Statistical Quality Control
Quality Control Charts using Excel
Learning Objectives

After this class the students should be
able to:

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
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Distinguish between controlled and uncontrolled
variation
Distinguish between variables and attributes
Determine control limits for several types of control
charts
Use graphics to create statistical control charts with
Excel
Interpret control charts
Create a Pareto chart
Time management

The expected time to deliver this
module is 50 minutes. 30 minutes are
reserved for team practices and
exercises and 20 minutes for lecture.
Statistical Quality Control
(SCQ)
Process

Any activity or set of activities that takes inputs
and create a product. The process for an industrial
plant takes raw materials and creates a finished
product is an example.
Statistical Quality Control (SQC) or
statistical process control (SPC)

The analysis of processes for improving quality.
Origins
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1924 - Walter A. Shewhart
W. Edwards Deming,
Controlled Variation


There is Variation that you can never eliminate it totally.
There are bound to be many small, unobservable,
chance effects that influence the outcome. this kind of
variation is said to be "in control," not because the
process operator is able to control the factors absolutely,
but rather because the variation is the result of normal
disturbances, called common causes, within the
process.
This type of variation can be predicted. In other words,
given the limitations of the process, each of these
common causes is controlled to the greatest extent
possible.
Uncontrolled Variation


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Variation that arise sporadically and for reasons
outside the normally functioning process, induced by a
special cause.
Special causes include differences between machines,
different skill or concentration levels of workers,
changes in atmospheric conditions, and variation in the
quality of inputs.
Unlike controlled variation, uncontrolled variation can
be reduced by eliminating its special cause.
Kind of Variation

Controlled


is native to the process, resulting from
normal factors called "common causes“
Uncontrolled

is the result of "special causes" and need
not be inherent in the process
Control Charts

As long as the points remain between the lower and upper
control limits, we assume that the observed variation is
controlled variation and that the process is in control
Control Chart

The process is out of control. Both the fourth and the twelfth
observations lie outside of the control limits, leading us to believe that
their values are the result of uncontrolled variation.
Control Chart

Even control charts in
which all points lie
between the control limits
might suggest that a
process is out of control.
In particular, the existence
of a pattern in eight or
more consecutive points
indicates a process out of
control, because an
obvious pattern violates
the assumption of random
variability.
Control Chart

The first eight
observations are
below the center
line, whereas the
second seven
observations all lie
above the center
line. Because of
prolonged periods
where values are
either small or
large, this process
is out of control.
Control Chart



Other suspicious patterns could appear in control
charts. Unfortunately, we cannot discuss them all
here.
Control chart makes it very easy for you to identify
visually points and processes that are out of control
without using complicated statistical tests.
This makes the control chart an ideal tool for the
shop floor, where quick and easy methods are
needed.
Chart and Hypothesis testing
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
The idea underlying control charts is closely related to
confidence intervals and hypothesis testing. The
associated null hypothesis is that the process is in
control; you reject this null hypothesis if any point lies
outside the control limits or if any clear pattern appears
in the distribution of the process values.
Another insight from this analogy is that the possibility
of making errors exists, just as errors can occur in
standard hypothesis testing. Occasionally a point that
lies outside the control limits does not have any special
cause but occurs because of normal process variation.
Variable and Attribute Charts
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
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Categories of control charts:
 those that monitor variables and
 those that monitor attributes.
Variable charts display continuous measures, such as weight,
diameter, thickness, purity, and temperature. Its statistical analysis
focuses on the mean values of such measures.
Attribute charts differ from variable charts in that they describe a
feature of the process rather than a continuous variable such as a
weight or volume. Attributes can be either discrete quantities, such as
the number of defects in a sample, or proportions, such as the
percentage of defects per lot.
Using Subgroups
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In order to compare process levels at various points in time, we usually
group individual observations together into subgroups.
The purpose of the subgroup is to create a set of observations in which
the process is relatively stable with controlled variation.
For example, if we were measuring the results of a manufacturing
process, we might create a subgroup consisting of values from the
same machine closely spaced in time.
A control chart might then answer the question "Do the averages
between the subgroups vary more than expected, given the variation
within the subgroups?"
The X Chart
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Each point in the x-chart displays the subgroup average
against the subgroup number: subgroup 2 occurring after
subgroup 1 and before subgroup 3.
As an example, consider a clothing store in which the
owner monitors the length of time customers wait to be
served. He decides to calculate the average wait-time in
half-hour increments. The first half-hour (for instance,
customers who were served between 9 a.m. and 9:30
a.m.) forms the first subgroup, and the owner records
the average wait-time during this interval. The second
subgroup covers the time from 9:30 a.m. to 10:00 a.m.,
and so forth.
The X Chart


It is based on the standard normal distribution.
The standard normal distribution underlies the
mean chart, because the Central Limit Theorem
states that the subgroup averages
approximately follow the normal distribution
even when the underlying observations are not
normally distributed.
The X Chart

The applicability of the normal distribution allows the
control limits to be calculated very easily when the
standard deviation of the process is known. 99.74% of
the observations in a normal distribution fall within 3
standard deviations of the mean (u). In SPC, this means
that points that fall more than 3 standard deviations from
the mean occur only 0.26% of the time. Because this
probability is so small, points outside the control limits
are assumed to be the result of uncontrolled special
causes.
Control Limits when s is known
LCL   
LCL   
3s
n
3s
n
s  Standard deviation
n  number of number of
observations in the subgroup
  mean
Calculating Control Limits
LCL  X 
LCL  X 
3s
n
3s
n
s  Standard deviation
n  number of number of
observations in the subgroup
X  mean of all of the subgroup averages
Go to Worksheet
The X Chart Example
Semester Score 1
Score 2
Score 3
Score 4
Score 5
1
97
89
80
81
82
2
74
100
94
65
86
3
85
100
88
62
65
4
100
91
77
67
71
5
83
92
88
79
75
6
72
79
85
100
78
7
80
83
93
88
96
8
80
100
100
79
84
9
87
70
84
96
83
10
75
77
84
75
85
11
55
95
89
100
100
12
75
73
100
72
78
13
75
100
89
66
100
14
69
88
100
84
84
15
100
84
95
80
92
16
91
100
99
77
79
17
92
90
93
87
90
18
82
80
80
79
76
19
54
89
97
84
71
20
83
66
69
100
82
The X Chart Example
To create a control chart of the teacher's scores:
1.
Click StatPlus > QC Charts > XBar Chart.
2.
Click the Subgroups in rows across columns option button.
3.
4.
1.
Click the Data Values button and select the range names Score -1
through Score 5. Click OK.
Click the Sigma Known checkbox and type 5 in the accompanying
text box.
Click the Output button and send the control chart to a new chart
Scores Control Chart
94.716
90.878
89.716
84.716
84.17
Values
are in
control
79.716
77.462
74.716
69.716
0
5
10
15
20
25
Analysis
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No mean score falls outside the control
limits. The lower control limit is 77.462,
the mean subgroup average is 84.17,
and the upper control limit is 90.878.
There is no evident trend to the data or
nonrandom pattern.
Then, there is no reason to believe the
teaching process is out of control.
Analysis
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
But, there is no evidence that this professor's
performance was better or worse in one
semester than in another.
The raw scores from the last three semesters
are misleading. A student might claim that
using a historical value for s also misleading,
because a smaller value for s could lead one
to conclude that the scores were not in control
after all.
Analysis
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One corollary to the preceding analysis
should be stated: Because even a single
professor experiences wide fluctuations
in student evaluations over time,
apparent differences among various
faculty members can also be deceptive.
You should use all such statistics with
caution.
Control Limits when
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s is unknown
In many instances, the value of s is not known. The normal
distribution does not strictly apply for analysis when s is
unknown. In this case we will use t-distribution instead
standard normal distribution.
Because SPC is often implemented on the shop floor by
workers who have had little or no formal statistical training
(and might not have ready access to Excel), the method for
estimating a is simplified and the normal approximation is
used to construct the control chart.
The difference is that when a is unknown, the control limits
are estimated using the average range of observations
within a subgroup as the measure of the variability of the
process.
Control Limits when

s is unknown
The control limits are
LCL  X  A2 R
UCL  X  A1 R
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R represents the average of the subgroup ranges, and
X is the average of the subgroup averages. Ai is a
correction factor that is used in quality-control charts.
There are many correction factors for different types
of control charts.
QC Correction Factors
n
2
3
4
5
6
7
8
9
10
11
12
13
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A2
d2
Dl
D2
D3
D4
1.88
1.023
0.729
0.577
0.483
0.419
0.373
0.337
0.308
0.285
0.266
0.249
1.128
1.693
2.059
2.326
2.534
2.704
2.847
2.970
3.078
3.173
3.258
3.336
0
0
0
0
0
0.204
0.388
0.547
0.687
0.811
0.922
1.025
3.686
4.358
4.698
4.918
5.078
5.204
5.306
5.393
5.469
5.535
5.594
5.647
0
0
0
0
0
0.076
0.136
0.184
0.223
0.256
0.284
0.308
3.268
2.574
2.282
2.114
2.004
1.924
1.864
1.816
1.777
1.744
1.717
1.692
Source: Adapted from "1950 ASTM Manual on Quality Control of Materials,"
American Society for Testing and Materials, in J. M. Juran, ed., Quality Control
Handbook (New York: McGraw-Hill, 1974), Appendix II, p. 39.
The X Chart: A Coating Process
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Using the data from a manufacturing firm that
sprays one of its metal products with a special
coating to prevent corrosion, create an X-chart
and analysis the results.
The company has just begun to implement SPC.
consequently, s is unknown. (20 minutes)
Click here to see data
Reference
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Data Analysis with Excel. Berk & Carey,
Duxbury, 2000, chapter 12, p. 475-488