Dr. Jerrell T. Stracener, SAE Fellow

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Department of Engineering Management, Information
and Systems
EMIS 7370/5370 STAT 5340
Probability and Statistics for Scientists and
Engineers
An Application of Probability
Statistics Statistical Quality Control
UPDATED 11/20/06
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
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Statistical Quality Control
Statistical Quality Control is an application of
probabilitistic and statistical techniques to
quality control
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Statistical Quality Control - Elements
Analysis of process capability
Statistical process control
Process improvement
Acceptance sampling
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Statistical Quality Control - Basic Concepts

• Quality begins with customer requirements
• Quality must be designed in. It cannot be
  inspected
• in!
• Quality depends on
• Parts selection and procurement
• Material
• Manufacturing/production processes
• Logistic processes
• .
• .
• .

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Statistical Quality Control - What is quality?

• Quality is meeting the customers needs over the
• life cycle of the product at the best value to
  the
• customer
• Quality has many dimensions
• Reliability
• Maintainability
• Performance
• Durability
• Conformance (to requirements and
• expectations)
• .
• .
• .

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Obstacles to Quality Improvement

• Making it happen
• - 99 agree that management is the problem
• not the workers
• - 35 of the problem is not invented here
• syndrome
• getting their attention and education
• resistance to change
• - 15 gaining management commitment
• - 14 communication
• getting the word out within the company
• Failure of management to understand variation

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Statistical Process Control - Definition
The application of statistical techniques is
to understand and analyze the variation in a
process. - Joseph Juran Quality Control
Handbook
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The Situation
In many situations, our knowledge is limited to
the information that can be obtained from data
that has been obtained or that will be obtained
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The Problem
The challenge is to obtain the maximum
information from the data and to arrive at the
most accurate conclusions
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Nature of Data

• Most data are characterized by variation, as
  opposed
• to deterministic, due to variation in
• Processes and materials
• Product/Manufacturing
• Inspection Measurement
• Operation
• Environment
• etc

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Need

• Methods and techniques are needed for analysis of
• data that account for
• Variation in the data
• Uncertainty in conclusion

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Statistics

• Statistics is the science of analyzing data and
• drawing conclusions
• Statistical methods and techniques that provide
• tools for
• - experimental design
• - analysis of data
• - making inferences

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Statistical Process Control (SPC)

• SPC is a powerful collection of problem-solving
• tools useful in achieving process stability and
• improving capability through the reduction of
• variability.
• SPC can be applied to any process
• Seven major tools
• 1. Histogram or stem and leaf display
• 2. Check sheet
• 3. Pareto chart
• 4. Cause and effect diagram
• 5. Defect concentration diagram
• 6. Scatter diagram
• 7. Control chart

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Statistical Process Control
Causes of Variation Assignable (special) -
Intermittent sources of variation that are
unpredictable. Signaled by violation of Western
Electric rules Common (natural) - Sources of
variation always present affecting all output
from a process Only management can affect
common causes of variation
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Statistical Process Control - Histograms

• Histograms - Questions to ask
• What is the shape of distribution?
• What would you expect shape to be?
• If computer generated, is data really normal?
• Is variation acceptable?
• Is the centering acceptable?
• Did you generate a histogram with and without
• outlier points?
• Did you include specification limits and process
• limits on the histogram?

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Statistical Process Control - Histograms

• The shape shows the nature of the distribution of
  the data
• The central tendency (average) and variability
  are easily seen
• Specification limits can be used to display the
  capability of the process

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Process Capability
Refers to the uniformity of the
process. Variability in the process is a measure
of the uniformity of the output. -
Instantaneous variability is the natural
or inherent variability at a specified time -
Variability over time
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Process Capability
A critical performance measure that
addresses process results relative to
process/product specifications. A capable
process is one for which the process outputs meet
or exceed expectation.
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Measures of Process Capability
Customary to use the six sigma spread in
the distribution of the product quality
characteristic
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Key Points

• The proportion of the process output that will
  fall
• outside the natural tolerance limits.
• Is 0.27 (or 2700 nonconforming parts per
  million)
• if the distribution is normal
• May differ considerably from 0.27 if the
• distribution is not normal

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Process Capability Measures or Indices
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Measure of Potential Process Capability, Cp
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Measure of Potential Process Capability, CpK
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Statistical Process Control
ppm parts per million Interpretation CpK lt
1 process not capable 1 ? CpK lt 1.5
process capable, monitor frequently CpK ?
1.5 process capable, monitor infrequently Pa
reto CpKs to attack worst problems Can only
convert CpK, Cp to ppm if distribution normal
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Interpretation of Cp
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Statistical Process Control
Impact of special causes on process capability
process stable
process unstable
time
time
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Statistical Process Flow Diagram

• Expresses detailed knowledge of the process
• Identifies process flow and interaction among the
  process steps
• Identifies potential control points

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Statistical Process Control - Pareto Diagram
20 16 12 8 4 0
100 80 60 40 20 0
Number of occurrences
Cumulative percent

• Identifies the most significant problems to be
  worked first
• Historically 80 of the problems are due to 20
  of the factors
• Shows the vital few

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Statistical Process Control - Scatter Plot

• Identifies the relationship between two variables
• A positive, negative, or no relationship can be
  easily detected

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Background of Six Sigma

• Six Sigma is a business initiative first
  espoused by
• Motorola in the early 1990s.
• Six Sigma strategy involves the use of
  statistical
• tools within a structured methodology for
  gaining
• the knowledge needed to achieve better, faster,
  and less expensive products and services than
  the competition.
• A Six Sigma initiative in a company is designed
• to change the culture through breakthrough
• improvement by focusing on out-of-the-box
• thinking in order to achieve aggressive, stretch
  goals.

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Motorolas Six Sigma Ten Steps
1. Prioritize opportunities for
improvement 2. Select the appropriate
team 3. Describe the total process 4. Perform
measurement system analysis 5. Identify and
describe the potential critical
process 6. Isolate and verify the critical
processes 7. Perform process and measurement
system capability studies 8. Implement optimum
operating conditions and control
methodology 9. Monitor processes over
time/continuous improvement 10. Reduce common
cause variation toward achieving six sigma
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Product Specification
Lower Specification Limit
Upper Specification Limit
Nominal Specification
x
Target (Ideal level for use in product) Tolerance
(Product characteristic)
(Maximum range of variation of the product
characteristic that will still work in the
product.)
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Traditional US Approach to Quality
(Make it to specifications)
No-Good
No-Good
Loss ()
Good
x
T
USL
LSL
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Setting Specification Limits on Discrete
Components
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Variability Reduction

• Variability reduction is a modern concept of
  design
• and manufacturing excellence
• Reducing variability around the target value
  leads
• to better performing, more uniform, defect-free
• product
• Virtually eliminates rework and waste
• Consistent with continuous improvement concept

Dont just conform to specifications
Reduce variability around the target
accept
reject
reject
target
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True Impact of Product Variability

• Sources of loss
• - scrap
• - rework
• - warranty obligations
• - decline of reputation
• - forfeiture of market share
• Loss function - dollar loss due to deviation of
• product from ideal characteristic
• Loss characteristic is continuous - not a step
• function.

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Representative Loss Function Characteristics
Loss
Loss
Loss
x
x
x
T
X nominal is best L k (x - T)2
X smaller is better L k (x2)
X larger is better L k (1/x2)
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Variability-Loss Relationship
LSL
USL
Target
Loss
savings due to reduced variability
Maximum loss per item
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Loss Computation for Total Product Population
X nominal is best L k (x - T)2
Loss
Loss
x
x
T
T
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Statistical Tolerancing - Convention
Normal Probability Distribution
0.00135
0.9973
0.00135
Nominal
LTL
UTL
?3?
?-3?
?
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Statistical Tolerancing - Concept
x
LTL
UTL
Nominal
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Caution
For a normal distribution, the natural tolerance
limits include 99.73 of the variable, or put
another way, only 0.27 of the process output
will fall outside the natural tolerance limits.
Two points should be remembered 1. 0.27
outside the natural tolerances sounds small,
but this corresponds to 2700 nonconforming
parts per million. 2. If the distribution of
process output is non normal, then the
percentage of output falling outside ? ? 3? may
differ considerably from 0.27.
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Normal Distribution - Example
The diameter of a metal shaft used in a
disk-drive unit is normally distributed with mean
0.2508 inches and standard deviation 0.0005
inches. The specifications on the shaft have been
established as 0.2500 ? 0.0015 inches. We wish
to determine what fraction of the shafts produced
conform to specifications.
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Normal Distribution - Example Solution

f(x)
0.2500
nominal
0.2508
x
0.2515 USL
0.2485 LSL
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Normal Distribution - Example Solution
Thus, we would expect the process yield to be
approximately 91.92 that is, about 91.92 of
the shafts produced conform to specifications.
Note that almost all of the nonconforming shafts
are too large, because the process mean is
located very near to the upper specification
limit. Suppose we can recenter the manufacturing
process, perhaps by adjusting the machine, so
that the process mean is exactly equal to the
nominal value of 0.2500. Then we have
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Normal Distribution - Example Solution
f(x)
nominal
0.2515 USL
0.2485 LSL
x
0.2500
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Normal Distribution - Example
Using a normal probability distribution as a
model for a quality characteristic with the
specification limits at three standard
deviations on either side of the mean. Now it
turns out that in this situation the probability
of producing a product within these
specifications is 0.9973, which corresponds to
2700 parts per million (ppm) defective. This is
referred to as three-sigma quality performance,
and it actually sounds pretty good. However,
suppose we have a product that consists of an
assembly of 100 components or parts and all 100
parts must be non-defective for the product to
function satisfactorily.
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Normal Distribution - Example
The probability that any specific unit of
product is non-defective is 0.9973 x 0.9973 x .
. . x 0.9973 (0.9973)100 0.7631 That
is, about 23.7 of the products produced
under three sigma quality will be defective. This
is not an acceptable situation, because many high
technology products are made up of thousands of
components. An automobile has about 200,000
components and an airplane has several million!
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Weibull Distribution - Example
The random variable X can modeled by a Weibull
distribution with ? ½ and ? 1000. The spec
time limit is set at x 4000. What is the
proportion of items not meeting spec?
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Weibull Distribution - Example
The fraction of items not meeting spec is
That is, all but about 13.53 of the items will
not meet spec.
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Statistical Process Control - Control Charts
Interpretation based on Western Electric
rules 1. Analyze the chart by separating it into
equal zones above and below the centerline
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Statistical Process Control - Control Charts
2. A process is out of statistical control
if (a) any point is above or below the control
limits (b) two out of three points in a row in
zone A or above (c) four out of five
points in a row in zone B or above (d)
eight in a row in zone C or above
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Statistical Process Control - Control Charts

• In general specification limits should not be on
• control charts
• Data must be displayed in time sequence
• Management controls the natural variation
  between
• the control limits
• Do not tweak the process

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Statistical Process Control - Control Charts
x
UCL
x
x
x
x
x
x
x
x
CL
x
x
x
x
x
LCL

• Helps reduce variability
• Monitors performance over time
• Allows process corrections to prevent rejections
• Trends and out-of-control conditions are
  immediately detected

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The Normal Distribution and the Control Charts
Upper Control Limit
m3s
m2s
m1s
Center LineProcess Average
m
m-1s
m-2s
m-3s
Lower Control Limit
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General Model for the Shewhart Control Chart
UCL ?W K?W Center Line ?W LCL ?W -
K?W where W is a statistic that measures a
quality characteristic ?W is the mean of W ?W
is the standard deviation of W K is the distance
of the control limits from the center line, in
multiples of ?W
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Types of Error that Can Occur When Using Control
Charts
Actual State of Process
Only Common Causes
Special Causes
Control Chart Indicates
OutofControl
A False Alarm
B Correct Decision
C Correct Decision
D Failure to Detect
Control
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Use of Control Chart

• Control charts are a proven technique for
  improving productivity
• Control charts are effective in defect prevention
• Control charts prevent unnecessary process
  adjustment
• Control charts provide diagnostic information.
• Control charts provide information about process
  capability

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Types of Control Chart
Measurement (variables)
Data
Counts (attributes)
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Types of Control Chart
X
One
Moving Range
Measurement (variables)
X
Multiple
R
S
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Types of Control Chart
p
Defectives
np
Counts (attributes)
c
Defects
?