Methods and Philosophy of Statistical Process Control

1
Chapter 4

• Methods and Philosophy of Statistical Process
  Control

2
4-1. Introduction

• Statistical process control is a collection of
  tools that when used together can result in
  process stability and variability reduction

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4-1. Introduction

• The seven major tools are
• 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

4
4-2. Chance and Assignable Causes of Quality
Variation

• A process that 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.
• The eventual goal of SPC is reduction or
  elimination of variability in the process by
  identification of assignable causes.

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4-2. Chance and Assignable Causes of Quality
Variation
See Figure 4-1
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4-3. Statistical Basis of the 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 Control
Chart
Basic Principles
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4-3. Statistical Basis of the Control
Chart

• 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 Control
Chart

• Relationship between hypothesis testing and
  control charts
• Assume that the true process mean is ? 74 and
  that the process standard deviation is ? 0.01.
  Samples of size 5 are taken giving a standard
  deviation of the sample average as

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4-3. Statistical Basis of the Control
Chart

• Relationship between hypothesis testing and
  control charts
• Control limits can be set at 3 standard
  deviations 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

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4-3. Statistical Basis of the Control
Chart

• Relationship between hypothesis testing and
  control charts
• Choosing the control limits is equivalent to
  setting up the critical region for testing
  hypothesis
• H0 ? 74
• H1 ? ? 74

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4-3. Statistical Basis of the Control
Chart

• Relationship between the process and the control
  chart

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4-3. Statistical Basis of the Control
Chart

• Important 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 from the
  process, variability will be reduced and the
  process will be improved.
• The control chart only detects assignable causes.
  Management, operator, and engineering action
  will be necessary to eliminate the assignable
  causes.

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Process Improvement Using the Control Chart
Input
Output
Process
Measurement System
Verify and follow up
Detect assignable causes
Implement corrective action
Identify root cause of problem
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4-3. Statistical Basis of the Control
Chart

• Types the control chart
• Variables Control Charts
• These charts are applied to data that follow a
  continuous distribution (measurement data).
• Attributes Control Charts
• These charts are applied to data that follow a
  discrete distribution (count data).

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4-3. Statistical Basis of the Control
Chart

• Type of Process Variability see Figure 4-6,
  pg. 162
• Stationary behavior, uncorrelated data
• Stationary behavior, autocorrelated data
• Nonstationary behavior

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4-3. Statistical Basis of the Control
Chart

• Type of Variability
• Shewhart control charts are most effective when
  the in-control process data is stationary and
  uncorrelated.

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4-3. Statistical Basis of the Control
Chart

• Popularity of control charts
• 1) Control charts are a proven technique for
  improving productivity.
• 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.

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Control charts are a proven technique for
improving productivity

• Scrap and rework will be decreased
• Then productivity increases
• The production rate goes up

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Control charts are effective in defect prevention

• Keeps the process in control
• Helps to do it right the first time
• It is rarely cheaper to sort good units from
  bad later on than to build it right initially
• Dont pay to make a nonconforming part

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Control charts prevent unnecessary process
adjustment

• Distinguishes between background noise and
  abnormal variation
• Human operators cannot do this
• Human operators will overreact to stimulus
• They will fix it when it is not broken and make
  the system much worse

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Control charts provide diagnostic information

• Experienced operator can see problems in the
  making and take corrective action

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Control charts provide information about process
capability

• From observing variation, process capability can
  be determined
• Very important when selling the product

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4-3.2 Choice of Control Limits

• General model of a control chart
• where L distance of the control limit from
  the center line
• mean of the sample statistic, w.
• standard deviation of the statistic, w.

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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
  lie within the control limits), then 1 - 0.9973
  0.0027 or 0.3 of the data can fall outside 3?
  (or 0.3 of the data lies outside the control
  limits).
• 0.0027 is the probability of a Type I error or a
  false alarm in this situation.

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4-3.2 Choice of Control Limits

• Three-Sigma Limits
• The use of 3-sigma limits generally gives good
  results in practice.
• If the distribution of the quality characteristic
  is reasonably well approximated by the normal
  distribution, then the use of 3-sigma limits is
  applicable.
• These limits are often referred to as action
  limits.

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.001 probability limits

• UK and Western Europe uses this
• L 3.09 (instead of L 3)
• UCL 74 3.09(.0045) 74.0139
• LCL 74 - 3.09(.0045) 73.9861

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4-3.2 Choice of Control Limits

• Warning Limits on Control Charts
• Warning limits (if used) are typically set at 2
  standard deviations from the mean.
• If one or more points fall between the warning
  limits and the control limits, or close to the
  warning limits the process may not be operating
  properly.
• Good thing warning limits often increase the
  sensitivity of the control chart.
• Bad thing warning limits could result in an
  increased risk of false alarms.

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4-3.3 Sample Size and Sampling Frequency

• In designing a control chart, both the sample
  size to be selected and the frequency of
  selection must be specified.
• Larger samples make it easier to detect small
  shifts in the process.
• Current practice tends to favor smaller, more
  frequent samples.

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Use of OC curves

• See Figure 4-9
• Say that the process mean shifts from 74.0000 mm
  to 74.0100 mm
• Read b, probability of the mean falling within
  the control limits
• Let n 5, 10, 15
• Then b(when n 5) .80
• Then b(when n 10) .55
• Then b(when n 15) .25

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Power P(detection)

• Probability of detecting the shift on the first
  sample following the shift to 74.01
• P(detection when n 5) .20
• P(detection when n 10) .45
• P(detection when n 15) .75

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P(detection)

• Probability of not detecting the shift within the
  first six samples following the shift to 74.01
• P(not detecting when n 5) .806 .262
• P(not detecting when n 10) . 556 .028
• P(not detecting when n 15) .256 0

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P(detection)

• Probability of detecting the shift within the
  first six samples following the shift to 74.01
• P(detecting when n 5) 1 - .262 .738
• P(detecting when n 10) 1 - .028 .972
• P(detecting when n 15) 1 - 0 1

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4-3.3 Sample Size and Sampling Frequency

• Average Run Length
• The average run length for in control (ARL0) is a
  very important way of determining the appropriate
  sample size and sampling frequency.
• Let p probability that any point exceeds the
  control limits by chance alone.
• Then, ARL0 1/p

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4-3.3 Sample Size and Sampling Frequency

• Illustration
• Consider a problem with control limits set at 3
  standard deviations from the mean. The
  probability that a point plots beyond the control
  limits is, again, 0.0027 (i.e., p 0.0027).
• Then the average run length for in control is
• ARL0 1/.0027 370

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4-3.3 Sample Size and Sampling Frequency

• What does the ARL0 tell us?
• The average run length for in control gives us
  the length of time (or number of samples) that
  should plot in control before a point plots
  outside the control limits.
• For our problem, even if the process remains in
  control, an out-of-control signal will be
  generated every 370 samples, on average.

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4-3.3 Sample Size and Sampling Frequency

• Average Time to Signal
• Sometimes it is more appropriate to express the
  performance of the control chart in terms of the
  average time to signal (ATS). Say that samples
  are taken at fixed intervals, h hours apart.

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ATS

• So, for ARL0, the ATS is
• ATS 370h
• And, if a sample is taken every 30 minutes
• ATS 370(.5) 185 hours

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Average run length for out-of-control

• ARL1 1/Power
• ARL1(when n 5 mean shifts to 74.01)
  1/.2 5
• On the average, five samples will be needed to
  detect the shift

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4-3.4 Rational Subgroups

• Subgroups or samples should be selected so that
  if assignable causes are present, the chance for
  differences between subgroups will be maximized,
  while the chance for differences due to these
  assignable causes within a subgroup will be
  minimized.

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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 sampling
  interval.
• Can be effective at detecting if the mean has
  wandered out-of-control and then back in-control.

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4-3.5 Analysis of Patterns on Control
Charts

• Nonrandom patterns can indicate out-of-control
  conditions
• Patterns such as cycles and trends are often of
  considerable diagnostic value (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-control situation.
• Run up a series of observations are increasing
• Run down a series of observations are decreasing

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Patterns Terminology
Only 6 points above the CL
UCL
Run down of 6
Run up of 5
CL
LCL
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Patterns Cycles
UCL
CL
LCL
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Patterns Over control
UCL
CL
LCL
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Patterns Improper limits set
UCL
CL
LCL
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4-3.5 Analysis of Patterns on Control
Charts

• Western Electric Handbook Rules (Should be used
  carefully because of the increased risk of false
  alarms)
• A process is considered out of control if any of
  the
• following 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.

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Some probabilities

• P(point between 0 and 1s) .34134
• P(point between 1s and 2s) .13591
• P(point between 2s and 3s) .02140
• P(point between 0 and -1s) .34134
• P(point between -1s and -2s) .13591
• P(point between -2s and -3s) .02140

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Some probabilities

• P(point between 0 and 2s) .47725
• P(point between 0 and 3s) .49865
• P(point between 0 and -2s) .47725
• P(point between 0 and -3s) .49865
• P(point between 1s) .68268
• P(point between 2s) .95450
• P(point between 3s) .99730

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Some probabilities Exact sequence
.341344
UCL
CL
LCL
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Some probabilities Exact sequence
(.0214)2(.34134)2
UCL
CL
LCL
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Some probabilities Any 3 from 0s to s, any 2
from 1s to 2s
(5!/3!2!)(.34134)3(.13591)2
UCL
CL
LCL
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Some probabilities Any 3 from 0s to s, any 2
from 1s to 2s, any 1 from -2s to 3s
(6!/3!2!1!)(.34134)3(.13591)2(.0214)
UCL
CL
LCL
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4-4. The Rest of the Magnificent Seven

• The control chart is most effective when
  integrated into a comprehensive SPC program.
• The seven major SPC problem-solving tools should
  be used routinely to identify improvement
  opportunities.
• The seven major SPC problem-solving tools should
  be used to assist in reducing variability and
  eliminating waste.

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4-4. The Rest of the Magnificent Seven

• Recall the magnificent seven
• 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

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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 under
  investigation.
• Can provide a useful time-oriented summary of
  data

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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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4-4. The Rest of the Magnificent Seven

• Cause and Effect Diagram
• Once a defect, error, or problem has been
  identified and isolated for further study,
  potential causes of this undesirable effect must
  be analyzed.
• Cause and effect diagrams are sometimes called
  fishbone diagrams because of their appearance
• See the example for the tank defects, Figure
  4-19, page 182

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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.

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Classroom exercise

• My internet connection wont work
• My PC is connected to a LinkSys router
• The LinkSys router is connected to a VoiceStream
  DSL modem
• Earthlink provides my DSL (thru BellSouth)
• The VoiceStream DSL modem is connected to a
  telephone wall jack
• Prepare a fishbone diagram (groups of 4)

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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 any
  useful information about the potential causes of
  the defects.

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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 potential
  relationship between the variables.
• The shape of the scatter diagram often indicates
  what type of relationship may exist.
• See example, Figure 4-22 on page 184.

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Assignment

• Be sure that you understand everything about b,
  the ARL, and the probability associated with
  control charts.

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End