Shewharts Theory of Chance Cause Systems of Variation

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Shewharts Theory of Chance Cause Systems of
Variation
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Shewharts Chance Cause System

• All empirical data are generated by some type of
  process.
• Walter Shewhart referred to these processes as
  chance cause systems of variation.

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Shewharts Chance Cause System
Shewharts Chance Cause System
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Shewharts Chance Cause System

• All chance cause systems are made up of two types
  of chance causes which are referred to as
  explainable causes and unexplainable causes.
• Therefore, chance cause systems can be thought of
  as satisfying the pseudo equation
• chance cause systems explainable causes
  unexplainable causes.

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Explainable Causes

• Explainable causes . . . lie outside the process,
  and they contribute significantly to the total
  variation observed in performance measures.
• The variation created by explainable causes is
  usually unpredictable, but it is explainable
  after it has been observed.

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The HD-2 Filler Case Study
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The HD-2 Filler Case Study
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Unexplainable Causes

• Unexplainable causes . . . May be unidentified
  explainable causes or they may be random causes
  that belong to, or are inherent in, the process.
• If they are common, unexplained causes then they
  produce random variation in the behavior of the
  performance measurement, but the variation is
  consistent and predictable. The variation
  associated with unexplained random causes of
  variation has a statistical identity.
• The random variation produced by unexplainable
  causes is often referred to as noise, because
  there is no real change in process performance.
• Noise cannot be traced to a specific cause, and
  it is therefore, although predictable, it is
  unexplainable.

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Unexplainable Causes

• The random nature of unexplainable cause
  variation lends itself to the application of the
  statistical methods, while the chaotic or
  pattened variation produced by explainable causes
  may not.
• Therefore, the statistical methods can be used
  to characterize the inherent process variation
  and to develop tools to identify the presence of
  explainable causes.

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Unexplainable Causes

• It may seem counter intuitive at first to claim
  that the objective of process control is to
  achieve a state where the process behaves in a
  random fashion.
• There are numerous instances where we depend upon
  random behavior to predict results.
• Card games like bridge or poker
• Games of chance in Las Vegas

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Shewharts Chance Cause System -The Woodpeckers
and the Mules!
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Shewharts General Chance Cause System

• The variation due to the woodpeckers in constant
  causes systems is unexplainable, but predictable!
• The variation due to mules in a general chance
  cause system is explainable, but generally
  unpredictable!

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Shewharts Theory of Chance Cause Systems of
Variation
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• Basic Statistical Concepts for Constant Cause
  Systems

Woodpeckers Only !
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Constant Cause Systems

• Chance cause systems that are made up only of
  unexplainable random causes are referred to as
  constant cause systems. That is,
• constant cause systems unexplainable random
  causes.

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Constant Cause Systems

• Constant cause systems are equivalent to the
  assumption of independent, identically
  distributed (IID) random variables.

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Constant Cause Systems

• Although both unexplainable and explainable
  causes create variation in the performance
  measure of interest, unexplainable causes create
  controlled variation while explainable causes
  create uncontrolled variation.

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Constant Cause Systems

• Shewhart defined controlled variation in the
  following way.
• A phenomenon will be said to be controlled when,
  through the use of past experience, we can
  predict, at least within limits, how the
  phenomenon may be expected to vary in the future.
  Here it is understood that prediction within
  limits means that we can state, at least
  approximately, the probability that the observed
  phenomenon will fall within given limits.
• Constant cause systems are therefore controlled
  cause systems.

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The Empirical Rule

• It a remarkable fact that the following
  relationships are approximately true for almost
  any process distribution associated with a
  constant cause system.
• 60 to 75 of the process output lies between µ
  -?? and µ ?
• 90 to 98 of the process output lies between µ -
  2? and µ 2 ?
• 99 to 100 of the process output lies between µ
  - 3? and µ 3? .

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The Empirical Rule
The Empirical Rule is the Foundation for Shewhart
Control Charts
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Understanding and Analyzing a Chance Cause System
of Variation
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Rational Subgrouping

• One of the most important concepts to understand
  and master in order to use use the data from
  analytic studies to their full potential is the
  notion of rational subgroups.
• The key to extraction of information from data is
  asking and answering the right questions.
• This can only be achieved by fully understanding
  and exploiting the structure of the data obtained
  from the process.

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Rational Subgrouping

• Shewhart made the following important observation
  regarding rational subgroups.
• Obviously, the ultimate object is not only to
  detect trouble but also to find it, and such
  discovery naturally involves classification. The
  engineer who is successful in dividing his data
  initially into rational subgroups based on
  rational hypotheses is therefore inherently
  better off in the long run than the one who is
  not thus successful.

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Understanding a Chance Cause System of Variation

• Many chance cause systems can be rationalized by
  hypothesizing what factors are potentially
  creating the observed variation in the data.
• Factor 1
• Factor 2
• .
• Factor K
• Time
• Unknown factors
• Random, Unexplained variation

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Understanding a Chance Cause System of Variation

• For example in the aseptic filler case study
• Factor 1 - Lane (1-4)
• Factor 2 - Phase (1-2)
• Time Order of production
• Unknown factors - pump effect
• Random, Unexplained variation

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The Concept of Rational Subgrouping

• The General Structure of Rational Subgroups

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Rational Subgrouping

• The fundamental concept of rational subgrouping
  is to study the variation observed across
  subgroups that are defined in a meaningful way
  relative to the variation observed within the
  subgroups, in order to answer important
  questions.
• Rational subgroups represent samples from the
  process organized in some meaningful way relative
  to a region of space, time, subprocess or
  product.

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Rational Subgrouping

• In general, the statistical analysis methods that
  are constructed from the data contained in the
  rational subgroups are designed to answer the
  following question
• Is the variation in the performance measure
  observed across subgroups greater than predicted
  based on the variation observed within the
  subgroups?

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Rational Subgrouping

• For a constant cause system the variation within
  a subgroup is the same as the variation across
  subgroups.
• Therefore, if the assumption of a constant cause
  system is correct, it should be possible to
  predict the behavior of summary statistics, like
  sample averages, ranges, and standard deviations,
  across subgroups based on the homogeneous
  variation observed within subgroups.

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Evidence of Explainable Causes

• Data from a constant cause system of variation
  will display random, unexplainable variation both
  within and across rational subgroups.
• The range of variation due to constant causes
  will be within predictable statistical limits.
• Nonrandom patterns of variation appearing within
  or across the rational subgroups, that can be
  meaningfully interpreted within the context of
  the cause system of variation, provide evidence
  that explainable causes are affecting the data.

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The AttendanceManagement Case Study

• The effective management of employee attendance
  is an important management responsibility.
  Within a data entry process, it is important that
  all 10 data entry specialists scheduled for work
  are present or the system becomes backlogged, and
  important deadlines are missed. The supervisors
  had raised important concerns about the level of
  absenteeism among the employees within the
  department.
• Each employees attendance rate, defined as the
  percent of scheduled hours actually worked, is
  recorded each pay period. The employees are paid
  on a bi-weekly basis, and the payroll department
  maintains the employee attendance data.
  Attendance data were available for 22 consecutive
  pay periods. The actual data for the ten
  employees are presented in Table 8.1.

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The AttendanceManagement Case Study
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The AttendanceManagement Case Study

• There are two organizations or structures of the
  data that were exploited to answer important
  questions concerning employee attendance using
  control charts. Table 8.2 presents the first
  structure which uses the 22 pay periods as the
  rational subgroups. The table entries Pij denote
  the recorded attendance rate for the ith employee
  for the jth pay period.
• The first question that was asked by management
  was whether or not the overall department
  attendance rate was changing over time i.e.,
  from pay period to pay period. To answer this
  question, the attendance rates were organized
  into 22 rational subgroups by pay period. The
  data within the subgroups were the attendance
  rates for the 10 employees for the pay period.
  An average chart was constructed using the pay
  period as the rational subgroup and the
  individual employee attendance rate as the basic
  data within the subgroup.

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The AttendanceManagement Case Study
Table 8.2. The First Organization of
the Employee Attendance Data
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The AttendanceManagement Case Study

• Figure 8.7 presents the control chart for the
  department attendance rate by pay period. Based
  on Figure 8.7, there is no evidence that the
  department attendance rate is changing over time.
  It appears to be in a reasonable state of
  statistical control around the average of 89.15.

Figure 8.7. Average Chart for the Department
Attendance Rate
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The AttendanceManagement Case Study

• The next question that was asked was whether
  there were differences in the attendance rates
  across the employees. To answer this question the
  data were reorganized using the employee as the
  rational subgroup. Table 8.3 presents the second
  organization of this data.

Table 8.3. The Second Organization of
the Employee Attendance Data
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The AttendanceManagement Case Study

• Figure 8.8 presents the average chart produced
  using the employee as the rational subgroup. This
  control chart compares the variation in
  attendance rates across employees to the
  variation observed over the 22 pay periods within
  an employee.

Figure 8.8. The Average Chart for Comparing
Employee Attendance Rates
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The AttendanceManagement Case Study

• This chart indicates that the variation in the
  averages across the employees is larger than
  expected compared to the variation in attendance
  within employees. The chart provides a clear
  signal that the attendance rates for employee 5
  and employee 10 fall outside of the expected
  range due to unexplainable cause variation, and
  they should be investigated.
• Management should work with these two employees
  on a localized basis in an attempt to discover
  the reasons for their low attendance in order to
  help them get back into the normal system.

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The Plastic Cup FlangeWidth Example

• The first generation HD-2 filler process was
  designed to simultaneously fill and seal
  preformed cups at four filling and sealing
  stations, so the original machine filled four
  cups at a time.
• The second generation machine was designed to
  simultaneously form cups and then fill the cups
  using 24 cup forming cavities, and 24 filling and
  sealing stations. This new design eliminated the
  need for an outside cup vendor, and increased the
  production capacity by a factor of 6.
• After the machine forms and fills the cup, the
  cup is sealed with a heat treated foil seal. The
  integrity of the product in the cup is dependent
  upon a good seal. The integrity of the seal is
  very dependent on the width of the cup flange
  because the heat treatment melts the flange and
  seats the foil seal into the melted flange.

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The Plastic Cup FlangeWidth Example

• The functional specification limits for the
  flange width are 4.5 mm  0.5 mm. The cup flange
  is created by the 24 cavities in the cup forming
  process. The geometry of the 24 cup forming
  cavities is presented in Figure 8.9.

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The Plastic Cup FlangeWidth Example
Figure 8.9. Cup Forming Cavity Geometry for the
24 Cavities
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The Plastic Cup FlangeWidth Example

• An acceptance test was conducted in which
  numerous performance characteristics of the
  machine were analyzed, including flange width.
  Two of the questions of interest were whether or
  not the flange width could be maintained in a
  state of statistical control during the
  production run, and whether or not the 24
  cavities significantly affect flange width. In
  order to answer these questions, the acceptance
  test was designed as follows.
• A nine-hour production run was scheduled under
  normal working conditions. At the beginning of
  each hour, n4 successive cups were sampled from
  each of the 24 cavities. This resulted in a
  total of N 9x24x4 864 flange width
  measurements. Three different organizations of
  the data were considered in order to answer the
  questions of interest.

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The Plastic Cup FlangeWidth Example

• The first structure for the data is presented in
  Table 8.4. Using this structure there are 216
  rational subgroups of size n4. The subgroups
  have been arranged so that the data for 800 A.M.
  are presented first for all 24 cavities, followed
  by the data for 900 A.M. for all 24 cavities,
  etc.
• The variation within the subgroups is the
  variation across four consecutive cups formed by
  the same cavity at the same point in time. This
  variation should reflect the inherent or
  unexplained variation in the process (i.e., the
  process noise).
• The variation across the subgroups is affected
  not only by the noise in the process, but also
  possibly by explainable causes due to cavity
  differences within a time period and explainable
  causes across time.

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The Plastic Cup FlangeWidth Example
Table 8.4. The First Structure of the Acceptance
Test Data 216 Rational Subgroups with N 4
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The Plastic Cup FlangeWidth Example

• Figure 8.10 is the average chart and Figure 8.11
  is the range chart produced from this
  organization of the data. It is clear from
  Figure 8.10 that the process was not in a state
  of statistical control during the production run.
  For example, the flange width increased
  significantly from subgroup 25 to 48 which
  represents the 900 a.m. and 1000 a.m. time
  frame.
• The flange width then decreased for subgroups 49
  through 96 which represents 1100 a.m. and 1200
  p.m. time frame. The range chart in Figure 8.11
  also indicates that the inherent process
  variation increased beginning with the 1000 a.m.
  subgroup.

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The Plastic Cup FlangeWidth Example
Figure 8.11. Range Chart for Flange Width - Data
Structure 1
Figure 8.10. Average Chart for Flange Width -
Data Structure 1
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The Plastic Cup FlangeWidth Example

• Figure 8.12 is the same range chart except only
  the first 48 subgroups (the 800 a.m. and 900
  a.m. data) were used to set the control limits.
  That chart clearly shows the process variation to
  be out of control during the production run.

Figure 8.12. Range Chart for Flange Width - Data
Structure 1 and Control Limits Set with 800
A.M. and 900 A.M. Data
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The Plastic Cup FlangeWidth Example

• These charts indicate that there are explainable
  causes of variation in the cause system affecting
  both the average flange width and the inherent
  variation in flange width. These explainable
  causes should be investigated and removed from
  the process if possible.
• Figure 8.13 is a histogram of the 864 flange
  width measurements with the upper and lower
  functional specification limits superimposed on
  the graph. Clearly, this process is not capable
  of meeting the functional specification limits.

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The Plastic Cup FlangeWidth Example
Figure 8.13. Comparison of the Histogram and
Functional Specification Limits for Flange Width
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The Plastic Cup FlangeWidth Example

• The second structure for the data is presented in
  Table 8.5. Here the rational subgroups presented
  in the first organization have simply been
  rearranged so that the nine time periods for
  cavity 1 are presented first, followed the nine
  time periods for cavity 2, etc.

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The Plastic Cup FlangeWidth Example
Table 8.5. The Second Structure of the
Acceptance Test Data 216 Rational Subgroups with
N 4
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The Plastic Cup FlangeWidth Example

• This organization allows an easy analysis of the
  performance, and comparison of, the individual
  cavities across time. The average chart for the
  first four cavities is presented in Figure 8.14.
• This control chart is reproduced in Figure 8.15
  which shows where each of the four cavities begin
  and end. Only four cavities are presented on the
  chart in order to see the patterns more clearly.
  In the actual analysis, all 24 cavities were
  studied.

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The Plastic Cup FlangeWidth Example
Figure 8.15. Control Chart for the First Four
Cavities - Beginning and End Points Identified
Figure 8.14. Control Chart for the First Four
Cavities
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The Plastic Cup FlangeWidth Example

• It is clear from Figure 8.15 that the flange
  width went out of control for each of the four
  cavities at 900 A.M. which indicates an
  explainable cause associated with the process
  that systematically affected all four cavities.
  (In fact, the complete control chart indicated
  that it affected all 24 cavities.) The cups are
  formed from plastic sheets which come in large
  rolls
• This shift in flange width was traced to a sheet
  splice (i.e., the plastic roll was changed over
  to a new roll). Similar shifts in the flange
  width occurred throughout the production run when
  new sheet rolls were spliced into the process.
  The explainable cause was traced to a change in
  sheet thickness. The thickness of the sheet
  rolls purchased from an outside vendor was not
  consistent from roll to roll, and the sheet roll
  vendor was contacted to discuss ways to improve
  the consistency of the sheet thickness.

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The Plastic Cup FlangeWidth Example

• The third organization of the same data,
  presented in Table 8.6, was designed to answer
  the question about the effects of the 24
  cavities. In this case the data were placed into
  24 subgroups defined by the 24 cavities. Since a
  sample of size n 4 cups was selected from each
  cavity for each of the 9 time periods, there are
  n 36 measurements per subgroup in this case.
• The variation within the subgroup includes the
  process noise plus the differences across time.
  The variation across subgroups includes the
  effects of cavities. Since the subgroup sample
  size is larger than 10, the average and standard
  deviation charts presented in Figures 8.16, 8.17,
  and 8.18 were used to analyze the data.

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The Plastic Cup FlangeWidth Example
Table 8.6. The Third Structure of the Acceptance
Test Data 24 Rational Subgroups with N 36
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The Plastic Cup FlangeWidth Example
Figure 8.17. Analysis of Cavity Effect on Flange
Width - Cavity Row Geometry Identified on the
Chart
Figure 8.16. Analysis of Cavity Effect on Flange
Width
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The Plastic Cup FlangeWidth Example

• Figure 8.16 is the initial average chart, and
  Figure 8.17 is the same chart with the
  information on the cavity row geometry described
  in Figure 8.9 included on the chart. There is a
  clear signal from these charts that the cavities
  are having a significant effect on the flange
  width. There is an obvious nonrandom pattern in
  the flange width across cavities in each row.

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The Plastic Cup FlangeWidth Example

• The flange width generally decreases across the
  6 cavities within each of the four rows. The
  decrease is associated with a column effect. The
  explainable cause was traced to an uneven
  distribution of heat in the forming plates that
  fit across the rows. The heat distribution
  system associated with the forming plates was
  redesigned to obtain a constant heat gradient
  across each of the four rows.

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The Plastic Cup FlangeWidth Example

• The standard deviation chart is presented in
  Figure 8.18. Since this organization of the data
  placed both the process noise and time effects
  into the rational subgroups, it was decided that
  no action should be taken based on this chart at
  this time.

Figure 8.18. Analysis of Standard Deviation of
Flange Width within Cavities