4-1. Introduction
Chapter 4 • Statistical process control is a
collection of tools that when used
Methods and Philosophy of
together can result in process stability
Statistical Process Control
and variability reduction
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4-2. Chance and Assignable
4-1. Introduction
Causes of Quality Variation
The seven major tools are
• A process that is operating with only chance
causes of variation present is said to be in
1) Histogram or Stem and Leaf plot
statistical control.
2) Check Sheet
• A process that is operating in the presence of
3) Pareto Chart
assignable causes is said to be out of control.
4) Cause and Effect Diagram
• The eventual goal of SPC is reduction or
5) Defect Concentration Diagram
elimination of variability in the process by
6) Scatter Diagram
identification of assignable causes.
7) Control Chart
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4-2. Chance and Assignable 4-3. Statistical Basis of the
Causes of Quality Variation Control Chart
Basic Principles
A typical control chart has control limits set at
values such that if the process is in control,
nearly all points will lie between the upper
control limit (UCL) and the lower control limit
(LCL).
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4-3. Statistical Basis of the 4-3. Statistical Basis of the
Control Chart Control Chart
Basic Principles Out-of-Control Situations
• If at least one point plots beyond the control
limits, the process is out of control
• If the points behave in a systematic or
nonrandom manner, then the process could
be out of control.
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4-3. Statistical Basis of the 4-3. Statistical Basis of the
Control Chart Control Chart
Relationship between hypothesis testing and
Relationship between hypothesis testing and
control charts
control charts
• Control limits can be set at 3 standard
• We have a process that we assume the true process
deviations from the mean.
mean is m = 74 and the process standard deviation
is s = 0.01. Samples of size 5 are taken giving a • This results in “3-Sigma Control Limits”
standard deviation of the sample average, x , as UCL = 74 + 3(0.0045) = 74.0135
s 0.01 CL= 74
sx = = = 0.0045
n 5 LCL = 74 - 3(0.0045) = 73.9865
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4-3. Statistical Basis of the 4-3. Statistical Basis of the
Control Chart Control Chart
Relationship between the process and the control
Relationship between hypothesis testing and chart
control charts
• Choosing the control limits is equivalent to
setting up the critical region for testing
hypothesis
H0: m = 74
H1: m ¹ 74
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4-3. Statistical Basis of the 4-3. Statistical Basis of the
Control Chart Control Chart
Important uses of the control chart Types the control chart
• Most processes do not operate in a state of statistical control.
• Variables Control Charts
• Consequently, the routine and attentive use of control charts will
identify assignable causes. If these causes can be eliminated from – These charts are applied to data that follow a
the process, variability will be reduced and the process will be
continuous distribution (measurement data).
improved.
• Attributes Control Charts
• The control chart only detects assignable causes. Management,
operator, and engineering action will be necessary to eliminate the
– These charts are applied to data that follow a
assignable causes.
• Out-of-control action plans (OCAPs) are an important aspect of discrete distribution.
successful control chart usage (see page 160).
• Refer to the process improvement cycle, Figure 4-5, page 160.
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4-3. Statistical Basis of the
Control Chart Stationary, uncorrelated
Type of Process Variability – see Figure
4-6, pg. 162
• Stationary behavior, uncorrelated data
• Stationary behavior, autocorrelated data
• Nonstationary behavior
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Stationary, correlated Non-stationary
50 100 150 200 250
12
-5
10
8
-10
6
-15
4
2
-20
50 100 150 200 250
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4-3. Statistical Basis of the 4-3. Statistical Basis of the
Control Chart Control Chart
Popularity of control charts
Type of Variability 1) Control charts are a proven technique for improving
productivity.
2) Control charts are effective in defect prevention.
• Shewhart control charts are most effective 3) Control charts prevent unnecessary process adjustment.
when the in-control process data is 4) Control charts provide diagnostic information.
stationary and uncorrelated. 5) Control charts provide information about process
capability.
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4-3.2 Choice of Control Limits 4-3.2 Choice of Control Limits
General model of a control chart “99.7% of the Data”
• If approximately 99.7% of the data lies within 3s
UCL = m W + Ls W
of the mean (i.e., 99.7% of the data should lie
Center Line = m W within the control limits), then 1 - 0.997 = 0.003
or 0.3% of the data can fall outside 3s (or 0.3%
LCL = m W - Ls W
of the data lies outside the control limits).
where L = distance of the control limit from the (Actually, we should use the more exact value
center line 0.0027)
m W = mean of the sample statistic, w. • 0.0027 is the probability of a Type I error or a
sW = standard deviation of the statistic, w. false alarm in this situation.
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Introduction to Statistical Quality Control, 4th Edition
4-3.2 Choice of Control Limits 4-3.2 Choice of Control Limits
Three-Sigma Limits Warning Limits on Control Charts
• Warning limits (if used) are typically set at 2 standard
• The use of 3-sigma limits generally gives good
deviations from the mean.
results in practice.
• If one or more points fall between the warning limits and
• If the distribution of the quality characteristic is the control limits, or close to the warning limits the
reasonably well approximated by the normal process may not be operating properly.
distribution, then the use of 3-sigma limits is • Good thing: warning limits often increase the sensitivity
applicable. of the control chart.
• These limits are often referred to as action limits. • Bad thing: warning limits could result in an increased
risk of false alarms.
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4-3.3 Sample Size and Sampling 4-3.3 Sample Size and Sampling
Frequency Frequency
Average Run Length
• In designing a control chart, both the
• The average run length (ARL) is a very
sample size to be selected and the
important way of determining the appropriate
frequency of selection must be specified. sample size and sampling frequency.
• Larger samples make it easier to detect • Let p = probability that any point exceeds the
small shifts in the process. control limits. Then,
• Current practice tends to favor smaller, 1
ARL =
more frequent samples.
p
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4-3.3 Sample Size and Sampling 4-3.3 Sample Size and Sampling
Frequency Frequency
Illustration What does the ARL tell us?
• The average run length gives us the length of
• Consider a problem with control limits set
time (or number of samples) that should plot in
at 3standard deviations from the mean.
control before a point plots outside the control
The probability that a point plots beyond
limits.
the control limits is again, 0.0027 (i.e., p =
• For our problem, even if the process remains in
0.0027). Then the average run length is
control, an out-of-control signal will be
1 generated every 370 samples, on average.
ARL = = 370
0.0027
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4-3.3 Sample Size and Sampling
4-3.4 Rational Subgroups
Frequency
Average Time to Signal
• Sometimes it is more appropriate to • Subgroups or samples should be selected
express the performance of the control so that if assignable causes are present, the
chart in terms of the average time to signal chance for differences between subgroups
(ATS). Say that samples are taken at fixed will be maximized, while the chance for
intervals, h hours apart. differences due to these assignable causes
within a subgroup will be minimized.
ATS = ARL ( h )
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4-3.4 Rational Subgroups 4-3.5 Analysis of Patterns on
Control Charts
Selection of Rational Subgroups
Nonrandom patterns can indicate out-of-control
• Select consecutive units of production.
conditions
– Provides a “snapshot” of the process. • Patterns such as cycles, trends, are often of considerable diagnostic
value (more about this in Chapter 5)
– Effective at detecting process shifts.
• Look for “runs” - this is a sequence of observations of the same type
• Select a random sample over the entire sampling (all above the center line, or all below the center line)
interval. • Runs of say 8 observations or more could indicate an out-of-control
situation.
– Can be effective at detecting if the mean has wandered – Run up: a series of observations are increasing
out-of-control and then back in-control. – Run down: a series of observations are decreasing
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4-3.5 Analysis of Patterns on
4-4. The Rest of the “Magnificent
Control Charts
Seven”
Western Electric Handbook Rules (Should be used
carefully because of the increased risk of false alarms) • The control chart is most effective when
integrated into a comprehensive SPC program.
A process is considered out of control if any of the
• The seven major SPC problem-solving tools
following occur:
should be used routinely to identify improvement
1) One point plots outside the 3-sigma control limits.
opportunities.
2) Two out of three consecutive points plot beyond the 2-
sigma warning limits. • The seven major SPC problem-solving tools
3) Four out of five consecutive points plot at a distance of 1- should be used to assist in reducing variability
sigma or beyond from the center line.
and eliminating waste.
4) Eight consecutive points plot on one side of the center line.
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4-4. The Rest of the “Magnificent 4-4. The Rest of the “Magnificent
Seven” Seven”
Recall the magnificent seven Check Sheets
1) Histogram or Stem and Leaf plot • See example, page 177 & 178
2) Check Sheet • Useful for collecting historical or current
3) Pareto Chart operating data about the process under
4) Cause and Effect Diagram investigation.
5) Defect Concentration Diagram
• Can provide a useful time-oriented summary of
6) Scatter Diagram
data
7) Control Chart
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4-4. The Rest of the “Magnificent
Seven”
Pareto Chart
• The Pareto chart is a frequency distribution (or histogram)
of attribute data arranged by category.
• Plot the frequency of occurrence of each defect type
against the various defect types.
• See example for the tank defect data, Figure 4-17, page
179
• There are many variations of the Pareto chart; see Figure
4-18, page 180
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Pareto Chart 4-4. The Rest of the “Magnificent
Seven”
40
Cause and Effect Diagram
30
• Once a defect, error, or problem has been identified and
isolated for further study, potential causes of this
20
Cylinder Heads
undesirable effect must be analyzed.
• Cause and effect diagrams are sometimes called fishbone
10
diagrams because of their appearance
• See the example for the tank defects, Figure 4-19, page
0
182
Size Surface Other
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4-4. The Rest of the “Magnificent
Example of Cause and Effect Diagram
Seven”
How to Construct a Cause-and-Effect Diagram (pg. 181) Activities Stops
• Define the problem or effect to be analyzed. Gas Station
Band practice
• Form the team to perform the analysis. Often the team
will uncover potential causes through brainstorming. Tau Beta Pi McDonalds
• Draw the effect box and the center line. Arrive late
to class
• Specify the major potential cause categories and join
them as boxes connected to the center line
Dog to the Vet
• Identify the possible causes and classify them into the Traffic
categories in step 4. Create new categories, if necessary. Kids to school
Parking
• Rank order the causes to identify those that seem most
likely to impact the problem. Family Commute
• Take corrective action.
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4-4. The Rest of the “Magnificent
4-4. The Rest of the “Magnificent
Seven”
Seven”
Defect Concentration Diagram
Scatter Diagram
• A defect concentration diagram is a picture of
the unit, showing all relevant views. • The scatter diagram is a plot of two variables
that can be used to identify any potential
• Various types of defects that can occur are
relationship between the variables.
drawn on the picture
• The shape of the scatter diagram often indicates
• See example, Figure 4-20, page 183
what type of relationship may exist.
• The diagram is then analyzed to determine if the
• See example, Figure 4-22 on page 184.
location of the defects on the unit provides any
useful information about the potential causes of
the defects.
Introduction to Statistical Quality Control, 4th Edition Introduction to Statistical Quality Control, 4th Edition
Uncorrelated and
(Positively) Correlated data
x[i-1]
x[i-1]
3
12
2 10
8
1
6
-2 -1 1 2 3 4
x[i]
-1
x[i]
2
-2 2 4 6 8 10 12
Introduction to Statistical Quality Control, 4th Edition