Post on 03-Feb-2020
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STATISTICAL QUALITY CONTROL
Majid Rafiee
Department of Industrial Engineering
Sharif University of Technology
rafiee@sharif.ir
How Statistical Process Control (SPC) works
Copyright Notice
• Parts (text & figures) of this lecture adopted from:
• Introduction to statistical quality control, 5th edition, by douglas C.
Montgomery , arizona state university
• Https://www.Wikipedia.Org
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Outline
4
Introduction
Chance & Assignable Cause of Quality Variation
Magnificent seven
Implementing SPC in a quality Improvement program
Application of SPC
Statistical basis of control chart
Outline
5
Introduction
Chance & Assignable Cause of Quality Variation
Magnificent seven
Implementing SPC in a quality Improvement program
Application of SPC
Statistical basis of control chart
Introduction
• Statistical quality control
• Monitoring various stages of production
• Monitoring of process
• Means to determine major source of observed variation
• Chance variation
• Inevitable
• Assignable cause
• Detected & corrected by appropriate tools
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Outline
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Introduction
Chance & Assignable Cause of Quality Variation
Magnificent seven
Implementing SPC in a quality Improvement program
Application of SPC
Statistical basis of control chart
Chance & Assignable Cause of Quality Variation
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Causes
Chance causes Assignable causes
Chance & Assignable Cause of Quality Variation
• Chance causes
• In random fashion
• inevitable
• Assignable causes
• Can be assigned to any particular cause
• Defective materials, defective labour, ..
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Chance & Assignable Cause of Quality Variation
• A process is operating with only chance causes of variation present
is said to be in statistical control.
• A process that is operating in the presence of assignable causes is
said to be out of control.
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Chance & Assignable Cause of Quality Variation
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Outline
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Introduction
Chance & Assignable Cause of Quality Variation
Magnificent seven
Implementing SPC in a quality Improvement program
Application of SPC
Statistical basis of control chart
Basic SPC Tools
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SPC tools
Histogram or steam-and-leaf
plot
Check sheet
Pareto chart
Cause & effect
diagram
Defect concentra
tion diagram
Scatter diagram
Control chart
Basic SPC Tools
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SPC tools
Histogram or steam-and-leaf
plot
Check sheet
Pareto chart
Cause & effect
diagram
Defect concentra
tion diagram
Scatter diagram
Control chart
Histogram or steam-and-leaf plot
• Displaying the relative density and shape of the data
• Giving the reader a quick overview of distribution
• Highlighting outliers
• Finding the mode
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Basic SPC Tools
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SPC tools
Histogram or steam-and-leaf
plot
Check sheet
Pareto chart
Cause & effect
diagram
Defect concentra
tion diagram
Scatter diagram
Control chart
Check sheet
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• It is a form used to collect data (quantitative or qualitative)
• In real time
• At the location where the data is generated
Check sheet
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Check sheet
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Basic SPC Tools
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SPC tools
Histogram or steam-and-leaf
plot
Check sheet
Pareto chart
Cause & effect
diagram
Defect concentra
tion diagram
Scatter diagram
Control chart
Pareto chart
• To assess the most frequently
occurring defects
• 80/20 rule
• ABC analysis
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Pareto chart
• To assess the most frequently
occurring defects
• 80/20 rule
• ABC analysis
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Various examples of Pareto charts
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Basic SPC Tools
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SPC tools
Histogram or steam-and-leaf
plot
Check sheet
Pareto chart
Cause & effect
diagram
Defect concentra
tion diagram
Scatter diagram
Control chart
Cause & effect diagram
• Visualization tool
• Categorizing the potential causes of a problem
• The design of the diagram looks like a skeleton of a fish
• Fishbone diagram
• Ishikawa diagram
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Cause & effect diagram
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How to Construct a Cause & effect diagram
• Define the problem to be analyzed
• Draw the effect box & the center line
• Specify major potential cause categories and join them to the center
line
• Identify possible causes & classify them into categories in previous
step
• Rank order the causes
• Take corrective actions
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Basic SPC Tools
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SPC tools
Histogram or steam-and-leaf
plot
Check sheet
Pareto chart
Cause & effect
diagram
Defect concentra
tion diagram
Scatter diagram
Control chart
Defect concentration diagram
• a graphical tool
• It is a drawing of the product
• all relevant views displayed
• Shows locations and
frequencies of various defects
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Adopted from: http://www.syque.com/improvement/Location%20Plot.htm
Defect concentration diagram
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Basic SPC Tools
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SPC tools
Histogram or steam-and-leaf
plot
Check sheet
Pareto chart
Cause & effect
diagram
Defect concentra
tion diagram
Scatter diagram
Control chart
Scatter diagram
• To identify type of
relationship between two
quantitative variables
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Basic SPC Tools
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SPC tools
Histogram or steam-and-leaf
plot
Check sheet
Pareto chart
Cause & effect
diagram
Defect concentra
tion diagram
Scatter diagram
Control chart
Control chart & Specifications
• a statistical process control tool used to determine if a
manufacturing or business process is in a state of control.
• Specifications
• Lower specification limit
• Upper specification limit
• Target or nominal values
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Control chart & Specifications
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Outline
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Introduction
Chance & Assignable Cause of Quality Variation
Magnificent seven
Implementing SPC in a quality Improvement program
Application of SPC
Statistical basis of control chart
Statistical Basis of Control chart & Specifications
• A point that plots within the control limits indicates the process is in
control
• No action is necessary
• A point that plots outside the control limits is evidence that the process
is out of control
• Investigation and corrective action are required to find and eliminate
assignable cause
• There is a close connection between control charts and hypothesis
testing
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Shewhart Control Chart Model
• Let w be a sample statistic that measures some CTQ
• Let L be the “distance” of the control limits from the center line
• Expressed in standard deviation units
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Photolithography Example
• Important quality characteristic in hard bake is resist flow width
• Process is monitored by average flow width sample of 5 wafers
• Process mean is 1.5 microns
• Process standard deviation is 0.15 microns
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Photolithography Example
𝝈𝒂𝒗𝒆𝒓𝒂𝒈𝒆 𝒐𝒇 𝒙 =𝝈𝒙
𝒏⇒ 𝝈𝒂𝒗𝒆𝒓𝒂𝒈𝒆 𝒐𝒇 𝒙 =
𝟎. 𝟏𝟓
𝟓= 𝟎. 𝟎𝟔𝟕𝟏
To assume x-bar is approximately normally distributed, we would
assume 𝟏𝟎𝟎 𝟏 − 𝜶 % of the sample fall btw 𝟏. 𝟓 + 𝒁𝜶
𝟐(𝟎. 𝟎𝟔𝟏𝟕) and
𝟏. 𝟓 − 𝒁𝜶
𝟐(𝟎. 𝟎𝟔𝟏𝟕)
𝑼𝑪𝑳 = 𝟏. 𝟓 + 𝟑(𝟎. 𝟎𝟔𝟕𝟏)
𝑪𝑳 = 𝟏. 𝟓
𝑳𝑪𝑳 = 𝟏. 𝟓 − 𝟑(𝟎. 𝟎𝟔𝟕𝟏)
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Photolithography Example
• Note that all plotted points
fall inside the control limits
• Process is considered to be
in statistical control
• Called three sigma
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Distribution of X-bar vs X
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Process improvement using control chart
• Using control charts consequently
• Identify assignable causes
• Eliminate causes
• Reducing variation
• Improving quality
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Out of Control Action Plan (OCAP)
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• a companion to the control chart
• reactions to out-of-control
situations
Out of Control Action Plan (OCAP)-1
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Out of Control Action Plan (OCAP)-2
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Types of Control Charts
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Control Charts
Variables Control Charts Attributes Control Charts
Review over Classifying Data On Quality Characteristics
• Variables
• Often continuous measurements
• Length, voltage, viscosity
• Following continuous distribution
• Attributes
• Usually discrete data
• Often taking the form of counts
• Following discrete distribution
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Types of Control Charts
• Variables Control Charts
• Applied to data following continuous distribution
• Attributes Control Charts
• Applied to data following discrete distribution
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Types of Control Charts
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Control Charts
Variables Control Charts Attributes Control Charts
Variables Control Charts
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Variables Control Charts
X-bar chart R chart S chart S**2 chart
Types of Control Charts
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Control Charts
Variables Control Charts Attributes Control Charts
Attributes Control Charts
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Attributes Control Charts
C chart U chart Np chart P chart
Control chart design
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Control chart design
selection of sample size & sampling frequency
control limits
Control chart design
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Control chart design
selection of sample size & sampling frequency
control limits
Types of Process Variability
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Process Variability
Stationary & uncorrelated
NonstationaryStationary & auto correlated
Types of Process Variability
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Process Variability
Stationary & uncorrelated
NonstationaryStationary & auto correlated
Stationary & uncorrelated Process Variability
• Data vary around a fixed mean
in a stable or predictable
manner
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Types of Process Variability
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Process Variability
Stationary & uncorrelated
NonstationaryStationary & auto correlated
Stationary & auto correlated Process Variability
• Successive observations are
dependent with tendency to
move in long runs on either
side of mean
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Types of Process Variability
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Process Variability
Stationary & uncorrelated
NonstationaryStationary & auto correlated
Nonstationary Process Variability
• process drifts without any
sense of a stable or fixed
mean
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Types of Errors
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Error Types
Type I Type II
Types of Errors
• Type I
• denoted by αlpha
• is the rejection of a true null hypothesis
• Type II
• denoted by Beta
• is the failure to reject a false null hypothesis
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Average Run Length (ARL)
• Average # of points that must be plotted before a point indicates an
out-of-control condition
• Let P be the probability that any point exceeds control limits
• If observations are uncorrelated, then
ARL = 1/P
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Average Run Length (ARL) in Control
• In 3-sigma
• P=0.0027 (alpha)
• The probability that a single point falls out of limits, when process is in
control
• Then the average run length of x-bar chart when process is in control
is
𝐴𝑅𝐿0 =1
𝑃=
1
0.0027= 370
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Average Run Length (ARL) Out of Control
• In our Photolithography Example
• Mean has been shifted to 1.75 instead of 1.5
• Process is out of control
• Probability of single point of x-bar being btw control limits via changed mean
is 0.5 (1-Beta)
• Then the average run length of x-bar chart when process is out of
control is
𝐴𝑅𝐿𝟏 =1
𝑃=
1
0. 𝟓= 𝟐
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Average Run Length (ARL) Distribution
• Distribution of run length for a Shewhart control chart is geometric
distribution
• Standard deviation is very large
• Geometric distribution is very skewed
• Mean of distribution (ARL) is not necessarily a very typical value of run length
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Average time to signal
• if samples are taken at fixed intervals of time that are h hours apart
ATS = ARL * h
• If the time interval btw samples is h=1 hour, the average time
required to detect the shift is
𝐴𝑇𝑆1 = 𝐴𝑅𝐿1ℎ = 2 1 = 2 ℎ𝑜𝑢𝑟𝑠
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Sample Size & Sampling Frequency
• If sampling frequency is fixed, then as sample size increases, out of
control ARL & ATS decreases
• E.g. if n = 10, instead of n=5 :
𝐴𝑅𝐿1=
1
𝑝
=1
0.9
= 1.11
𝐴𝑇𝑆1 = 𝐴𝑅𝐿1ℎ = 1.11 1 = 1.11 ℎ𝑜𝑢𝑟𝑠
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Sample Size & Sampling Frequency
• If it became important to detect shift in approximately first hour
after it occurred, two control chart designs would work:
• Design1
• Sampling size: n=5
• Sampling Frequency: every half hour
• Design2
• Sampling size: n=10
• Sampling Frequency: every hour
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Control chart design
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Control chart design
selection of sample size & sampling frequency
control limits
Choice of Control Limits
• 3-Sigma Control Limits
• Probability of type I error is 0.0027
• Probability Limits
• Type I error probability is chosen directly
• For example, 0.001 gives 3.09-sigma control limits
• Warning Limits
• Typically selected as 2-sigma limits
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Choice of Control Limits
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rational subgroup
• Subgroups or samples should be selected so that
• If assignable causes are present, chance for differences between
subgroups will be maximized
• Chance for difference due to assignable causes within a subgroup will be
minimized.
• Two general approaches for constructing rational subgroups:
• Consecutive units
• Random sample of all process output over sampling interval
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consecutive units
• Sample consists of units produced at the same time
• Primary purpose is to detect process shifts
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random sample
• random sample of all process output over sampling interval
• Sample consists of units that are representative of all units
produced since last sample
• Often used to make decisions about acceptance of product
• Effective at detecting shifts to out-of-control state and back into
in-control state between samples
• we can often make any process appear to be in statistical control
just by stretching out the interval between observations in the
sample.
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Different Approaches for Sampling
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Analysis of Patterns on Control Chart
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• Pattern is very nonrandom in
appearance
• 19 of 25 points plot below the center
line, while only 6 plot above
• Following 4th point, 5 points in a row
increase in magnitude, a run up
• There is also an unusually long run
down beginning with 18th point
Cyclic Pattern
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how to detect nonrandom patterns?
• Western Electronic Handbook, Set of decision rules
• 1. Any single data point falls outside the 3σ-limit from the centerline
• 2. Two out of three consecutive points fall beyond the 2σ-limit, on the
same side of the centerline
• 3. Four out of five consecutive points fall beyond the 1σ-limit, on the
same side of the centerline
• 4. Nine consecutive points fall on the same side of the centerline
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Discussion of Sensitizing Rules for Control Chart
• Suppose that analyst uses k decision rules & that criterion I has type
I error probability 𝛼𝑖 , then the overall type I error based on k test is:
𝛼 = 1 −ෑ
𝑖=1
𝑘
(1 − 𝛼𝑖)
• Q) What are the consequences of increasing overall error type I?
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Control Chart Application
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Control Chart Application
Phase I Phase II
Control Chart Application
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Control Chart Application
Phase I Phase II
Phase I
• Phase I is a retrospective analysis of process data to construct trial
control limits
• Charts are effective at detecting large, sustained shifts in process
parameters, outliers, measurement errors, data entry errors, etc.
• Facilitates identification and removal of assignable causes
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Control Chart Application
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Control Chart Application
Phase I Phase II
Phase II
• In this phase, the control chart is used to monitor the process
• Process is assumed to be reasonably stable
• Emphasis is on process monitoring, not on bringing an unruly
process into control
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Outline
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Introduction
Chance & Assignable Cause of Quality Variation
Magnificent seven
Implementing SPC in a quality Improvement program
Application of SPC
Statistical basis of control chart
Implementing SPC
• Elements of a successful SPC program
• Management leadership
• A team approach
• Education of employees at all levels
• Emphasis on reducing variability
• Measuring success in quantitative terms
• A mechanism for communicating successful results
throughout the organization
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Outline
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Introduction
Chance & Assignable Cause of Quality Variation
Magnificent seven
Implementing SPC in a quality Improvement program
Application of SPC
Statistical basis of control chart
Application of SPC
• Industrial applications
• Nonmanufacturing applications
• Sometimes require ingenuity
• Mostly do not have a natural measurement system
• The observability of the process may be fairly low
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process mapping
• Flow charts & operation process charts are particularly useful in
developing process definition & process understanding.
• This is sometimes called process mapping.
• Used to identify value-added versus nonvalue-added activity
• To eliminate non-value added activities
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Flow chart
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Back up
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