# Shewhart�s Theory of Chance Cause Systems of Variation

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```					Shewhart’s Theory of Chance Cause Systems of
Variation

1
Shewhart’s 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.

2
Shewhart’s Chance Cause System

The Shewhart Model of a
Chance Cause System
of Variation
Group 1                Group 2              Group K
M Units from           M Units from         M Units from the
the Process            the Process             Process
Produced at            Produced at            Produced at
Time Point 1           Time Point 2          Time Point K

Data
Production                ...                                           ...
Process

Explainable Causes
of Variation -            UE       UE
Explainable Causes
UE                     Sampling Mules
of Variation -                             UE                                            UE                       Sampling Cause System
Production Mules                   W                                       E                  UE                  Creating the Sample of
UE            Figure 1     E        E         UE
Figure 4                     E           UE                                                                       N Units from the Batch
UE
E   E                     UE  UE                        E
E       E        Unexplainable Causes of
M     E          E        W       W                                            Variation -
E                       UE                                                   Sampling Woodpeckers
M

E Fi      E        W    Unexplainable Causes
of Variation -                                                                     UE
Explainable Causes       UE
g                       Production Woodpeckers                          of Variation -
u                                                                                                      UE
Measurement Mules                                       Y
r                                                                                                 UE
M                                                                                   E   E              W
Unexplainable Causes
3
Production  Cause System                                                      Figure 2                      of Variation -
E        E
Measurement Woodpeckers
Creating the Fundamental
Variation Across and Within
Groups of Widgets                                   Sample of Size N
Units from the Batch                        Measurement Cause
System Creating the
Observed Data Y

Shewhart’s Chance Cause System
3
Shewhart’s 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.

4
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.

5
The HD-2 Filler Case Study

Control Chart for Lane 1                                                                               Control Chart for Lane 4
245                                                                                                     245

Gross Fill Weight - Grams   3.0SL=241.0                                                                        3.0SL=241.0
240                                                                                                     240

X=238.0                                                                            X=238.0

235                                                                                      -3.0SL=235.0   235                                                                 -3.0SL=235.0

Evidence of an assignable
Evidence of an assignable cause - a saw tooth pattern
cause - periodic low weights
230                                                                                                     230
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100                                             5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Cup                                                                                                         Cup

6
The HD-2 Filler Case Study

Control Chart for Lane 4 - Phase 1                                                              Control Chart for Lane 4 - Phase 2
245                                                                                             245

Gross Fill Weight - Grams   3.0SL=241.0                                                                         3.0SL=241.0
240                                                                                             240

X=238.0                                                                             X=238.0

235                                                                              -3.0SL=235.0                                                                        -3.0SL=235.0
235

230                                                                                             230
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100                                      5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Cup                                                                                              Cup

7
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.

8
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.

9
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

10
Shewhart’s Chance Cause System -
The Woodpeckers and the Mules!

11
Shewhart’s 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!

12
Shewhart’s Theory of Chance Cause Systems of
Variation

Unexplainable Explainable
Random,
Predictable     Constant
Causes
Unpredictable                Assignable
Causes

13
Basic Statistical
Concepts for
Constant Cause
Systems

Woodpeckers Only !
14
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.

15
Constant Cause Systems

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

16
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.

17
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.

18
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 .

19
The Empirical Rule

99%-100%

90%-98%

60%- 75%

m-3     m-2   m-      m      m+   m+2   m+3

The Empirical Rule is the Foundation for
Shewhart Control Charts
20
Understanding and Analyzing a Chance Cause
System of Variation

21
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.

22
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.”

23
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

24
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

25
The Concept of Rational Subgrouping

The General Structure of Rational Subgroups

Rational Subgroups

1        2        …        K
1   X11      X21       …        XK1
Observations
2   X12      X22       …        XK2
within
Subgroups       …   …        …         …       …
n   X1N      X2N       …       XKN
Average within
X1       X2       …        XK
Subgroups
Range within
R1       R2       …        RK
Subgroups

Standard Deviation
S1       S2       …        SK
within Subgroups

26
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.

27
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?”

28
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.

29
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.

30
The Attendance
Management 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 employee’s 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.

31
The Attendance
Management Case Study
PAY                                           EMPLOYEE
PERIOD     1        2        3        4          5        6        7        8          9        10
1        100.00   100.00    96.39    70.00      10.00   100.00   100.00   100.00     100.00     0.00
2        100.00   100.00    90.00   100.00      77.21   100.00   100.00   100.00     100.00     0.00
3        100.00   100.00   100.00   100.00      44.44   100.00   100.00   100.00     100.00   100.00
4        100.00   100.00   100.00   100.00      60.00   100.00   100.00   100.00     100.00    98.10
5        100.00   100.00   100.00    80.00      89.74   100.00   100.00    97.22     100.00    69.03
6         74.94   100.00    80.00   100.00     100.00   100.00   100.00   100.00     100.00   100.00
7        100.00    20.00   100.00   100.00     100.00   100.00   100.00   100.00     100.00   100.00
8        100.00   100.00    83.13   100.00      50.00   100.00   100.00    78.95      94.12    60.00
9         74.91   100.00   100.00   100.00     100.00   100.00   100.00    71.43     100.00    82.98
10       100.00   100.00   100.00   100.00     100.00    68.42   100.00   100.00      87.80    84.81
11       100.00   100.00   100.00    77.78      88.89   100.00   100.00   100.00     100.00    87.50
12        87.50    73.73    80.00   100.00      80.00   100.00   100.00   100.00      90.00    77.78
13       100.00   100.00   100.00   100.00      50.00    88.32   100.00    80.00     100.00    59.75
14       100.00    85.71   100.00   100.00      60.00    88.89   100.00    86.88     100.00    33.05
15       100.00    88.89    63.64   100.00      87.50   100.00   100.00   100.00     100.00    80.00
16        60.00   100.00    90.00   100.00      90.00   100.00   100.00   100.00     100.00   100.00
17       100.00    75.00   100.00   100.00      60.00   100.00   100.00   100.00     100.00    43.25
18       100.00    12.50    95.00   100.00      55.00   100.00    70.00    75.00     100.00    70.00
19       100.00    54.55   100.00   100.00      89.97    88.69    96.38    67.50     100.00   100.00
20       100.00    71.53   100.00    90.00      88.89   100.00    80.00   100.00     100.00   100.00
21       100.00    75.00   100.00   100.00      53.62    97.81    95.00       0.00   100.00    82.22
22       100.00    62.50   100.00   100.00      62.50    88.65    82.92       0.00   100.00   100.00
2
32
The Attendance
Management 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.

33
The Attendance
Management Case Study

Table 8.2. The First Organization of the
Employee Attendance Data

Rational Subgroup - Pay Period

Employee          1                2              …    22

1             P1 1             P1 2                P1 22

2             P2 1             P2 2                P2 22

10            P10 1           P10 2                P10 22

34
The Attendance
Management 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%.

Control Chart for Department Attendance Rate
0
10
90                                                        5
X=89.1
80
E A E
A TT NDA NC R T

70                                                   LCL=69.49
60
50
40
E

30
20
0
1
0

PA Y PERI OD                                  0 12 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 20 2122

Figure 8.7. Average Chart for the
Department Attendance Rate                                                        35
The Attendance
Management 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

Rational Subgroup - Employee

Pay Period          1            2            3          …          10

1             P1   1        P 21        P 31                  P 10   1

2             P1   2       P2   2       P3   2                P 10   2

22             P1   22      P2   22      P3   22               P 10   22

36
The Attendance
Management 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.

Comparison of Employee Attendance Rates
0
10
90                                                5
X=89.1
E

80
ATT ND ANC RAT

LCL=77.24
70
E

60
50
40
E

30
20
0
1
0

E
EMPLOY E                 1   2   3   4   5   6   7   8   9   0
1

Figure 8.8. The Average Chart for
Comparing Employee Attendance Rates
37
The Attendance
Management 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.

38
The Plastic Cup Flange
Width 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.

39
The Plastic Cup Flange
Width 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.

40
The Plastic Cup Flange
Width Example

Column Column Column Column Column Column
1      2      3      4      5      6

Row 1           1      2       3     4      5      6

Row 2           7      8       9     10    11     12

Row 3           13     14      15    16     17    18

Row 4           19     20      21    22     23    24

Machine Direction

Figure 8.9. Cup Forming Cavity Geometry for the 24 Cavities
41
The Plastic Cup Flange
Width 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, n=4 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.

42
The Plastic Cup Flange
Width Example

The first structure for the data is presented in Table 8.4. Using this structure there
are 216 rational subgroups of size n=4. The subgroups have been arranged so that
the data for 8:00 A.M. are presented first for all 24 cavities, followed by the data
for 9:00 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.

43
The Plastic Cup Flange
Width Example

Table 8.4. The First Structure of the Acceptance Test Data
216 Rational Subgroups with N = 4

Time
8:00 A.M.            9:00 A.M.        …       4:00 P.M.
Cavity
1    2       ...   24    1      2    ...   24       1    2    ...   24
1     X    X             X     X      X          X        X    X           X
2     X    X             X     X      X          X        X    X           X
Replicates
3     X    X             X     X      X          X        X    X           X
4     X    X             X     X      X          X        X    X           X
Average           X1   X2                                                             X216
Range            R1   R2                                                             R216

44
The Plastic Cup Flange
Width 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 9:00 a.m. and 10:00 a.m. time frame.

   The flange width then decreased for subgroups 49 through 96 which represents
11:00 a.m. and 12:00 p.m. time frame. The range chart in Figure 8.11 also
indicates that the inherent process variation increased beginning with the
10:00 a.m. subgroup.

45
The Plastic Cup Flange
Width Example

.
15
5.0

UCL = 4.849                                                    UCL = 10.55
.
10

4.5                                            X = 4.512
R = 0.4624
0.5
LCL = 4.175

4.0                                                                                                           LCL = 0.000
0.0

10               1 3 5 7 9           0
30 50 70 90 10 1 0 1 0 1 0 1 0 21                          1
0                1 3 5 7 9          0
30 50 70 90 10 1 0 1 0 1 0 1 0 21
Subgroup                                                         Subgroup

Figure 8.10. Average Chart for Flange Width -                   Figure 8.11. Range Chart for Flange Width -
Data Structure #1                                              Data Structure #1

46
The Plastic Cup Flange
Width Example
Figure 8.12 is the same range chart except only the first 48 subgroups (the 8:00 a.m.
and 9:00 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.

1.5

1.0
RA N GE

UCL=0.6321
0.5

R=0.2770

0.0                                                                LCL=0.000

10   30   50   70   90   110   130   150   170   190   210

SUB GROUP

Figure 8.12. Range Chart for Flange Width - Data Structure #1
and Control Limits Set with 8:00 A.M. and 9:00 A.M. Data
47
The Plastic Cup Flange
Width 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.

48
The Plastic Cup Flange
Width Example

USL                 LSL
0
10
Frequency

50

0
3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25   5.50
Flange Width

Figure 8.13. Comparison of the Histogram and
Functional Specification Limits for Flange Width
49
The Plastic Cup Flange
Width 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.

50
The Plastic Cup Flange
Width Example

Table 8.5. The Second Structure of the Acceptance Test Data
216 Rational Subgroups with N = 4

1                          2                …             24

Time        8 am 9 am       ...   4 pm 8 am 9 am       ...   4 pm       8 am 9 am      ...   4 pm

1     X    X               X    X    X                X          X    X               X

2     X    X               X    X    X                X          X    X               X
Replicates
3     X    X               X    X    X                X          X    X               X
4     X    X               X    X    X                X          X    X               X

Average         X1   X2                                                                        X216

Range          R1   R2                                                                        R216

51
The Plastic Cup Flange
Width 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.

52
The Plastic Cup Flange
Width Example

C VI T 1
A Y           C VI T 2
A Y       C VI T 3
A    Y      A Y
C VI T 4
5.0                                                             5.0
UCL = 4.849                                                              UCL = 4.849

X = 4.512                                                                X = 4.512
4.5                                                             4.5

LCL - 4.175                                                              LCL - 4.175

4.0                                                             4.0

2   6   1
0   1
4   1
8   22   26   30   34                           2   6      0
1      1
4   8
1     22   26   30   34

Subgroup                                                              Subgroup

Figure 8.14. Control Chart for the                            Figure 8.15. Control Chart for the First Four
First Four Cavities                                   Cavities - Beginning and End Points Identified

53
The Plastic Cup Flange
Width Example

It is clear from Figure 8.15 that the flange width went out of control for each of the
four cavities at 9:00 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.

54
The Plastic Cup Flange
Width 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.

55
The Plastic Cup Flange
Width Example
Table 8.6. The Third Structure of the Acceptance Test Data
24 Rational Subgroups with N = 36
Cavity
Time           Replicates
1     2     ...   24
1         X     X            X
2         X     X            X
8:00 AM
3         X     X            X
4         X     X            X
1         X     X            X
2         X     X            X
9:00 AM
3         X     X            X
4         X     X            X
...
1         X     X            X
2         X     X            X
4:00 PM
3         X     X            X
4         X     X            X
Average             X1    X2          X24
Range               S1    S2          S24
56
The Plastic Cup Flange
Width Example

4 .8                                                                         4.8
O
R W1            O
R W2      O
R W3     O
R W4

4. 7                                                                         4.7
Flange Width

Flange Width
UCL = 4.652                                                                      UCL = 4.652
4. 6                                                                         4.6
= = 4.512                                                                        =
4. 5                                            X                                                                                X = 4.512
4.5

4.4                                             LCL = 4.372                  4.4
LCL = 4.372

4.3                                                                          4.3
2   4   6   8               8
10 12 14 16 1 20 22 24                                      2    4     6   8       2
10 1 14 16 18 20 22 2 4

Cavity                                                                                Cavity

Figure 8.16. Analysis of Cavity Effect on                                   Figure 8.17. Analysis of Cavity Effect on
Flange Width                                                    Flange Width - Cavity Row Geometry
Identified on the Chart

57
The Plastic Cup Flange
Width 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.

58
The Plastic Cup Flange
Width 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.

59
The Plastic Cup Flange
Width 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.

0.4
UCL = 0.3791
Standard Deviation

0.3
S = 0.2788

0.2
LCL = 0.1784

2   4   6   8   10    12   14   16   18   20 22 24

Cavity

Figure 8.18. Analysis of Standard Deviation
of Flange Width within Cavities
60

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