type n errors
increase which in turn decreases the power of the control charts. Di scussions of these
types of errors can be found in many texts including Hayes (17), Savage (18), Hair ( 19),
and Miller (20). Previous subobjectives of this research attempt to find control limits
which have smaller type I errors than existing methods, therefore, it is important to
evaluate the effect of asymmetrical limits on the power of the individual measurement X
control charts.
Section 2: GENERAL DATA
2.1: The Data
The primary source of data used to develop asymmetrical control limits consist
of values generated from the Pearson Type ill fam ily of distributions with location
29


parameter c=O (gamma distributions). Random variates are generated from the normal
distribution, the lognormal distribution, and the gamma distribution to evaluate the
performance of the asymmetrical control limits developed from the Pearson Type liT
family of distributions.
2.2: The Criteria for the Admissibility of the Data
The criteria for the admissibility of the data used for this research is as follows:
• Only values generated from the Pearson Type III family of distributions with
location parameter c=O (gamma distributions) are utilized in the development of the
mathematical models for t I, t:2, and t~.
• Only a (shape parameter) values greater than or equal to the value of 1.0 are applied
to the Pearson Type III family of distributions with location parameter c=O (gamma
distributions).
• Only ~ (scale parameter) values equal to I, 2, and 5 are applied to the Pearson Type
III family of distributions with location parameter c=O (gamma distributions).
2.3: The Research Methodology
The method of research used in this study is based on numeri cal data. Since the
data are numeric, quantitative methodology is utilized to conduct this research.
Section 3: SPECIFIC TREATMENT OF THE DATA FOR EACH SUBOBJECTIVE
3.1: Subobjective one:
Statement of the SubO~jective: Develop mathematical models representative of
the upper and lower control limits for asymmetrical distributions based on the shape
30
parameter (a) and the scale parameter (~) from the Pearson Type ill family of
distributions with location parameter c=O (gamma distributions) so often encountered in
industry.
The Data Needed: The data needed for this subobjective cons ist of values
generated from the Pearson Type ill family of distributions with location parameter c=O
(gamma distributions). The values generated from this distribution include individual
measurements X, as taken from integrating the distribution, and range values, as
produced from randomly generated observations.
The Location of the Data: The Pearson Type ill family of di stribu tions with
location parameter c=O (gamma distributions) have the following probability density
function (pdf):
All data needed in generating mathematical models representative of the upper and
lower control limits for individual measurement X and moving range n=2 mR control
charts are produced from this function.
The Means of Obtaining the Data: The control limits required for sllbobj ective
one are obtained by integrating the above function (the specific treatment of the functi on
is explained in the steps below). Mathcad for windows release 4.02 is utilized to
perform the necessary integration of the Pearson Type m family of distributions with
location parameter c=O (gamma distributions) and Minitab for Windows release 10.5 is
used to generate random variates.
3]

Treatment of the Data: The treatment of the data is expl ained separately for the
individual measurement X control limits and the moving range n=2 mR control limits .
The explanations are as follows on subsections 3.1.1 and 3.1.2.
3.1.1: Individual measurement X control chart limits:
Individual measurement X control charts based on the normal distribution have
upper and lower control limits set at +/ 3Gx units above and below the average of a set
of data. When these control limits are applied to a process having a normal distribution,
there is a probability of approximately 0.00135 that a point will fall beyond the upper
control limit and a probability of 0.00 135 that a point will fall below the lower control
limit. To stay consistent with normal probabi Iity theory of statistical process control, the
Pearson Type III family of distributions with location parameter c=O (gamma
distribu tions) is evaluated against the same probabilities of a point fa lling outs ide
control limits. This evaluation is described in the following paragraphs.
3.1.1.1: The upper colltrollimif
The following steps describe the methodology for generating a mathematical
model for the asymmetrical upper control limit on the individual measureme nt X control
chart.
1) The value of the upper control limit for the Pearson Type III family of distributions
with location parameter c=O (gamma distribution) is located by integrating the
distribution on Mathcad. An upper control limit (UCL) is generated which leaves a tail
area of 0.00 135 beyond the limit. The UCL for the Pearson Type III family of
distributions with location parameter c=O (gamma distribution) is denoted by (UCL) in
32
the equations below. The limit is evaluated in this manner for all combinations of 0:
(shape parameter) = 1, 5(5) 135 and P (scale parameter) = 1, 2, and 5. As demonstrated
in Appendix I, the a values represent a range of skewed distribution s from exponential
to approximately normal (since the gamma distribution cannot generate an exact normal
distribution). The UCL's are expressed as a multiple of ax units (tl ) to the right of the
mean of the distribution. The following equations are used to generate the limits:
"OU r ", ,[il .00135 = [ r(a) * x *e dx  I ~ (g iven a and ~. find UCLJ
O"x = Ja * f32
X(mean) = a* f3
UCLa*!3
11 = multiple of ax units from the mean = r:.:.::;:(i2
va * f3
(eq.31)
(eq. 32)
(eq. 33)
(eq.34)
The multiple of ax units from the mean (lJ) generated in this step are paired with the ir
associated a and P values and recorded as demonstrated in the table (Table 31) on the
following page.
2) A stati stical software package (Minitab for Windows release 1 O.S)' which fea tures
regression software, is used to generate different multiple regression models for
predicting the tl value with the predictors a and ~. There are differe nt mathematical
models which can predict the 11 values, so, by trial and error, a model is found which
does a good job of predicting tl but may not be the only model that can be used. A
mode] is chosen that has a high adjusted multiple coefficient of determi nation R 2,
33
Table 31: ax Units From the Mean (tl)
ax units from the mean (t l )
ex ~ = 1 ~=2 ~=5
1 5.6080 ).6080 ).6080
5 4.2005 4.2005 4.2005
135 3.2305 3.2305 3.2305 I
34
The adjusted multiple coefficient of detennination (R 2) is a sample statistic that
demonstrates how well the mathematical model fits the data; therefore, it represents a
measure of adequacy of the model. The R2 is defined as:
3) A global F test is used to test the validity of the multiple regression model selected.
The null hypothesis of this F test is:
where An is the distance from the integrated 11 value to the cOITesponding tl value
calculated from the multiple regression equation. The n = I, 2, 3, ... , k represent the A
for the respective a.. = 1,5(5) 135.
The null hypothesis is tested against the alternative hypothesis
Ha : at least one of the A parameters does not equal zero
The test statistic is defined by
F = (R2!k)! { (l_R2 )![n(k+ I)]}
and the rejection region by
F > Fa'. (k, n(k+l»
where
k is the number of A parameters in the multiple regression model excluding the
constant term ~.
n is the number of integrated tl values used to generate the multiple regression
model.
35

A/S are the distances from the integrated t[ value to the corresponding t[ values
calculated from the multiple regression equation.
a' is the significance level.
3.1.1.2: The lower control limit
The following steps describe the methodology for generatin g a mathematical
model for the asymmetrical lower control limit on the individual measurement X control
chart.
1) The value of the lower control limit for the Pearson Type ill family of distributions
with location parameter c=O (gamma distribution) is located by integrating the
distribution on Mathcad. A lower control limit (LCL) is generated which leaves <l lail
area of 0.00] 35 below the limit. The LCL of the Pearson Type 1II family of
distributions with location parameter c=O (gamma distribution) is denoted by (LCL) in
the equations below. The limit is evaluated in this manner for all combinalions of a
(shape parameter) = 1,5(5)135 and ~ (scale parameter) = 1,2, and 5. The LCL's are
expressed as a multiple of ax units (t 2) to the let'l of the mean of the distri hution. The
following equations are used to generate the limits:
(eq. 35)
(eq. 32)
mean = a* f3 (eq. 33)
36
a*13  LCL
t2 = Ox units away from the mean = ~2
"a* [3
(eq . 36)
The t2 values are placed in a table similar to Table 31 in step I of section 3. 1. 1.
Steps 2) and 3) are the same for the lower cont.rol limit as staled earlier for the
upper control limit.
3.1.2: Moving range n = 2 mR upper control chart limits:
The moving range 11=2 mR control charts are commonly used in industry.
Unfortunately, they may produce inaccurate representations of the process if the
underlying process distribution is nonnormal. As seen from previolls research by Oyon
(1995), moving range n = 2 mR control chalts fall well short of achieving ARLs of 100
(the assumed ARL for industry acceptance in the research) for moving ranges of skewed
distributions. The result of the poor performance of these charts is the appearance of
many false outofcontrol signals. To llse the moving range n=2 mR control charls
appropriately, there must be a method for setting control limits that more accurately
predicts the stability of the process. This portion of the research sets control limits based
on the location (t3) of the upper control limits as a multiple of the standard deviation of
the ranges. The (t3) values for the moving range n=2 mR control charts for skewed
distributions are evaluated as follows :
I) The value of the upper control limit for the moving range n = 2 mR control charl~
based on the Pearson Type III family of di stributions with location parameter c=()
(gamma distribution) is located by simulating values of the distribution from
Minitab. Two columns of k = 60,000 randomly generated observations are produced
for aU values of a (shape parameter) = 1, 5, 10. 15, 20, ... , 135. The selection of the
37
i •• •• »
~
1
of • J • I .).
2
)
04
)
number of subgroups, k = 60,000, is found in Appendix B. The scale parameter is
not evaluated in generating this mathematical model for the ranges because it does
not affect the value (t3) as demonstrated in Appendix A.
The upper control limit for the mR chart is based on ranges of subgroup size two as
is common with Shewhart 's mR control charts. Ranges of subgroup size 11=2 can be
used instead of mR values. This is demonstrated in Appendix H.
2) The observations are grouped in the following manner:
Where Xi represents the first column of k = 60,000 observations and Yj represents the
second column of k = 60,000 observations.
3) The range for each pair of data is found using the following equation:
4) The average range, R, is found wi th the equation:
LR,
R = '
k
where k = 60,000. The value of R is found for each value of a (shape parameter) = I,
5(5) 135.
5) The standard deviation of the ranges, O'R, is found with the following equation:
(j R 
where k = 60,000. The value of (j R is found for each value of ex. (shape parameter) = I,
5(5)135.
38
• I • •.. •• :• •• J • I ) • 2
J )

6) An upper control limit is generated for each a which leaves a tail area of 0.0027 (see
introduction) outside the limit. In order to accomplish this step, the following equation
is used:
(eq.37)
To locate the upper control limit, the values from steps 4 and 5 are applied to
this equation and an appropriate (t3) value is found. This is accomplished by increas ing
the value of (t3) by 0.0001 until 0.27% of the ranges are outside the control limits.
The limit is evaluated in this manner for all values of a (shape parameter) = I, 5.
10, 15,20, ... , 135. The results are expressed as a multiple of (jR units (t:l) to the right
of the mean range of the distribution. The multiple of (jR unils from the mean (( 3)
generated in this step are paired with their associated a values as demonstrated in the
table (Table 32) on the following page.
Steps 7) and 8) are the same as steps 2) and 3) for the upper ancl lower control
limits of the individual measurement X control chart.
3.2: Subobjective two:
Statement of the SubObjective: Evaluate the performance of the individual
measurement X and moving range n=2 mR control charts, based on the average run
length CARL) using the Pearson type In family of distributions with location parameter
c=O (gamma distribution) control limits determined from suhobjective I. The control
charts are evaluated against an ARL that. is acceptable for practical application in
industry and compared with methods having symmetrical control limits. An ARL that is
acceptable for practical application in industry means that the average run length CARL)
39
• ) • J
•J )
Table 32: (j'x Units From the Mean (t3) for the UCL of the Range Chart
(j'x units from the mean (t3) for the
UCL of the range c hart
a ~=I
I 4.9826
5 4.11 26
10 3.9821
135 3.6919
40
•• i ·
•• I
I

for each control chart is a minimum of 100 observations. An ARL of 100 i equivalent
to a 1 % chance of a type I error when the process is tn a state of statisti cal control.
The Data Needed: The data for subobjective two consist of randomly generated
variates from the normal, lognormal. and gamma distributions. Indi vidual measurement
X and moving range n=2 mR control limits are also needed for the normal Shewhart,
Oyon's symmetrical Pearson type ill (c=O), and asymmetrical Pearson type ill (c=O)
control charts.
The Location of the Data: The location of the data for subobjective two is as
follows:
• A random variate generator is utilized to generate values from the norm al, lognormal,
and gamma distributions.
• Symmetrical individual measurement X and moving range n=2 mR control limit
equations based on the normal distribution produced by Dr. Shewharl are rOllnd ill
various quality control texts including Wheeler and Chambers (1992), Burr ( 1(53),
and Duncan (1986). These equations can be found in step 6 be low.
• Symmetrical individual measurement X and moving range 11=2 mR control limi t
equations are produced using the d ' ~ , d'3 , and D'4 values approximated by the
Pearson Type ill family of distributions with location parameter c=O (gamma
distribution) from previolls research by Jose Oyon (1995). These equations can be
found in step 8 below.
• Asymmetrical individual measurement X and moving range n=2 mR control limit
equations are produced using the mathematical models generated for the "t" values
41
•• I
• t
t
I
I • t
• •• I
•J )

approximated by the Pearson Type ill family of distributions with location
parameter c=O (gamma di stribution). The mathematical models are produced in subobjective
one above.
Means of Obtaining the Data: The lndividuaI measurement X and moving range
n=2 mR control limits are obtained through the calculation of the symmetrical and
asymmetrical control limit equations. Equations 311 through 313 are for the normal
control limits, equations 318 through 320 are for the Oyon control limit s, and
equations 321 through 323 are for the asymmetrical control limits.
Treatment of the Data: The following is a detailed procedure to achieve subobjective
two:
1) Fi ve process distributions are selected to reprcsen t unknown parent d istri bu tions.
The distributions are chosen to represent a variety of process distributions that occur in
industry. The five process distributions selected are as follows :
• Normal (40, 102)
• Lognormal (0, 12)
• Gamma(a= 1.5, ~ = 1)
• Chisquare (df = 4)
• Exponential (~ = 1)
2) One set of k' = 50 observations IS gellerated from one of the five distributions
selected in the previous step.
42
•,I
t
I
t
I ,•
t
~ • I
I • )

3) The average (X) is calculated for the 50 observation (the 50 observations for which
the control limits are calculated are referred to as k'). The average is obtained us ing the
following equation:
(eq. 38)
4) The moving range n=2 is calculated for the k' = 50 observations. The moving range
n=2 is calculated by grouping the observations into subgroups of two consecutive
measurements and then applying those subgroups to the following equation:
mR. =IX 1 Xl J 1+ I
(eq. 39)
5) The average moving range is calculated from the 49 moving ranges calculated in slep
4 for the k' = 50 observations using the following equation:
k I
ImRy
 . y=1
mR = '(,)k'
J
(eq.310)
6) Control limits based on the normal distribution are calculate l. The individual
measurement X and moving range n=2 mR control chart limits are calculated with the
following equations:
UCLx = X+2.66(mR) (eq.311)
LCL. = X  2.66(mR) (eq. 312)
(eq.313)
7) In order to calculate the control chart constants based on the Pearson Type III family
of distributions with location parameter c=O (gamma di stribution), the shape parameter
43
,I•
t
,I
I
f • ~
(a) has to first be estimated. The following was written by Jose Oyon (1995) in regards
to estimating the parameters a and ~ for the Pearson Type ill family of distributions
with location parameter c=O (gamma distribution):
"In order to get the Pearson type III with c=O (gamma) control
chart constants d2, d3, and D4 to be used in setting control iiinits f or
each run of (k') observations, Pearson type III parmneters ex and f3 have
to be estimated from the (k') observations generated.
"Since the process distribution is supposedly unkn own, the idea
is to fit the data with a Pearson type III with c=O distribution hv
estimating the parameters ex and f3 from the (k ') data values (gamma
distribution assumption as the underlying process distribution).
According to Fisher (I, p.332), the method of moments is inefficient to
estimate parameters of a gamma distribution, except f or (I disfrihllfion
closely resembling the normal distribution. Kendall and Stuart (2, p.3R)
show that the efficiency of the estimated shape parameter ex 0/ a !!,mnma
distribution In the method of moments mav be as low 22 percent.
Therefore, Fisher (I, p.332) and Law and Kelton (I 5, p.331) recornmend
the method of maximum likelihood estimation (MLE) in order to estimate
the parameters ex and f3 of type If! from the data.
"The difficulty in applying the method of maximum likelihood
estimation to estimate the parameters ex and f3 of the gamma distrihution
is that closed expressions for the maximum likelihood estimators a and
44
f3 cannot be obtained analytically. Therefore, numerical methods must
be used to estimate the parameters a and [3 of the gamma distribution.
"Choi and Wette (9, p.683) developed a numerical technique of
fhe maximum likelihood method to estimate the parameters of the
gamma distribution. This method is recommended by Law and Kelton
(15, p. 331) to estimate a and [3. Therefore, this method is the one to be
used in this (subobjective two) to estimate a and f3 from the data ill
order to fit a Pearson type III distribution with location parameter c=O
(gamma distribution). "
The maximum likelihood method stated above utilizes a T stati stic to estinlate
the parameters (J. and ~ (15 , p.331). The T statistic is obtained with the following
equation as given by Law and Kelton (15, p.410):
[
_ LlnXj ]J
T= InX  k (eq.314)
Using the T stati stic from the above equ ation, the estimator a can be obtai nec!
using Table 6.19 in Law and Kelton (15, p. 411). A reproduction of this table is
included in Appendix C of this research.
8) The control chart constants d'2, d' ,), and 0'4 based on the Pearson Type ill family of"
distributions with location parameter c=O (gamma distribution) are calculated for the k'
= 50 observations. The following mathematical models (as produced by research from
Jose Oyon (1995)) are used to generate the constants:
45

d'2 = 0.64282+0'(1J77~1e{).5a ) +0.3573~ l_e2(1) +o.024S~ l_eD.'(1) (eq. 315)
d') = 0.859457 +O.2%4{e{l) +O.2~e{)·5<I) +O.475~e2(1) (eq.316)
0 4 = 3.28976+ 1. 87(x)7( e a) + O.l366~ e o.l(1) (eq.317)
9) Symmetrical control limits based on the Pearson Type m family of distributions
with location parameter c=O (gamma distribution) are calculated. The individual
measurement X and moving range n=2 mR control chart limits are calculated using the
following equations:
(eq . 318)
(eq . 319)
(eq. 320)
10) The tl, t2, and t3 values are calculated for the Pearson Type III family of
distributions with location parameter c=O (gamma distribution) for the k' = 50
observations. The parameter (ex) designated from step 7 of subobjective two is used (0
estimate the tl, t2, and 13 values using the mathematical models generated from s ubobjective
one of this research.
46
11) Asymmetrical control limits based on the Pearson Type III family of distribution:;
with location parameter c=O (gamma distribution) are calcul ated. The indi vidual
measurement X and moving range n=2 mR control chart limits are calculated with the
following equations:
(eq. 321)
(eq. 322)
(eq. 323)
where D',=(l+t,(:::)
12) For the three sets of control limits (normal, Oyon, and asymmetrical), random
variates are generated until a value falls outside each set control limits. A run lengtil
(number of values generated before an ooe sign al) is recorded for each cont rol chart.
13) Steps 2) through 12) are repeated 1,000 times for each of the five parent
distributions stated in step one of this subobjecti ve. An average run length (ARL) ror
each of the five distributions is calculated using the following equation:
ARL= LCRL)
1,000
(eq.324)
14) A variance of the run length (VRL) for each of the five distribu tions is calcul ated
with the following equation:
47
(eq.325)
VRL= ;=1
999
15) The J 000 run lengths are stored and presented on a histogram for each of the five
parent distributions. The data from steps 13) and 14) of subobjective two are
grouped according to the parent distributions of the random vari ates and pl aced in a
table for easy reference. The table (Table 33) is illustrated on the following page.
3.3: Subobj1ective three:
Statement of the SubObjective: Compare the power of the individual
measurement X and moving range n=2 mR control charts using the Pearson Type HI
c=O asymmetrical control limits with those methods having symmetrical control limits.
The power, in this case, refers to the ability of the control charts to detect shifts in
process location of 0.5, 1.0, 1.5, 2.0,2.5, and 3.0crx units.
The Data Needed: The data for subobjective two consist of randomly generaLed
variates from the normal, lognormal, and gamma distributions. Individual measureme nt
X and moving range n=2 mR control limits are also needed for the normal Shewhart.
symmetrical Pearson type III (c=O), and asymmetrical Pearson type m (c=O) control
charts.
The Location of the Data: The location of the data for subobjecti ve two is as
follows :
• A random variate generator is utilized to generate values from the normal , lognormal,
and gamma distributions.
48

Table 33: Control Chart ARLs and VRLs
Individual Measurement X Moving Range
Parent
Distribution Shewhan Symmetrical Asymmetrical Shewhnn Symmetrical Asymmetrical
Normal ARL
(40, 102)
VRL
as 51. dey"
Lognormal ARL
(0. ]2)
VRL ,
as 51. dev . II
Gamma ARL
(1.5,1)
VRL
as 51. dev.
Chisquare ARL
(df = 4)
VRL
as SI. dey"
Exponential ARL
( 1 )
VRL
as st. dev.
49

• Symmetrical individual measurement X and moving range n=2 mR control limit
equations based on the normal di stribution produced by Dr. Shew hart are found in
various quality control texts including Wheeler and Chambers (1992), BUIT (1953),
and Duncan (1986). These equations can be found in step 6 of section 3.2.
• Symmetrical individual measurement X and moving range n=2 mR control limit
equations are produced using the d' 2, d' 3, and D' 4 values approximated by the
Pearson Type III family of distributions with location parameter c=O (gamma
distribution) from previous research by lose Oyon (1995). These equations can be
found in step 8 of section 3.2.
• Asymmetrical individual measurement X and moving range n=2 mR control limit
equations are produced using the mathematical models generated for the tJ, t2, and t1
values approximated by the Pearson Type ill family of distributions with location
parameter c=O (gamma distribution), The mathematical models are produced in subobjective
one above.
Means of' Obtaining the Data: All data used for this subobjective are obtained
from the data generated in subobjective two. The normal, Oyon, and asymmetrical
control limits calculated in subobjective two are adjusted to represent a mean shift in
subobjective three. The random variates generated in subobjective two are used to
evaluate the adjusted control limits.
50
Treatment of the Data: This subobjective evaluates the ability of the control
charts to detect shifts in the mean on the individual measurement X control charts. The
following is a detailed procedure to achieve subobj ective three:
I) The theoretical standard deviation is found for each of the five parent di stributions
listed in step one of subobjective two. The following equations are used to find the
standard deviations:
Exponential, Gamma, and Chisquare:
(eq. 326)
Lognormal:
(eq.327)
Normal: The standard deviation for the normal distribution is taken from the definition
of the distribution's parameters. The normal distribution used in this research is a N(40,
102) ; therefore, the mean is 40 and the standard deviation is 10.
2) Shifts in the process mean of +/ 0.5, 1.0. 1.5, 2.0, 2.5, and 3.0ax are simul ated by
adjusting the individual measurement X control chart limits generated in subobjective
two. The control limits are adjusted as follows:
UCL(adjusted) = UCL  Ll * (a x )
LCL(adjusted) = LCL Ll * (ax )
where Ll is the process mean shift as a multiple of ax.
51
(eq. 328)
(eq. 329)
3) For the set of ten adjusted control limits, random variates are generated until a value
falls outside each set of control limits. The number of values generated before the value
faU outside the limits (run length (RL») is recorded.
4) Steps 2) and 3) are repeated 1 ,000 times for each of the five parent distributions
stated in step one of subobjective two. An average run length CARL) for each of the
shifts in the five distributions is calculated using the following equation:
L,CRL)
ARL= I, 0 0 0
(eq. 330)
5) The data from step 4 of subobjective three are grouped according to the parent
distributions of the random variables and placed in a table for easy reference. The
table (Table 34) is illustrated on the following page.
52
Table 34: ARLsNRLs for Shifts in the Process Mean
A RLs for Shifts in the Process Mean
Parent +0.5 + 1.0 + I.S +2.0 +2.5 +3.0 0.5 1.0  1.5 2.0 2.5 3.0
Distribution
Nonnal Nonna]
(40,102)
Symme trical
Asymmetrical I
Lognormal Normal
,
(0. 12)
Symmetrical ,
Asymmetrical
Gamma Normal
(J .5, 1)
Symmetrical
I Asymmetrical
Chisquare Normal
(df = 4)
Symmetrical
Asymmetrical
Exponential Normal
(I)
Symmetrical ,
A. ymmetrical
53
CHAPTER 4
RESULTS AND ANALYSIS
The results of this thesis research are presented following the three subobjecti
ves described in chapter 1.
Section 1: SUBOBJECTIVE ONE
The first subobjective is to develop mathematical models representative of the
upper and lower control limits for asymmetrical distributions based on the shape
parameter (a) and the scale parameter (P) from the Pearson Type ill family or
distributions with location parameter c=O (gamma distributions) so often encountered in
industry.
1.1: Individual Measurement X Control Chart Limits:
Following the steps in Section 3.1.1.1 and 3.1.l.2, described in detail in Chapter
3 ( pg. 38), the values of the individual measurement X upper and lower control I il11il5
for the Pearson Type III family of distributions with location parameter c=O (gamma
distribution) are located by integrating the distribution on MathCad for Windows re lease
4.02. The value of the upper control limit is expressed as a multiple of O"x units from the
mean. An upper and lower control limit is generated which leaves a tail area of 0.00135
beyond each limit. The limit is evaluated in this manner for all combinations of a
(shape parameter) = 1, 5(5) 135 and P (scale parameter) = I, 2, and 5.
The table on the following page, Table 4.1: Gamma Distribution Upper & Lower
Control Limits, demonstrates the results of integrating the gamma distribution. As can
be seen from Table 4. J, the values for ~ appear to have I ittle or no effl'ct 011 the upper
54
 
Table 41: Gamma Distribution Upper & Lower Control Limits
Sigma units from Sigma units from
average on the average on the Total sigma
skew (upper) tail nonskew (lower) tail spread
for 0.00135 for 0.00135 for 0.0027
Alpha Sigma /3 = J /3 = 2 f3 = 5 f3 = J f3 = 2 f3 = 5
135 11.6190 .................. 3.2305 · 2.7718 . · 60023
130 11.4018 3.2350 . · 2.7675 2.7675 · 6.0025
• ••• • •••••••••••• r
125 11.1803 3.2395 . · 2.7630 2.7630 6.0025 ·.·.··r·.··· .. · ...
120 10.9545 .................. 3.2445 3.2445 · 2.7582 2.7582 · 6.0027
115 10.7238 .................. 3.2497 3.2497 · 2.7530 2.7530 2.7530 60027
110 10.4881 ................... 3.2555 3.2555 · 2.7475 2.7475 27475 6.0030
105 10.2470 ................ .. 3.2615 3.2615 3.2615 2.7416 2.7416 2.7416 60031
100 10.0000 ............. , .... 3.2680 3.2680 3.2680 2.7354 2.7354 2.7354 60034
95 9.7468 .................. 3.2750 3.2750 3.2750 2.7285 2.7285 2.7285 6 0035
90 9.4868 .................. 3.2825 3.2825 32825 2.7211 2.7211 2.7211 6 0036
85 9.2195 .................. 3.2908 3.2908 3.2908 2.7132 2.7132 2.7132 6.0040
80 8.9443 3.2997 3.2997 3.2997 2.7045 2.7045 2.7045 6.0042 , ..... ......... ...
75 8.6603 .............. .... 3.3095 3.3095 3.3095 2.6949 2.6949 2.6949 6.0044
70 8.3666 ........... ... .... 3.3205 3.3205 3.3205 2.6843 2.6843 2.6843 6.0048
65 80623 .. , ................ 3.3328 3.3328 3.3328 2.6725 2.6725 2.6725 6.0053
60 7.7460 ........ ........... 3.3464 3.3464 3.3464 , 2.6594 2.6594 2.6594 6.0058
55 7.4162 .................. 3.3617 3.3617 3.3617 ' 2.6444 2.6444 2.6444 6.0061
50 7.0711 ....., . . ...... ... 3.3795 3.3795 3.3795 2.6273 2.6273 2.6273 6.0068
45 6 .. 7082 .................. 3.4000 3.4000 3.4000 2.6075 2.6075 2.6075 6.0075
40 6.3.246 3.4245 3.4245 3.4245 2.5840 2.5840 2.5840 6.0085 , .................
35 5.9161 .................. 3.4540 3.4540 3.4540 2.5559 2.5559 2.5559 6.0099
30 5A772 ............. ...... 3.4905 3.4905 3.4905 2.5211 2.5211 2.5211 6.0116
25 5.0000 .................. 3.5375 3.5375 35375 2.4765 2.4765 2.4765 6.0140
20 4.4721 .. , ............... 3.6010 3.6010 3.6010 2.4166 2.4166 2A166 6.0176
15 3.8730 .................. 3.6940 3.6940 3.6940 2.3297 2.3297 2. 3297 6.0237
10 3.1623 3.8505 3.8505 3.8505 · 2.1870 2,1870 21870 6.0375 , .......... , .......
5 2.2361 ................... 4.2005 4.2005 4.2005 1.8820 1.8820 1.8820 6.0825
1 1.0000 ............. , .... 5.6080 5.6080 5.6080 0.9986 0.9986 0.9986 6. 6066
55
and lower control limits for the gamma distribution when expressed as a multiple of O'x
units. Following step 2 in 3.1.1.1 and 3.1 . 1.2. regression models are generated in
Minitab for Windows release 10.5. This statistical software package is used to generate
different multiple regression models for predicting the t) and t2 values with predictors ex
(shape parameter) and ~ (scale parameter). There are different mathemat ical models
which can predict the t) and t2 values. By trial and error, a model is found which does a
good job of predicting tJ and t2' The models found may not be the only models that can
be used.
The output from Minitab can be found in Appendix D: Regression Output For
Control Limits. The best t1 and t2 regression models, based on R2, for the upper and
lower control limits are as follows:
t1 = 3.23 + 3.19*e('CX) + 0.852*e(·oICX) + 0.442*e(·O.2S(x)
t2 = 2.77  1.81 *e(·a)  0.751 *e(·OJ a)  0.438* e(·O.025a)
A global F test is used to test the validity of the multiple regression models. The F tes ts
for the upper and lower control limits are shown in sections 1.1.1 and 1.1.2.
1.1.1: The upper control limit
The model for t) has a multiple coefficient of determin ation R2 of 99.9%. The
global F test is used to test the validity of the upper control (tJ) limit mUltiple regression
model as indicated in Chapter 3 section 3.1.1.1 step 3.
From the Minitab output (Appendix D  Regression Output For Control Limits
the value for the test statistic F is:
F = 6935.38
56
c
Using a significance level a' = 0.0 I, the rejection region for the test is defined by the
critical value Fc a'(k. n(k+ I». From an F tabl e, this critical value is:
Fc 0.01 (3 . 23) = 4.765
Clearly the null hypothesis Ho: Al = 1..2 = A3 = ... = Ak =0 is rejected since the
value of the F statistic is greater than the critical value Fc:
F > FCa'(k. n(k+I))
6935.38 > 4.765
Therefore, it is concluded that one can be very confident that this model is useful
in predicting tl.
1.1.2: The lower control limit
The model for t2 bas a multiple coefficient of determination R2 of 99.8%. The
global F test is used to test the validity of the lower control (t2) limit multiple regression
model as indicated in Chapter 3, section 3.1. 1.2, step 3.
From the Minitab output (Appendix D  Regress ion Output For Control Limits),
the value for the test statistic F is:
F = 5465.84
Using a significance level a' = 0.0 I. the rejection region for the test is defined by the
critical value Fc a ' (k. n(k+l )} From all F table, th is critical value is:
Fc 001 (3 .23 ) = 4.765
Clearly the null hypothesis Ho: AI = A2 = Al = ... = Ak =0 is rej ected since the
value of the F statistic is greater than the critical value Fc:
57
F> FCu'(k. n(k+l»)
5465.84> 4.765
Therefore, it is concluded that one can be very confident that this model is useful in
predicting t2'
1.2: Moving Range n = 2 Upper Control Chart Limits:
Following the steps in Section 3.1.2, described in detail in Chapter 3 ( pg. 3 J 5\
the values of the moving range n = 2 upper control chart limit for the Pearson Type III
family of distributions with location parameter c=O (gamma distribution) are located by
simulating values in MathCad for Windows release 4.02. The value of the upper control
limit is expressed as a multiple of (jR units from the average range. An upper contro[
limit is generated which leaves a tail area of 0.0027 beyond the upper limit. The limit is
evaluated in this manner for all combinations of a (shape parameter) = 1, 5(5) 135. As
demonstrated in Appendix A, the values for ~ appear to have little or 110 effect all the
upper control limit for the gamma distribution when expressed as a mulliple of (jl{ units.
The table on the following page, Table 4.2: Gamma Distribution Upper Control Limits
For Moving Range, demonstrates the results of simulating the gamma distribution.
Following step 7 in 3.1.2, a regression model is generated in Minitab for
Windows release 10.5. This statistical software package is used to generate different
multiple regression models for predicting the t, values with the predictor ex (sh ape
parameter). There are different mathematical models which can predict the t3 values, so,
by trial and error, a model is found which does a good job of predicting t, but may not
be the only model that can be used.
58
Table 42: Gamma Distribution Upper Control Limits For Moving Range
I a t3 @ ~ = 1 I
1 4.9826
5 4.1126
10 3.9821
15 3.8825
20 3.8271
25 3.7939
30 3.7458
35 3.7430
40 3.7119
45 3.7168
50 3.7105
55 3.7103
60 3.6884
65 3.7213
70 3.7091
75 3.7065
80 3.7155
85 3.6922
90 3.6955
95 3.6603
100 3.7005
105 3.7107
110 3.6967
115 3.7065
120 3.6903
125 3.6927
130 3.6614
135 3.6919
59
The output from Minitab can be found in Appendix D: Regression Output For
Control Limits. The best t3 regression model, based on R2, for the upper control limits is
as follows:
t~ = 3.68 + 1.88*e(U) + O.564*el OIU) + O.0969*e(·0025U)
A global F test is used to test the validity of the muhiple regression models. The F test
for the upper and lower control limits are as follows :
The model for t3 has a multiple coefficient of determination R2 of 99.7%. The
global F test is used to test the validity of the upper control (t3) limit multiple regress ion
model as indicated in Chapter 3 section 3.1.2, step 8.
From the Minitab output (Appendix D  Regression Output For Control Limits) ,
the value for the test statistic F is:
F = 2892.98
Using a significance level a' = 0.01, the rejection region for the test is defined by the
critical value Fc u·(k. n(k+ I )). From an F tabl e. this critical value is:
Fc (Jom. 23) = 4.765
Clearly the null hypothesis Ho: AI = 1~2 = A.'. = ... = Ak =0 is rejected since the
value of the F statistic is greater than the critical value Fc:
F> FCU· Ck. n(k+ l»
2892.98 > 4.765
Therefore, it is concluded that one can be very confident that this model is useful
in predicting t3.
60
Section 2: SUBOBJECTIVE TWO
The second subobjective is to evaluate the performance of the individual
measurement X and moving range n=2 mR control charts, based on the average run
length (ARL) using the Pearson type III family of distributions with location parameter
c=O (gamma distribution) control limits determined from subobjective 1. The control
charts are evaluated against an ARL that is acceptable for practical application in
industry and compared with methods having symmetrical control limits. An ARL that is
acceptable for practical application in industry means that the average run length (ARL)
for each control chart is a minimum of LOO observations. An ARL of 100 is equivalent
to a I % chance of a type I error when the process is in a state of stati stical control.
Following the steps in section 3.2, five process distributions were selected to
represent unknown parent distributions. The distributions were chosen to represent a
variety of process distributions that occur in industry. The five process distributions
selected are as follows:
• Normal (40, 102)
• Lognomlal (0, 12)
• Gamma (ex = 1.5, ~ = 1)
• Chisquare (df = 4)
• Exponential (~ = I)
A Turbo Pascal (version 6.0) program was written to perform steps 2 through 13
of section 3.2 (Chapter 3) . The Turbo Pascal program for the Chisquare (dr = 4)
6]
distribution can be found in Appendix E. To generate random variates from each of the
five parent distributions, random variates were first generated from the uniform
distribution. The random uniform variates were generated according to Marse and
Roberts' random number generator found In Appendix F. Based on the numbers
generated from the unjform di stribution, random vari ates for each of the parent
distributions were then generated according to the following al gorithms, as
recommended by Law and Kelton (lS, p. 48493):
2.1: Normal (40, 102) Algorithm:
The algorithm used to generate Normal (40, 102) random variates is known as
the polar method.
Algorithm:
2.1.1 Generate UJ and U2 as lID U(O,l), let Vi = 2Ui I for i = 1,2, . .. and let W =
2.1.2 If W > 1, go back to step I . Otherwi se, let Y = ~ (  2 In W IW) , XI =
V I Y, and X2 = V 2 Y. Then X I and X2 are lID N(O, 1) randol11 vari ates.
2.1.3 Given that Y  N(O, I), X  N(!l,cr2) can be obtained by using X = ~ + crY .
2.2: Lognormal (0, 12) Algorithm:
A special property of the lognormal di stribution is that if Y  N(!l,cr2) then
e Y  LN(~,cr\ Therefore, Lognormal variates can be generated based on Normal
variates from the algorithm above (Chapter 4, section 2. 1).
Algorithm:
62
2.2.1 Generate Y ~ N (~,cr\
y 2.2.2 Return X = e .
2.3: Gamma (a = 1.5, B = 1) Algorithm:
Random Gamma variates are typically generated according to three cases : 0 < a
< 1; a = 1; and a> 1. Since a = 1.5, the case for a > 1 will be used. According to Law
and Kelton (5, p. 489), ''There are several good algorithms for the case 0: > I ."
However, they recommend a method due to Cheng (22) referred to as the GB
algorithm.
Algorithm:
2.3.1 Generate U, and U2 as IID U(O, I ).
y 7 2.3.2 Let V = a In [U,/(lU,)j, Y = ae , Z = U,U2, and W = b + qV  Y.
2.3.3 If W + d  8Z >= 0, return X = Y. Otherwise, proceed to step 4.
2.3.4 If W >= In Z, return X = Y. Otherwise, go back to step I.
where:
a=l/.j(2a 1)
b = a  In 4
q = a + 1Ia
e =4.5
d = I + In 8
2.4: Chisquare (df = 4) Algorithm:
The Chisquare distribution is a Gamma distribution with shape parameter a =
df/2 and scale parameter ~ = 2. Therefore, the algorithm used to generate Gamma (a =
63
2, ~ = 2) will be used to generate the Chisquare distribution. The algorithm in section
4.2.3 for the case ex> 1 will be the one used for Chisquare (df = 4).
2.5: Exponential <B = 1) Algorithm:
The Gamma distribution with shape parameter ex = 1 and scale parameter ~ is an
exponential distribution with mean ~ . The algorithm used to generate Gamma variates
(ex = I, ~) is based on the inverse transform method.
Algorithm:
2.5.1 Generate U ~ DCO, I),
2.5.2 Return X = ~ In (U).
Based on the algorithms above and steps 2 through 13 from section 3.2 (Chapter
3), 1000 run lengths were generated for each of the five parent distributions. The
program output for the Normal distributions can be found in Appendix G. The output
consists of 1000 run lengths based on the individual measureme nt X and moving range
n=2 mR control charts for Shewhart, Oyon's symmetrical, and Ankney's asymmetri cal
control limits.
Average run lengths (ARLs) and variance of run lengths (VRLs) were calculated
for each of the distributions. The ARLs and VRLs can be found in Table 4.3: Control
ChaJ1 ARLs and VRLs on the following page. The 1000 run lengths are also presented
on a histogram for each of the five parent distributions according to their relative control
limits. These histograms, figures 41 through 415, are on the following pages.
64
Table 43: Control Chart ARLs and VRLs for No Mean Shift
Ideal ARL = 00 Indi vid ual Measuremenl X Moving Range
Acceptable ARL = 100
Parent
Distribution Shewhar Oyon Asymmetrical Shewhar Oyol1 Asymmetrical
t I
Normal ARL 1118.9 1427.2 141.9 213.4 277.0 4011.7
(40,102)
VRL 4693.9 6783.5 224.0 495.9 817.4 33429.5
as sl. dey. I
Lognormal ARL 33.0 40.7 161.0 33.9 56.0 148.2
(0. 12)
VRL 45.0 58.6 285.8 48.1 87.2 273 .2
as Sf.. dey.
Gamma ARL 58.3 80.1 1885.7 57.0 130.1 1418.9
(1.5, I) I
VRL 83.3 126.2 12566.5 78.4 284.7 5256.1
as 51. dey.
Chisquare ARL 72.2 99.0 2079.1 70.1 144.1 1550.9
(df = 4)
VRL Imu 192.2 9075 .6 108.0 :'127.3 7198.2
as 51. dev.
Exponenlial ARL 49.7 77.8 1758 .0 49.1 143.0 1713.9
( I )
VRL 75.7 136.5 8897.2 73.8 3 18.1 6846.8
as 51. dey.
65
500
400
300
200
100
o
XNormal Distribution 
Shewhart limits
,n ,n ,n ,M,n ,M,'"
o 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0
N ~ 0 v 00 N ~ 0
.. N N (Y')
Individual Measurement X Run lengths
,n
450
400
350
300
250
200
150
100
50
o
R  Normal Distribution 
Shewhart Limits
o
I.()
Hhl.ll n ,n ,n n ,,..
o
I.()
N
o
I.()
C"l
o
I.() v
Moving Range Run Lengths
o
I.()
I.()
it
Figure 41: Normal Distribution Run Lengths  Shewhart Control Limits
500
400
300
200
100
o
X  Normal Distribution 
Oyon limits
~,n,n.n, ... ,n , ... ,
o
o
N
o
o
~
oo
o
o
o v 8
00
o
o
N
N
o
o
~
N
o
o
o
C'l
Individual Measurement X Run Lengths
,n
400
350
300
250
200
150
100
50
o
R  Normal Distribution 
Oyon limits
o
U)
~ hJJL D D ,D ,n . ["'J
o
li)
o
Ll)
N
o
U)
C')
o
U) v
o
li)
li)
Moving Range Run Lengths
Figure 42: Normal Distribution Run Lengths  Oyon Control Limits
66
~~rn
500
400
300
200
100
o
o
If)
x  NormalDistribution 
Asymmetrical limits
lJ D,[] 0 ........ , ........
o
U1
o
If)
(\j
o
U1
C')
o
If)
'<t
o
U1
U1
Individual Measurement X Run Lengths
n
350
300
250
200
150
100
50
o
R  Normal Distribution 
Asymmetrical limits
IitU,[L[] ,(] ,n ,n , .. ,,., ,,., , ...
o 000 000 0
o 0 0 0 0 0 0 00
C\I «) 0 '<t co (\j <D
~ ~  N N M
Moving Range Run Lengths
Figure 43: Normal Distribution Run Lengths  Asymmetrical Control Limits
X Lognormal Distribution 
Shewhart limits
350
300
250
200
150
100
50
o
o
Hto 0 1"1 D ,[] ..., .... []
o
U1
o
r o
(J)
Individual Measurement X Run Lengths
R  Lognormal Distribution·
Shewhart limits
350
300
250
200
150
100
'
50
o ~nnn , .... ~ II
o
(0')
o
U1
o
(J)
Moving Range Run Lengths
Figure 44: Lognormal Distribution Run Lengths  Shewhart Control Limits
67
x  Lognormal Distribution 
Oyon limits
500
400
300
200
100
o
o
C\J
Jl.n.n.n ._
o
<.0
o
o
~
o
C\J
C\J
Individual Measurement X Run Lengths
Q) a
~
R  Lognormal Distribution·
Oyon limits
350
300
250
200
150
100
50
o Hl" .ocncn n ~ ... ~ j]
L() o
Moving Range Run Lengths
Q)
<5
:2
Figure 45: Lognormal Distribution Run Lengths  OyOIl Control Limits
x Lognormal Distribution 
Asymmetrical limits
500 ~~
400
300
200
100
0+U+U~~~~~~~~~~~
o
L()
o
11l
o
11l
C\J
o
11l
"<t
Individual Measurement X Run Lengths
R  Lognormal Distribution
 Asymmetrical limits
300 ~.
250
200
150
100
50
o ~~~~~~~~~~~~
o
(')
o
C1l
o
11l
~
o
~
N
o
r..
N
Moving Range Run Lengths
Figure 46: Lognormal Distribution Run Lengths  Asymmetrical Control Limits
68
400
350
300
250
200
150
100
50
o
x  Gamma Distribution 
Shewhart limits
o
C\I
].1 ..
JI 'illl!. n n ,", ....
o
(J:)
o
o
o 0
<t co
...
o
C\I
C\I
Individual Measurement X Run Lengths
,n
450
400
350
300
250
200
150
100
50
o
R  Gamma Distribution 
Shewhart limits
l{)
N
~.IUI n ,n ,_
l{) l{) l{) I.()
..... C\I ..... N
~ ~ N
Moving Range Run Lengths
Figure 47: Gamma Distribution Run Lengths  Shewhart Control Limits
400
350
300
250
200
150
100
50
o
x Gamma Distribution 
Oyon limits
I.()
(\J
• ~ II n n ,n n .... , ... , ... ,O
l{) I.()
C\I .....
C\I N
Individual Measurement X Run Lengths
450
400
350
300
250
200
150
100
50
o
R  Gamma Distribution 
Oyon limits
o v
Htn,ll.no ....
o
N
00 0 o co <D
C\I N (')
Moving Range Run Lengths
Figure 48: Gamma Distribution Run Lengths  Oyon Control Limits
69
,"

n
600
500
400
300
200
100
o
x  Gamma Distribution 
Asymmetrical limits
rn n n n,n , ... , ~
a
a
C')
o
o
cYl
a
o
U) .
o
o
.
N
o
o
l"N
o
o
C')
C')
Individual Measurement X Run Lengths
n
R  Gamma Distribution 
Asymmetrical limits
500 ~,
400~ ~ 1;
300 ~ 1~
200
100 ~"4 •• ~;
a
o
C')
a
a
cYl
a
a
Ifl .
a a
N
o
o
l"N
Moving Range Run Lengths
o
o
C')
C')
Figure 49: Gamma Distribution Run Lengths  Asymmetrical Control Limits
x  ChiSquare Distribution 
Shewhart limits
350
300
250
200
150
100
50
o
o
N
...
iHl1J D ,n ,n ,n .. ,n
o
<.D
o
o
.
o
N
N
Individual Measurement X Run Lengths
,U
R  ChiSquare Distribution 
Shewhart limits
350
300 
250
200
150
100 
50
0
0
N
r l
0 0
<.D 0 .
o
<Xl
o
N
N
Moving Range Run Lengths
Figure 410: Chisquare Distribution Run Lengths  Shewhart Control Limits
70
x  ChiSquare Distribution 
Oyon limits
350
300
250
200
150
100
50
o
l!)
N
l[) .....
r:1... ...n Ilii. nnn.n , ,,," , rt .... ,U
l!)
N
l{) l[)
N .....
N N
Individual Measurement X Run Lengths
500
400
300
200
100
o
R  ChiSquare Distribution 
Oyon limits
o
l[)
Il.n,n " .......
o
l!)
N
Moving Range Run Lengths
,0
Figure 411: Chisquare Distribution Run Lengths  Oyon Contro.! Limits
x  ChiSquare Distribution 
Asymmetrical limits
500~,
400
300
200
100
o o
('"J
o 0 0 Q)
000 <;
N ~ ~ :::;;!
o 0 o 0
C]) l{)
Individual Measurement X Run Lengths
500
400
300
200
100
o
R ChiSqua,re Distribution 
Asymmetrical limits
,n n n n ....... n
000
000
('"J C]) l{)
000 Q)
00 0 a
N ~ ~ :::;;!
Moving Range Run Lengths
Figure 412: Chisquare Distribution Run Lengths  Asymmetrical Control Limits
71
x Exponential Distribution 
Shewhart limits
300 r,
250 +rn~
200 W.1
150
100
50
O+U+U+U+U~~~~~~~~~
o 0
~ C')
o
l1J
Individual Measurement X Run Lengths
R  Exponential Distribution·
Shewhart limits
300 r~
250
200
150
100 ~;1;~~~
50
0 +U~~~~~~~~~~~~
o
C')
o
l1J
o
I'
o
en
Moving Range Run Lengths
Figure 413: Exponential Distribution Run Lengths  Shewhart Control Limits
x  Exponential Distribution 
Oyon limits
350
300
250
200
150
100
50
o
o
N
o
(0
illl ,n,n n n ,.,
o
o
o
<Xl
o
N
N
, ~ it
Individual Measurement X Run Lengths
R  Exponential Distribution'
Oyon Ilimits
500
400
300
200
100
a
o
l{)
U O. n . n "' , [1
o
L{)
N
o
Ii)
"<t
o
L{)
l{)
Moving Range Run Lengths
Figure 414: Exponential Distribution Run Lengths  Oyon Control Limits
72
0
x Exponential Distribution·
Asymmetrical limits
450
400
350
300
250
200
150
100
50
o
a
Ii)
~n.n . n . n ...... .., ... ....
a
Ii)
a
Ii)
N
a
II)
Ii)
Individual Measurement X Run Lengths
.
R  Exponential Distribution
 Asymmetrical limits
400
350
300
250
200
150
100
50
o
a
a
N
T"rll .n . n . n .., ...
a
o
to
o
o
o
a
a
'<t
a
a
to
a
a
N
N
Moving Range Run Lengths

~
Figure 415: Exponential Distribution Run Lengths  Asymmetrical Control Limits
/
73
2.6: Analysis of the Normal Distribution; No Mean Shift:
Individual measurement X control chart limits:
The data in Table 4.3: Control Chart ARLs and VRLs for No Mean Shift
indicate that the asymmetrical individual measurement X control limits are acceptable
for practical use in industry when the underlying process distribution is normal and
there is no shift in the process mean. An ARL that is acceptable for practical application
in industry means that the average run length for the controJ chart is a minimum of 100
observations. Although the asymmetrical controJ limits are acceptable, they do not work
as well as individual measurement X controJ limits produced by Shewhart or Oyon.
When the underlying process distribution is normal and there is no shift in the process
mean, the ARLs for Shewhart and Oyon individual measurement X control limits are
approximately 1119 and 1427 observations, respectively. The asymmetrical control
limits have an ARL of 141.9 observations.
Based on the histograms in Figures 4.1, 4.2, and 4.3 for individual measurement
X control limits, the difference in the performance between the asymmetrical and
symmetrical control limits is not so prevalent. The median run length for Shewharl
limits is between 200 and 300 observations, Oyon is between 200 and 300 observations,
and the asymmetrical limits are between 100 and 150. The differences in nlll lengths
between the symmetrical and asymmetrical control charts are much smaller than when
comparing ARLs. The median run length for the asymmetricaI limits is only 150 to 200
observations less than that of the symmetrical limits. The symmetrical individual
74
measurement X control limits perform better than those that are asymmetrical whether
comparing ARLs or median RLs .
Moving range (n=2) mR control chart limits:
The data in Table 4.3 also indicate that the moving range (n=2) mR control chart
limits based on the Pearson type ill family of distributions work well when the
underlying process distribution is Normal. The ARL for the asymmetrical control chart
limits perform better than the Shewhart and Oyon limits. When the underlying process
distribution is normal and there is no shift in the process mean, the ARLs for Shewhart
and Oyon mR control limits are approximately 213 and 277 observations, respectively.
The asymmetrical mR control limits have an ARL of 4012 observations.
The median RLs also indicate that the asymmetrical mR limits exceed the
performance of the Shewhart and Oyon limits. From Figures 4.1, 4.2, and 4.3 for mR
control limits, the median run length for Shewhart limits is between 75 and 125
observations, Oyon is approximately 100 obse rvations. anclthe asymmetrical limits are
between 500 and 700 observations. The differences in median run lengths between the
symmetrical and asymmetrical control charts are smaller than when comparing ARLs.
The median run length for the asymmetrical limits is 425 to 575 observations greater
than that of the symmetrical limits. The asymmetrical mR control limits perform better
than those that are symmetrical whether comparing ARLs or median RLs with no shift
in the mean.
75
2.7: Analysis of NonNormal Distributions; No Mean Shift:
The data in Table 4.3 for the lognormal, gamma, chisquare, and exponenti al
distributions indicate that the asymmetrical control chart limits perform better than the
Shewhart and Oyon limits when the underlying distribution is nonnormal with no mean
shift. The asymmetrical limits outperform the other limits on both the individu al
measurement X and mR control charts. In all cases, the asymmetrical ARLs exceed 100
observations when there is no shift in the mean. For the highly skewed di stribution s.
lognormal and exponential, the Shewhart and Oyon limits fall well short of 100
observations. In these cases, the Shewhart and Oyon individual measurement X lim its
have ARLs of 33 and 41 for the lognormal distribution and 50 and 78 observation s for
the exponential distribution.
The median run lengths follow the same pattern as the ARLs when there is no
shift in the process mean. The median RLs for the asymmetrical limits exceed 100
observations in all but one instance. The medi an RL for the lognormal mR
asymmetrical control limits falls between 75 and 105. The median run length comes
very close to 100 but falls short. The median RLs for the Shewhart and Oyon limits are
less than 30 observations when the underlying distribution is log normal. The
asymmetrical limits perform better than the symmetrical limits when there is 110 shift in
the process mean, even though the median RL fall s short of 100 obse rvati ons.
76

Section 3: SUBOBJECTIVE THREE
The third subobjective is to compare the power of the individual measurement
X and moving range n=2 mR control charts using the Pearson Type III c=O
asymmetrical control limits with those methods having symmetrical control limits. The
power, in this case, refers to the ability of the control charts to detect shifts in process
location of 0.5, J .0, 1.5, 2.0, 2.5, and 3.0crx units.
To perform the third subobjective, a Turbo Pascal (version 6.0) program was
written to perform steps 1 through 5 of section 3.3 (Chapter 3). The Turbo Pascal
program is the same program referred to in section 4.2, page 49. The program for the
Chisquare (df = 4) distribution can be found in Appendix E and the program output for
the Normal distribution is in Appendix G. The output consists of 1000 run lengths based
on the individual measurement X and moving range n=2 mR control charts for
Shewhart, Oyon's symmetrical, and Ankney's asymmetrical control limits at process
mean shifts of 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0crx units.
Average run lengths CARLs) were calculated for each of the distributions
according to the shift in the process mean . The ARLs can be found in Tahle 4.4:
ARLsNRLs for Shifts in th Process Mean on the following page.
77
..
Table 44: ARLs/VRLS for Shifts in the Process Mean
Ideal ARL = I ARLs NRLs for Shifts in the Process Mean
Parent Shifl 7 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 0.5  1.0  1.5 2.0 2.5 :1.0
Distributio
Nonnal
(40. 10' ) Normal 437.51 84.2/ 25.3/ 9.)1 4.31 2.3/ 402.21 91.11 25.4/ 83/ 4. 1/ 2.21
1476.9 205.4 86.7 15.6 7,6 2.3 1490.9 230.1 63 .9 12.9 5.6 2.1
Symmelric.1 485.9/ 93.31 28.3/ 9.71 4.51 2.41 450.61 98.01 28.4/ 9.91 4.31 231
1582.4 217.3 104.6 16.1 7.7 2.9 1571.6 239 .0 72 .4 31.8 6.1 :l .O
Asymmetrical 518.2/ 11 55.9/ 764.01 122.91 25.3/ 10.61 39.21 12.11 5.21 2.81 1.71 1.31
1059.6 4846.4 6946.3 698.4 106.5 59J 66.7 14.S 5.7 2.6 1.2 0.7
Lognormal
(0. I', Nonn.1 13.21 4.31 1.51 1.01 1.01 1.01 58 .71 19.21 2. 1/ 1.01 1.01 1.01
IS.6 7.4 2.4 0.4 0.0 0.0 82.4 99.9 31.8 0.2 0.1 0.1
Symmetrical 18. 11 6.41 2. 11 1.21 1.01 1.01 76.71 54 .61 1211 '201 101 1.01
31.7 IU 4.2 14 0.2 0.0 110.5 203 .9 119.9 31 8 0.1 01
Asymmetrical 96.51 5411 26.31 12.71 551 2.RI 1.21 1.11 1.01 1.0/ 1.01 1.01
214.6 170.8 94.0 51.0 28.9 21.7 0.6 0.2 0.2 01 0. 1 on
Gamma
(1.5.1) Normal 33.91 21.01 11.91 6.91 4.11 2.51 103.41 185.51 255.81 19201 7931 4.71
49.5 31.7 17.2 10.8 5.2 3.3 16R.9 3 11.7 500.0 K'i14 8.'14.4 <)6.7
Symmetrical 45.91 2~ . 51 16.61 9.61 5.51 :UI 147.11 265 .31 4.12.111 459.61 362.91 229.51
67.1 45.0 25.0 15 .0 9.5 4 .4 250 463.3 BK4.6 129 1.4 2165. 1 2470.2
Asymmetrical 105901 639.61 W'J.()f 194 .21 117.21 65 .01 5.81 2.11 1.51 1.21 1. 11 1.11
.' 806A 2619.8 13K 1.9 733.4 538.9 26H.J 6.6 1.6 0.8 0.5 0.4 0.2
Chi·square
(dr = 4) Normal 40.21 24.91 13.SI 7.41 4.31 2.61 118.11 2501 327.91 2 IH . ~1 '15 . 11 1.71
64.6 4).1 24.9 11.5 5.1 3.1 219.6 414 .2 710.3 l)4~.4 12H2 .. ~ I IA
Symmetri cal 52.51 30.11 18.21 10.21 .1 .61 1.41 181.41 322.71 54.\1/ 605 .71 41301 120.51
86.4 49.1 33.2 19.8 8.2 4.4 310.7 542.5 109H 2 146.5 2Wn .5 26~ :U
Asymmetrical 1237.81 525.41 30931 160.41 91.11 53.01 15.51 2.9/ 1.71 1.31 1.21 1.11
7071.7 2225.3 19 15.1 765 .8 460.3 4 19 .4 57.9 2.6 1.1 0.7 D.S 03
Exponenti al
(J) Normal 29 .21 18.61 10.71 6.61 4.11 2.51 82 .01 D 6.31 19(1.21 167 II (,9 II 5 41
46.4 35.3 15.5 10.1 5.8 3.5 131.4 206.4 .154.5 551'1 476.0 ID ')
Symmetri cal 46.01 28. 11 17 .41 10.41 6 .. 11 .UI 12.1.21 2 J 2.21 34 5.61 41 \i . 1I 403 r)f 21601
75.4 49.4 16.0 17.0 10.4 5 R 2(}6.3 .156.3 6 14.3 g4 1. 1 12'>0. I 1567 .3
ASYl1unetricai 1358.51 872.51 512 .SI 297.41 192.0/ 122 .HI 2.61 1.61 J IJI 1.21 1.11 1.11
4947 .R 3954.3 1503.8 7963 602.2 400.1 2.1 1.0 0.6 0.4 0.4 0.2
78
3.1: Analysis of Negative shifts in the mean (shifts to the left):
The data in Table 4.4: ARLs/VRLs for Shifts in the Process Mean , indicate th at
the asymmetrical individual measurement X and mR control limits do a very good job
of detecting negative shifts in the process mean . Even at a very small shift of 0.5
standard deviations, the asymmetrical control limits are very sensitive to the detection of
shifts. Regardless of the underlying distribution, the asymmetrical limits detect a 0.5
standard deviation in less than 40 observations, a 1.0 shift in less than 13 observation s,
and a  1.5 shift in less than 6 observations. For large shifts in the process mean of 2.5
and 3.0 standard deviations, the asymmetrical limits detect the shift within the first two
observations. The asymmetrical limits are much more sensitive to negative shifts than
the symmetrical limits. It can be concluded that the asymmetrical control limits do a
good job of detecting negative shifts in the mean regardless of the underlying
distribution .
3.2: Analysis of Positive shifts in the mean (shifts to the right):
The data in Table 4.4: ARLs/VRLs for Shifts in the Process Mean. lI1dicate th at
the asymmetrical individual measurement X and mR control limits do not do a good job
of detecting positive shifts in the process mean. The symmetrical control limits are more
sensitive to detecting positive shifts. The only underlying distribution for which the
asymmetrical limits appear to be effective in detecting positive shifts is the lognormal
distribution. Although the asymmetrical limits appear to be moderately effective in this
case, the symmetrical limits still perform better.
79
...

CHAPTERS
CONCLUSIONS AND RECOMMENDATIONS
Section 1: CONCLUSIONS & RECOMMENDA TJONS
This section consists of conclusions and recommendations for this thesis
research. Using the results and analysis generated in Chapter 4, the performance of the
asymmetrical control limits are compared to that of Shewhart' sand Oyon' s control chart
limits. The following conclusions are made based on the information in Tables 4.3,
Table 4.4, and the analysis in Chapter 4, sections 2.6,2.7,3.1, and 3.2.
• The performance of the individual measurement X symmetrical control charts is
much better than that of the asymmetrical charts when the underlying distributi on is
normal and there is no shift in the mean. This conclusion is supported in Chapter 3,
section 2.6: Individual measurement X control chart limits, page 421.
• The performance of the moving range (n=2) asymmetrical control charts is much
better than that of the symmetrical charts when the underlying distribution is nonnal
and there is no shift in the mean. This conclusion is supported in Chapte r 3, section
2.6: Moving range (n=2j mR control chart limits, page 422.
• The performance of the asymmetrical controJ charts is better than th at of the
symmetrical charts when the underlying distribution is nonnormal and there is no
shift in the mean. This conclusion is supported in Chapter 3, section 2.7, page 423.
• The performance of the asymmetrical control charts is better than that of the
symmetrical charts when there is a negative shift in the mean, regardless of the
80

underlying distribution. This conclusion is supported in Chapter 3, section 3.1, page
426.
• The asymmetrical control charts do not do a good job of detecting positive shifts in
the mean regardless of the underlying distribution. This conclusion is supported in
Chapter 3, section 3.2, page 426.
The asymmetrical limits perform well when there is no mean shift and the
underlying distribution is nonnormal. The problem with the asymmetrical control
charts is that they do not do a good job of detect.ing positive shifts in the process mean.
In general, control charts for skewed distributions are most useful for detecting positive
shifts in the mean. According Irving Burr (1953), " ... causes of nonnormality is that the
distribution may be unable to go beyond a certain point, such as zero (5, p.80) ... " As
indicated by this statement, negative shifts in the mean will not occur because the
inability to go beyond this point (zero in this research). Shifts in the mean will, in most
cases, be positive. Based all the conclusion that the asymmetrical control limi1s do not
do a good job of detecting positive shifts in the mean, the author recommends the
asymmetrical control limits developed in this research not be used.
81

Section 2: RESEARCH CONTRIBUTIONS
• This thesis research provides empirical equations to calculate approximately the
correct asymmetrical control chart constants tl, t2, and t:\ when the underlying
process distribution is a gamma distribution with shape parameter ex and scale
parameter ~ .
• This thesis research provides empirical evidence that the asymmetrical gamma
control charts (X and mR) perform better than the normal curve and symmetri cal
gamma control charts eX and mR) when the distribution has a marked departure
from normality (represented in this research by skewed distributions) and there is no
shift in the mean. However, more research is needed in this area since the
asymmetrical control charts lack the power to detect positive shi fts in rhe process
mean . In this regard, this research opens avenues for future research providing
improved methodology for setting control limits (X and mR) under skewed
circumstances.
82

Section 3: FUTURE RESEARCH
The fact that the asymmetrical control charts lack the power to detect positive
shifts in the process mean suggests that more research is needed in this area. It is the
author's belief that the inability to detect positive shifts in the mean is due to the
following three factors:
I. The use of 0.00135 of the observations falling outside the upper or lower
control limits when setting those limits, regardless of the skew of the unde rlying
distributions.
2. The empirical nature of the study (Number of observations).
3. The ability to accurately estimate the parameters ex and ~ from the unknown
underlying distributions.
Additional research is recommended in setting asymmetrical control limit eX
and mR) based on the method for determining the location of the upper and lower
control limits. The upper and lower control limits in this research are determined based
on 0.00135 of the observations falling beyond each limit; regardless of the skew of Lhe
underlying distributions. The upper and lower control limits can be determined by
varying the percent of outlying observations with the shape parameter a. When the
underlying distribution is skewed, a higher percentage can be allotted to the upper
control limit so that it is not set so far out on the tail; meanwhile, a lower percentage can
be allotted to the lower control limit since the process will not produce values less th an
a specified lower value. For example, when the distribution is exponentiaJ, set the lower
bound at 0.0000 and use all 0.0027 on the upper limit. The control chart will not be as
83
sensitive to negative shifts in the process mean as the limits in this research but will
become more sensitive to positive mean shifts.
This approach can be demonstrated with the theoretical run lengths for the
exponential distribution under the cases where limits are onesided and based on fa lse
alarm rates of 0.0027, 0.0050, and 0.0100 when the re is no shift in the process mean .
Table 5.1 : Theoretical Run Lengths for Exponential Distribution on the following page
demonstrates the theoretical ARLs of these limits. As seen Table 5.1. the ones ided
control limits perform much better than the twosided limits developed in this research .
The onesided limits have the power to detect positive shifts in the mean while
maintaining an acceptable false alarm rate when no mean shift is present.
The onesided asymmetrical control limits detect shifts in the process mean
bctter than the twosided asymmetrical control limits. The onesided limits, howcver,
do not detect shifts in the process mean as well as the symmetrical control limits. As can
be seen from Tables 5.1 and 5.2, at a sigma shift of 3.0. the theoretical ARL for the oncsided
asymmetrical control limits is 4.98 while the symmetrical control limits pick up
the shift in 2.72. Although the symmetrical control limits perform better than the onesided
symmetrical control limits when the underlying di stribution is exponential, both
the symmetrical and asymmetrical control limits have good performance.
The onesided asymmetrical control limits have a much better false alarl1l rate
than the symmetrical limits. Based on the criteria defined in this research, the
symmetrical control limits are not acceptable for practical use in indu stry because the
run
84
Table 51: Theoretical Run Lengths for Exponential Distribution
Positive Set Limits @ 0.00135 Set Upper Limits @ 0.0027
Shift Upper Lower ARL Upper ARL
0.0 0.001350 0.001350 370.37 0.002700 370.37
0.5 0.002225 0.000000 449.44 0.004452 224.62
1.0 0.003668 0.000000 272.63 0.007339 136.26
1.5 0.006048 0.000000 165.34 0.012]01 82.64
2,0 0.009972 0.000000 100.28 0.01995 1 50,]2 I
2.5 0.016441 0.000000 60.82 0.032893 30.40
3,0 0.027106 0.000000 36.89 0.054231 18.44
Positive Set Upper Limit @ 0.005 Set Upper Limit @ 0,0 I 0
Shifl Upper ARL Upper ARL
0,0 0,005000 200,00 0,010000 ] 00.00
0,5 0,008243 121 J2 0,016487 60,65
1.0 0,013550 73,80 0.027182 36,79
1.5 0,022407 44.63 0.044816 22,31
2.0 0,036942 27.07 0,073888 13 ,53
2,5 0,060907 16.42 0.121821 8,21
3,0 0,100419 9.96 0.200849 4,98
85

lengths for the symmetrical limits are less than 100. Based on the theoretical ARL the
asyrnmetricallimits do have a run length of 100.
There is a tradeoff between the asymmetrical and symmetrical control limits.
The tradeoff is a matter of economics. Compared to the symmetrical control limits. the
asymmetrical control limits do not have as much power but do have a smaller false
alarm rate. If the cost of defects is significantly larger than the cost of readjusting the
process mean, than a higher false alarm rate would be more desirable than the inability
to detect a shift in the mean. In this case the symmetrical control limits would be more
desirable. If the cost of readjusting the process mean involves a much higher cost than
the cost of defects, a lower false alarm rate would be more desirable than the power to
detect a shift. In this case the onesided asymmetrical control limits are more desirable.
The selection and use of the control limits is dependent on the economics of the process.
The empirical nature of this research also affects the results. The control limits
for this research are set on fifty observations per run . Using sLlch a small number of
observations creates variation in the control limits which generates ARLs th at are Ilot
representative of those dictated by theory. This is apparent by comparing the theoretical
results in Table 5. J to the results shown in Table 4.4: ARLs for Shifts in the Process
Mean. As can be seen, the run lengths generated in this research are much higher than
what theory states.
Increasing the number of observations used in setting control limits improves the
performance of the control charts under shifts in the process mean. Table 5.2: ARLs
For Control Limits Set On Different Number Of Observations demonstrates the ability
86
to produce control limits which are more representative of the underl yi ng distriburioll by
increasing the number of observations. Table 5.2 consis ts of ARLs for the exponential
distribution when control limits are based on SO, 100, sao, and 1000 observations. By
observation, it can be seen that increasing the number of observations greatly improves
the performance of the control charts.
87
Table 52: ARLs For Control Limits Set On Different Number Of Observations
Shift > 0.0 + 0.5
Observations Shewhart Oyon Asymmetrical Shew Oyon Asymm
50 49.72 77.85 1758.01 29.24 46.00 1358.52
100 44.13 62.74 859.88 26.56 40.38 784.67
SOD 39.54 55.74 490. 14 24.86 35.10 54 1.75
1000 40.06 56.78 422.14 23.14 35. 10 509.04
theoretical 54.60 54.60 370. 34 33.12 33.12 449.44
Shift > + 1.0 + 15
Observations Shew Oyon Asymm Shew Oyon Asymm
50 18.62 28.08 872.54 10.68 17.36 512.77
100 16.01 24.48 471.82 9.33 14.60 290.45
500 15.05 21.99 329 .31 8.91 12.96 205.68
1000 14.65 20.95 30701 8.91 12.9K 187.19
theoretical 20.089 20.089 272.63 12.18 12.18 165.34
Shift> + 2.0 + 2.5
Observations Shew Oyon Asymm Shew Oyon Asyrnm
50 6.60 10.42 297.36 4.05 6.33 19 1.99
100 5.73 8.56 185.03 3.64 5.27 107.08
500 5.34 7.88 122.05 3.32 4.79 70.30
1000 5.28 7.76 114.83 3.14 4.65 67.83
th eoretical 7.39 7.39 100.28 4.48 4.48 60.82
Shift > + 3.0
Observations Shew Oyon ASYITlI11
50 2.52 3.79 122.77
100 2.19 3.23 67.81
500 1.96 2.96 43. 13
1000 1.95 2.76 42.95
theoretical 2.72 2.72 36.89
88
REFERENCES
I. Fisher, R.A. "On the mathematical foundations of theoretical statistics. " Philos.
Trans. Roy. Soc. London Ser. A 222: 309368; 1922.
2. Kendall, M.; Stuart, A. The Advanced Theory of Statistics 4d ed. Charles Griffin &
Company Limited, 1979.
3. Schilling, E.G.; Nelson, P.R. "The effect of nonnormality on the control limits of
X(bar) charts." Journal of Quality Technology 8 (October 1976): ] 838.
4. Wheeler, DJ., Chambers, D.S. Understanding Statistical Process Contral2et eel.
Knoxville, Tennessee: SPC Press. 1992.
5. Burr, I.W. Engineering Statistics and Qualit'.1 Control. New York: McGrawHilI.
Inc., 1953.
6. Duncan, AJ. Quality Control and Industrial Statistics Sci ed Homewood, IJiinois:
Irwin, Inc., 1986.
7. Juran, J .M. Quality Control Handbook I ci ed. New York: McGrawHi II, Inc., 1951.
8. Oyon, J.R. The Effects of Skewed Distributions In Individual Measurement X {lnd
Moving Range n=2 mR Control Charts. M.S. Thesis. Oklahoma State University,
May 1995.
9. Deming W.E. The New Economics For Industry, Government, and Education.
Cambridge, MA: Massachusetts (nstitute of Technology Center For Advanced
Engineering Study, 1993.
89
c
10. Juran, 1.M. editor. A History of Managing For Quality. Milwaukee, WI: ASQC
Quality Press, 1995.
11. Joiner, B.L. Fourth Generation management. New York: McGrawHili , Inc .. 1994.
12. American Society of Quality Control. ANSIIASQC Standard A I  Definitions,
Symbols, Formulas, and Tables for Control Charts. Milwaukee, WI: ASQC, 1978.
13. Shewhart, W.A. Economic Control of Quality of Manufactured Products. New
York: D. Van Nostrand Co .. Inc .. 1931.
14. Wadsworth, H.M., Stephes, K.S., Godfrey , A.B. Modern methods for Qua/itv
Control and Improvement. New York: John Wiley and Sons, 1986.
15. Law, A.M., Kelton, W.o. Simulation Modeling and Analysis. New York: McGrawHill,
Inc., 1991.
16. ASTM Special Technical Publication ISD. AS7M manual on presentation of'data
and control chart analysis. Philadelphia: ASTM, 1976.
17. Hayes, W.L., Winkler, R.L. Statistics; Probability, Inference, ({nd Decision New
York: Holt, Rinehart and Wilson, 1970.
18. Savage, 1.R. Statistics: Uncertainty and Behavior. Boston: Houghton Mifflin,
1968.
19. Hair, J.F., Anderson, R.E. , Tatum, R.L., Black, W.e. Mulrivariate Data Analysis:
With Readings 4d ed. London: Prentice Hall, 1995.
20. Miller, 1., Freund, lE., Johnson, R.A. Probahility and Statistics for Enf?in eers 4d
ed. Englewood Clif