Notes on Introduction to Statistical Quality
Control by Montgomery
Qianqian Shan
Contents
1 Quality Improvement in The Modern Business Environment 2
1.1 The Meaning of Quality and Quality Improvement . . . . . . . 2
1.2 Statistical Methods for Quality Control and Improvement . . . 3
2 The DMAIC Process 3
3 Methods and Philosophy of Statistical Process Control 4
3.1 Statistical Basis of the Control Chart . . . . . . . . . . . . . . 4
3.2 Choice of Control Limits . . . . . . . . . . . . . . . . . . . . . 5
3.3 Choice of Sample Size and Sampling Frequency . . . . . . . . 5
3.4 Sensitizing Rules for Control Charts . . . . . . . . . . . . . . . 5
4 Cumulative Sum and Exponentially Weighted Moving Aver-
age Control Charts 6
4.1 Cumulative Sum Control Chart . . . . . . . . . . . . . . . . . 6
4.1.1 Basic Principals: The Cusum Control Chart . . . . . . 6
4.1.2 The Tabular or Algorithmic Cusum for Monitoring the
Process Mean . . . . . . . . . . . . . . . . . . . . . . . 6
4.1.3 Improving Cusum Reponsiveness for Large Shifts . . . 7
4.1.4 Cusums for Other Sample Statistics . . . . . . . . . . . 7
4.2 The Exponentially Weighted Moving Average Control Chart . 8
1
1 QUALITY IMPROVEMENT IN THE MODERN BUSINESS ENVIRONMENT
1 Quality Improvement in The Modern Busi-
ness Environment
1.1 The Meaning of Quality and Quality Improvement
Dimensions of Quality
1. Performance (Will the product do the intended job?)
2. Reliability (Hwo often does the product fail?)
3. Durability (How long the product lasts), the effective service life of the
product .
4. Serviceability (How easy is it to repair the product?) How quickly and
economically a repair or routine maintenance activity can be accom-
plished.
5. Aesthetics (What does the product look like?) Visual appeal of the
product.
6. Features (What does the product do?), features beyond the basic per-
formance of competition.
7. Perceived Quality (What is the reputation of the economy or its prod-
uct?), the reputation is directly influenced by failures of the product
that are highly visible to the public or that require product recalls, and
by how the customer is treated when a quality related probelm with
the product is reported.
8. Conformance to Standards (Is the product made exactly as the designer
intended?)
Quality is inversely proportional to variability.
Quality improvement is the reduction of variability in processes and
products.
Quality Engineering Terminology
Every product possesses a number of elements that jointly describe what
the user or consumer thinks of as a quality, and these parameters are often
called quality characteristics or critical-to-quality(CTQ) characteristics with
several types:
1. Physical: length, weight, voltage, viscosity
2
2 THE DMAIC PROCESS
2. Sensory: taste, appearance, color
3. Time Orientation: reliability, durability, serviceability
1.2 Statistical Methods for Quality Control and Im-
provement
Three major areas,
1. Statistical process control (SPC), control chart is one of the primary
techniques of SPC.
2. Design of experiments, a designed experiement is helpful in discovering
the key variables influenceing the quality characteristics of interest in
the process (by systematically varying the controllable input factors in
the process and determining their effects on the output parameters).
3. Acceptance sampling, closely connected with inspection and testing of
product.
The primary objective of quality engineering efforts is the systematic
redution of variability in the key quality characteristics of the product.
2 The DMAIC Process
DMAIC is a structured problem-soloving precedure widely used in quality
and process improvement. DMAIC fors an acronym for the five steps, Define,
Measure, Analyze, Improve and Control.
1. Define: A project chapter that contains a description of the project
and its scope, the start and anticipated completion dates, an initial
description of both primary and secondary metrics that will be used
to measure success and how those metrics aligh with business unit and
coporate goals, the potential benefits to the customer, the potential
benefits to the organization, mielstones that should be accomplished
during the project, the team members . . .
2. Measure: evaluate and understand the current state of process. This
involves collecting data on measures of quality, cost, and throughput/-
cycle time.
3
3 METHODS AND PHILOSOPHY OF STATISTICAL PROCESS CONTROL
3. Analyze: use the data from measure step to begin to determine the
cause-and-effect relationships in the process and to understand different
sources of variability.
4. Improve: Creative thinking about the specific changes that can be made
in the process and other things that can be done to have the desired
impact on process performance. Develop a solution to the problem and
to pilot test the solution.
Pilot test: a form of confirmation experiment, it evaluates and docu-
ments the solution and confirms the solution attains the project goals.
5. Control: complete all remaining work on the project and to hand off
the improved process to the process owner along with a process control
plan and other necessary procedures to ensure that the gains from the
project will be institutionalized.
3 Methods and Philosophy of Statistical Pro-
cess Control
Seven major tools:
1. Histogram or stem and leaf plot
2. Check sheet
3. Pareto chart
4. Cause-and-effect diagram
5. Defect concentration diagram
6. Scatter diagram
7. Control chart
3.1 Statistical Basis of the Control Chart
There is a close connection between control charts and hypothesis testing in
the sense that the control chart is a test of the hypothesis that the process is
in a state of statistical control. A point within the control limit is equivalent
to failing to reject the hypothesis of statistical control, while outside the
control limit is equivalent to rejecting the hypothesis of statistical control.
4
3 METHODS AND PHILOSOPHY OF STATISTICAL PROCESS CONTROL
However, in testing statistical hypotheses, we usually check the validity
of assumptions, whereas control charts are used to detect departures from an
assumed state of statistical control.
Hypothesis testing framework is useful in analyzing the performance of a
control chart such as probability of Type I error of control chart (conclude
out of control (H
1
) when it’s really in control) or Type II error (conclude in
control (H
0
) when it’s really out of control).
3.2 Choice of Control Limits
It’s standard practice to determine the control limits as a multiple of standard
deviation of the statistic plotted on the chart. The multiple is usually 3.
3.3 Choice of Sample Size and Sampling Frequency
When choosing the sample size, keep in mind the size of the shift that we are
trying to detect. If the shift is relatively large, then we use smaller sample
sizes than those that would be employed if the shift of interest were relatively
small.
Or use average run length(ARL) to evaluate the decisions regarding sam-
ple size and sampling frequency with,
ARL =
1
p
,
where p is the probability that any point exceeds the control limits.
3.4 Sensitizing Rules for Control Charts
Several criteria may be applied to a control chart to determine whethere the
process is out of control,
standard action signal:
1. One or more points outside of the control limits
2. Two of three consecutive points outside the two-sigma warning limits
but still inside the control limits
3. Four of five consecutive points beyond the one-sigma limits
4. A run of eight consecutive points on one side of the center line
5. Six points in a row steadily increasing or decreasing
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4 CUMULATIVE SUM AND EXPONENTIALLY WEIGHTED MOVING AVERAGE
CONTROL CHARTS
6. Fifteen points in a row in zone C (1-sigma from center line)
7. Fourteen points in a row alternating up and down
8. Eight points in a row on both sides of the center line with none in zone
C
9. An unusual or nonrandom pattern in the data
10. One or more points near a warning or control limit.
4 Cumulative Sum and Exponentially Weighted
Moving Average Control Charts
Shewhart control charts are useful in phase I implementation, where the
process is likely to be out of control and experiencing assignable causes that
result in large shifts. It’s also useful in the diagnostics of bringing an unruly
process into statistical control, because the patterns on these charts often
provide guidance regarding the nature of the assignable cause.
However, Shewhart control chart relatively insensitive to small process
shifts, say, on the order of 1.5σ or less. This potentially makes Shewhart
control charts less useful in phase II monitoring, where the process tends
to operate in control, reliable estimates of the process parameters are avail-
able and assignable causes do not typically result in large process upsets or
disturbances.
4.1 Cumulative Sum Control Chart
4.1.1 Basic Principals: The Cusum Control Chart
The cusum chart directly incorporates all the information in the sequence
of sample values by plotting the cumulative sums of the deviations of the
sample values from a target value/
4.1.2 The Tabular or Algorithmic Cusum for Monitoring the Pro-
cess Mean
The tabular cusum works by accumulating derivations from µ
0
that are above
target with on statistic and accumulating derivations from µ
0
that are below
target with another statistic ,
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4 CUMULATIVE SUM AND EXPONENTIALLY WEIGHTED MOVING AVERAGE
CONTROL CHARTS
C
+
i
= max[0, x
i
− (µ
0
+ K) + C
+
i−1
]
C
−
i
= max[0, (µ
0
− K) − x
i
+ C
−
i−1
]
where K is called the reference vlaue (allownance or slack value), and it’s
often chosen about halfway between the target and the out of control value.
And starting values C
+
0
= C
−
0
= 0. If either C
+
0
or C
−
0
exceed the decision
interval H, the process is considered to be out of control. A reasonable value
for H is five times the process standard deviation σ.
The graphical display for the tabular cusum is called cusum status charts.
The cusum can be thought of as a weighted average, where weights are
stochastic or random. It’s usually recommended that parameters K, H are
selected to provide a good average run length performance.
Standardized cusum can be adopted as 1) many cusum charts cna now
have the same value of k and h , and the choices of these parameters are
not scale dependent , 2) a standardized cusum lead naturally to a cusum for
controlling variability.
4.1.3 Improving Cusum Reponsiveness for Large Shifts
Use combined Cusum-Shewhart procedure for on-line control with the She-
whart control limit should be located approximately 3.5 standard deviations
from the center line or target value. An out-of-control signal on either (or
both) charts constitutes an action signal.
4.1.4 Cusums for Other Sample Statistics
When working with count data and the count rate is very low, it’s frequently
more effective to form a cusum using the time between events to detect an
increase in the count rate. When the number of counts is generated from a
Poisson distribution, the time between these events will follow an exponential
distribution.
C
−
i
= max[0, K − T
i
+ C
−
i−1
],
where K is the reference value and T
i
is the time that has elapsed since that
last observed count.
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4 CUMULATIVE SUM AND EXPONENTIALLY WEIGHTED MOVING AVERAGE
CONTROL CHARTS
4.2 The Exponentially Weighted Moving Average Con-
trol Chart
EWMA is a good alternative to the Shewhart chart when we are interested in
detecting small shifts. The performance is approximately equivalent to that
of the cumulative sum control chart, and it’s typically used with individual
observations.
z
i
= λx
i
+ (1 − λ)z
i−1
,
where λ ∈ (0, 1] and the starting value z
0
= µ
0
is the process target.
EWMA can be viewed as a weighted average of all past and current
observations, it’s very insensitive to the nomality assumption. And therefore
an ideal control chart to use with individual observations.
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