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Introduction to Statistical Quality Control, 4th Edition
Chapter 4
Methods and Philosophy ofStatistical Process Control
Introduction to Statistical Quality Control, 4th Edition
4-1. Introduction
• Statistical process control is acollection of tools that when usedtogether can result in process stabilityand variability reduction
Introduction to Statistical Quality Control, 4th Edition
4-1. Introduction
The seven major tools are
1) Histogram or Stem and Leaf plot2) Check Sheet3) Pareto Chart4) Cause and Effect Diagram5) Defect Concentration Diagram6) Scatter Diagram7) Control Chart
Introduction to Statistical Quality Control, 4th Edition
4-2. Chance and AssignableCauses of Quality Variation
• A process that is operating with only chancecauses of variation present is said to be instatistical control.
• A process that is operating in the presence ofassignable causes is said to be out of control.
• The eventual goal of SPC is reduction orelimination of variability in the process byidentification of assignable causes.
Introduction to Statistical Quality Control, 4th Edition
4-2. Chance and AssignableCauses of Quality Variation
Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control Chart
Basic PrinciplesA typical control chart has control limits set atvalues such that if the process is in control,nearly all points will lie between the uppercontrol limit (UCL) and the lower control limit(LCL).
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Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control Chart
Basic Principles
Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control Chart
Out-of-Control Situations
• If at least one point plots beyond the controllimits, the process is out of control
• If the points behave in a systematic ornonrandom manner, then the process couldbe out of control.
Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control Chart
Relationship between hypothesis testing andcontrol charts
• We have a process that we assume the true processmean is � = 74 and the process standard deviationis � = 0.01. Samples of size 5 are taken giving astandard deviation of the sample average, , asx
0045.0501.0
nx ��
�
��
Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control Chart
Relationship between hypothesis testing andcontrol charts
• Control limits can be set at 3 standarddeviations from the mean.
• This results in “3-Sigma Control Limits” UCL = 74 + 3(0.0045) = 74.0135 CL= 74 LCL = 74 - 3(0.0045) = 73.9865
Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control Chart
Relationship between hypothesis testing andcontrol charts
• Choosing the control limits is equivalent tosetting up the critical region for testinghypothesis
H0: � = 74 H1: � � 74
Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control ChartRelationship between the process and the control
chart
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Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control ChartImportant uses of the control chart• Most processes do not operate in a state of statistical control.• Consequently, the routine and attentive use of control charts will
identify assignable causes. If these causes can be eliminated fromthe process, variability will be reduced and the process will beimproved.
• The control chart only detects assignable causes. Management,operator, and engineering action will be necessary to eliminate theassignable causes.
• Out-of-control action plans (OCAPs) are an important aspect ofsuccessful control chart usage (see page 160).
• Refer to the process improvement cycle, Figure 4-5, page 160.
Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control Chart
Types the control chart• Variables Control Charts
– These charts are applied to data that follow acontinuous distribution (measurement data).
• Attributes Control Charts– These charts are applied to data that follow a
discrete distribution.
Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control Chart
Type of Process Variability – see Figure4-6, pg. 162
• Stationary behavior, uncorrelated data• Stationary behavior, autocorrelated data• Nonstationary behavior
Introduction to Statistical Quality Control, 4th Edition
Stationary, uncorrelated
Introduction to Statistical Quality Control, 4th Edition
Stationary, correlated
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Introduction to Statistical Quality Control, 4th Edition
Non-stationary
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Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control Chart
Type of Variability
• Shewhart control charts are most effectivewhen the in-control process data isstationary and uncorrelated.
Introduction to Statistical Quality Control, 4th Edition
4-3. Statistical Basis of the Control ChartPopularity of control charts
1) Control charts are a proven technique for improvingproductivity.
2) Control charts are effective in defect prevention.3) Control charts prevent unnecessary process adjustment.4) Control charts provide diagnostic information.5) Control charts provide information about process
capability.
Introduction to Statistical Quality Control, 4th Edition
4-3.2 Choice of Control Limits
General model of a control chart
where L = distance of the control limit from thecenter line
= mean of the sample statistic, w. = standard deviation of the statistic, w.
WW
W
WW
LLCLLineCenter
LUCL
����
��
����
W�
W�
Introduction to Statistical Quality Control, 4th Edition
4-3.2 Choice of Control Limits
“99.7% of the Data”• If approximately 99.7% of the data lies within 3�
of the mean (i.e., 99.7% of the data should liewithin the control limits), then 1 - 0.997 = 0.003or 0.3% of the data can fall outside 3� (or 0.3%of the data lies outside the control limits).(Actually, we should use the more exact value0.0027)
• 0.0027 is the probability of a Type I error or afalse alarm in this situation.
Introduction to Statistical Quality Control, 4th Edition
4-3.2 Choice of Control Limits
Three-Sigma Limits
• The use of 3-sigma limits generally gives goodresults in practice.
• If the distribution of the quality characteristic isreasonably well approximated by the normaldistribution, then the use of 3-sigma limits isapplicable.
• These limits are often referred to as action limits.
Introduction to Statistical Quality Control, 4th Edition
4-3.2 Choice of Control Limits
Warning Limits on Control Charts
• Warning limits (if used) are typically set at 2 standarddeviations from the mean.
• If one or more points fall between the warning limits andthe control limits, or close to the warning limits theprocess may not be operating properly.
• Good thing: warning limits often increase the sensitivityof the control chart.
• Bad thing: warning limits could result in an increasedrisk of false alarms.
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Introduction to Statistical Quality Control, 4th Edition
4-3.3 Sample Size and Sampling Frequency
• In designing a control chart, both thesample size to be selected and thefrequency of selection must be specified.
• Larger samples make it easier to detectsmall shifts in the process.
• Current practice tends to favor smaller,more frequent samples.
Introduction to Statistical Quality Control, 4th Edition
4-3.3 Sample Size and Sampling Frequency
Average Run Length• The average run length (ARL) is a very
important way of determining the appropriatesample size and sampling frequency.
• Let p = probability that any point exceeds thecontrol limits. Then,
p1ARL �
Introduction to Statistical Quality Control, 4th Edition
4-3.3 Sample Size and Sampling Frequency
Illustration• Consider a problem with control limits set
at 3standard deviations from the mean.The probability that a point plots beyondthe control limits is again, 0.0027 (i.e., p =0.0027). Then the average run length is
3700027.01ARL ��
Introduction to Statistical Quality Control, 4th Edition
4-3.3 Sample Size and Sampling Frequency
What does the ARL tell us?• The average run length gives us the length of
time (or number of samples) that should plot incontrol before a point plots outside the controllimits.
• For our problem, even if the process remains incontrol, an out-of-control signal will begenerated every 370 samples, on average.
Introduction to Statistical Quality Control, 4th Edition
4-3.3 Sample Size and Sampling Frequency
Average Time to Signal• Sometimes it is more appropriate to
express the performance of the controlchart in terms of the average time to signal(ATS). Say that samples are taken at fixedintervals, h hours apart.
)h(ARLATS �
Introduction to Statistical Quality Control, 4th Edition
4-3.4 Rational Subgroups
• Subgroups or samples should be selectedso that if assignable causes are present, thechance for differences between subgroupswill be maximized, while the chance fordifferences due to these assignable causeswithin a subgroup will be minimized.
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Introduction to Statistical Quality Control, 4th Edition
4-3.4 Rational Subgroups
Selection of Rational Subgroups
• Select consecutive units of production.– Provides a “snapshot” of the process.– Effective at detecting process shifts.
• Select a random sample over the entire samplinginterval.– Can be effective at detecting if the mean has wandered
out-of-control and then back in-control.
Introduction to Statistical Quality Control, 4th Edition
4-3.5 Analysis of Patterns on Control Charts
Nonrandom patterns can indicate out-of-controlconditions
• Patterns such as cycles, trends, are often of considerable diagnosticvalue (more about this in Chapter 5)
• Look for “runs” - this is a sequence of observations of the same type(all above the center line, or all below the center line)
• Runs of say 8 observations or more could indicate an out-of-controlsituation.
– Run up: a series of observations are increasing– Run down: a series of observations are decreasing
Introduction to Statistical Quality Control, 4th Edition
4-3.5 Analysis of Patterns on Control Charts
Western Electric Handbook Rules (Should be usedcarefully because of the increased risk of false alarms)
A process is considered out of control if any of thefollowing occur:1) One point plots outside the 3-sigma control limits.2) Two out of three consecutive points plot beyond the 2-
sigma warning limits.3) Four out of five consecutive points plot at a distance of 1-
sigma or beyond from the center line.4) Eight consecutive points plot on one side of the center line.
Introduction to Statistical Quality Control, 4th Edition
4-4. The Rest of the “Magnificent Seven”
• The control chart is most effective whenintegrated into a comprehensive SPC program.
• The seven major SPC problem-solving toolsshould be used routinely to identify improvementopportunities.
• The seven major SPC problem-solving toolsshould be used to assist in reducing variabilityand eliminating waste.
Introduction to Statistical Quality Control, 4th Edition
4-4. The Rest of the “Magnificent Seven”
Recall the magnificent seven 1) Histogram or Stem and Leaf plot
2) Check Sheet3) Pareto Chart4) Cause and Effect Diagram5) Defect Concentration Diagram6) Scatter Diagram7) Control Chart
Introduction to Statistical Quality Control, 4th Edition
4-4. The Rest of the “Magnificent Seven”
Check Sheets• See example, page 177 & 178• Useful for collecting historical or current
operating data about the process underinvestigation.
• Can provide a useful time-oriented summary ofdata
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Introduction to Statistical Quality Control, 4th Edition Introduction to Statistical Quality Control, 4th Edition
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
Introduction to Statistical Quality Control, 4th Edition
Pareto Chart
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Cylinder Heads
Introduction to Statistical Quality Control, 4th Edition
4-4. The Rest of the “Magnificent Seven”
Cause and Effect Diagram
• Once a defect, error, or problem has been identified andisolated for further study, potential causes of thisundesirable effect must be analyzed.
• Cause and effect diagrams are sometimes called fishbonediagrams because of their appearance
• See the example for the tank defects, Figure 4-19, page182
Introduction to Statistical Quality Control, 4th Edition
4-4. The Rest of the “Magnificent Seven”
How to Construct a Cause-and-Effect Diagram (pg. 181)• Define the problem or effect to be analyzed.• Form the team to perform the analysis. Often the team
will uncover potential causes through brainstorming.• Draw the effect box and the center line.• Specify the major potential cause categories and join
them as boxes connected to the center line• Identify the possible causes and classify them into the
categories in step 4. Create new categories, if necessary.• Rank order the causes to identify those that seem most
likely to impact the problem.• Take corrective action.
Introduction to Statistical Quality Control, 4th Edition
Example of Cause and Effect Diagram
Arrive lateto class
CommuteFamily
Activities Stops
Dog to the Vet
Kids to school
Traffic
Parking
Band practice
Tau Beta Pi
Gas Station
McDonalds
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Introduction to Statistical Quality Control, 4th Edition
4-4. The Rest of the “Magnificent Seven”
Defect Concentration Diagram• A defect concentration diagram is a picture of
the unit, showing all relevant views.• Various types of defects that can occur are
drawn on the picture• See example, Figure 4-20, page 183• The diagram is then analyzed to determine if the
location of the defects on the unit provides anyuseful information about the potential causes ofthe defects.
Introduction to Statistical Quality Control, 4th Edition
4-4. The Rest of the “Magnificent Seven”Scatter Diagram• The scatter diagram is a plot of two variables
that can be used to identify any potentialrelationship between the variables.
• The shape of the scatter diagram often indicateswhat type of relationship may exist.
• See example, Figure 4-22 on page 184.
Introduction to Statistical Quality Control, 4th Edition
Uncorrelated and(Positively) Correlated data
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