Statistical Process Control

1
Statistical Process Control

• Sample output of process and
• make inferences about its state
• Demonstrate that the
• distribution of process output
• is known and unchanging
• Plot and monitor over time
• Use statistical tests to detect shifts
• and anomalies and react to them quickly
• Use statistical evidence to guide and
• confirm process improvements

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Evolution from Inspection to SPC
SPC.
SPC.
SQC/SPC.
3
Statistical Process Control Topics

• Introduction to Variability
• Control Charts
• General info
• x-bar Charts, R Charts
• p Charts, C Charts
• Type 1 and Type 2 Errors
• Process Capability Analysis

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Variation
The less variation, the better off we are.
Improve Capabilities
Common cause variation Inherent in the
system Assignable cause variation
Event-related, special (assignable special)
Analyze and Act
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Analyze and Act React to Assignable Causes

• Note unusual variation diagnosed by using a
  common test to evaluate individual data points
• Identify cause by noting what change in the
  process occurred at that point in time
• Eliminate cause or build in the cause
• Monitor performance to verify the effect of the
  fix
• Generally, assignable causes cause points outside
  of control limits!

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Improve Process Capabilities Drive Out Common
Causes

• Variation is inherent in the system
• Dont react to individual points (this is
  tampering)
• Analyze possible factors affecting variation (use
  Cause and Effect Diagram)
• Work to reduce variation Make an improvement,
  that is, introduce a special cause
• Monitor performance to verify the effect of the
  intended improvement

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X-bar and R charts

• Sample output of process - parameter
• of interest is continuously variable
• Plot one chart to track sample means and
• another one to track sample ranges (variation)
• Use statistical evidence to detect changes
• and improve the process to better position
• the mean and to reduce variation

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Underlying Assumptions

• process mean m and standard deviation s when the
  process is in control
• process may go out of control in two possible
  ways
• mean shifts to m1, with standard deviation
  unchanged
• standard deviation shifts to s1, with mean
  unchanged
• sample means are normally distributed (when in or
  out of control, because either
• process output measurements on individual units
  are normally distributed when in or out of
  control
• OR Central Limit Theorem applies
• n gt 30 OR
• If distribution unimodal or symmetric, then much
  smaller ns are acceptable to assume normality (n
  on the order of 4).

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Basic Probabilities Concerning the Distribution
of Sample Means
Std. dev. of the sample means
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Estimation of Mean and Std. Dev. of the
Underlying Process

• use historical data taken from the process when
  it was known to be in control
• usually data is in the form of samples
  (preferably with fixed sample size) taken at
  regular intervals
• process mean m estimated as the average of the
  sample means (the grand mean)
• process standard deviation s estimated by
• standard deviation of all individual samples
• OR mean of sample range R/d2, where
• sample range R max. in sample minus min. in
  sample
• and d2 value from look-up table (Appendix
  Table B)

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Example Estimation of Mean and Std. Dev. of the
Underlying Process
Estimate of the process mean m 2.3 Estimate
of the process std. dev. (1) Combined std.
dev. of all 30 points s 1.1 OR (2) s
R/d2 (n5) 2.7/2.326 1.2
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Determination of Control Limits

• For the x-bar chart
• - Center Line grand mean
• - Control Limits usually use
• - Can analyze process capability based on the
  specification limits
• For the R chart
• - Center Line average range
• - Control Limits
• Use Table C (Appendix)
• Alternative Use an Economic Approach
• - Consider the cost impact of out-of-control
  detection delay (Type 2 error), false alarm (Type
  1 error) and sampling costs
• - Difficult to estimate costs

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Ex. Two Machines -Process Capability Analysis and
x-bar and R-charts
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X-bar vs. R charts

• R charts monitor variability Is the variability
  of the process stable over time? Do the items
  come from one distribution?
• X-bar charts monitor centering (once the R chart
  is in control) Is the mean stable over time?
• gtgt Bring the R-chart under control, then look
• at the x-bar chart

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How to Construct a Control Chart

• 1. Take samples and measure them.
• 2. For each subgroup, calculate the sample
  average and range.
• 3. Set trial center line and control limits.
• 4. Plot the R chart. Remove out-of-control
  points and revise control limits.
• 5. Plot x-bar chart. Remove out-of-control
  points and revise control limits.
• 6. Implement - sample and plot points at standard
  intervals. Monitor the chart.

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Type 1 and Type 2 Error
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Common Tests to Determine if the Process is Out
of Control

• One point outside of either control limit
• 2 out of 3 points beyond UCL - 2 sigma
• 7 successive points on same side of the central
  line
• of 11 successive points, at least 10 on the same
  side of the central line
• of 20 successive points, at least 16 on the same
  side of the central line
• Shows particular pattern