Quality and Statistical Process Control
The concept of quality has been with us since the beginning of time. As early as the
creation of the world described in the Bible in Genesis, God pronounced his creation
"good"-- e.g., acceptable quality. Artisans' and craftsmen's skills and the quality of their
work are described throughout history. Typically the quality intrinsic to their products
was described by some attribute of the products such as strength, beauty or finish.
However, it was not until the advent of the mass production of products that the
reproducibility of the size or shape of a product became a quality issue.
Quality, particularly the dimensions of component parts, became a very serious issue
because no longer were the parts hand-built and individually fitted until the product
worked. Now, the mass-produced part had to function properly in every product built.
Quality was obtained by inspecting each part and passing only those that met
specifications. This was true until 1931 when Walter Shewhart, a statistician at the
Hawthorne plant at Western Electric, published his book Economic Control of Quality of
Manufactured Product (Van Nostrand, 1931). This book is the foundation of modern
statistical process control (SPC) and provides the basis for the philosophy of total quality
management or continuous process improvement for improving processes. With
statistical process control, the process is monitored through sampling. Considering the
results of the sample, adjustments are made to the process before the process is able to
produce defective parts.
Processes and Process Variability
The concept of process variability forms the heart of statistical process control. For
example, if a basketball player shot free throws in practice, and the player shot 100 free
throws every day, the player would not get exactly the same number of baskets each day.
Some days the player would get 84 of 100, some days 67 of 100, some days 77 of 100,
and so on. All processes have this kind of variation or variability.
This process variation can be partitioned into two components. Natural process variation,
frequently called common cause or system variation, is the naturally occurring fluctuation
or variation inherent in all processes. In the case of the basketball player, this variation
would fluctuate around the player's long-run percentage of free throws made. Special
cause variation is typically caused by some problem or extraordinary occurrence in the
system. In the case of the basketball player, a hand injury might cause the player to miss
a larger than usual number of free throws on a particular day.
Statistical Process Control
Shewhart's discovery statistical process control or SPC is a methodology for charting the
process and quickly determining when a process is "out of control" (e.g., a special cause
variation is present because something unusual is occurring in the process). The process
is then investigated to determine the root cause of the "out of control" condition. When
the root cause of the problem is determined, a strategy is identified to correct it. The
investigation and subsequent correction strategy is frequently a team process and one or
more of the TQM process improvement tools are used to identify the root cause. Hence,
the emphasis on teamwork and training in process improvement methodology.
It is management's responsibility to reduce common cause or system variation as well as
special cause variation. This is done through process improvement techniques, investing
in new technology, or reengineering the process to have fewer steps and therefore less
variation. Management wants as little total variation in a process as possible--both
common cause and special cause variation. Reduced variation makes the process more
predictable with process output closer to the desired or nominal value. The desire for
absolutely minimal variation mandates working toward the goal of reduced process
variation.
The process above is in apparent statistical control. Notice that all points lie within the
upper control limits (UCL) and the lower control limits (LCL). This process exhibits only
common cause variation.
The process above is out of statistical control. Notice that a single point can be found
outside the control limits (above them). This means that a source of special cause
variation is present. The likelihood of this happening by chance is only about 1 in 1,000.
This small probability means that when a point is found outside the control limits that it is
very likely that a source of special cause variation is present and should be isolated and
dealt with. Having a point outside the control limits is the most easily detectable out-ofcontrol condition.
The graphic above illustrates the typical cycle in SPC. First, the process is highly variable
and out of statistical control. Second, as special causes of variation are found, the process
comes into statistical control. Finally, through process improvement, variation is reduced.
This is seen from the narrowing of the control limits. Eliminating special cause variation
keeps the process in control; process improvement reduces the process variation and
moves the control limits in toward the centerline of the process.
Types of Out-of-Control Conditions
Several types of conditions exist that indicate that a process is out of control. The first of
these we have seen already—having one or more points outside the  3 limits as shown
below:
Extreme Point Condition
This process is out of control because a point is either above the UCL or below the UCL.
This is the most frequent and obvious out of control condition and is true for all control
charts.
Control Chart Zones
Control charts can be broken into three zones, a, b, and c on each side of the process
center line.
A series of rules exist that are used to detect conditions in which the process is behaving
abnormally to the extent that an out of control condition is declared.
Two of Three Consecutive Points in Zone A or Outside Zone A
The probability of having two out of three consecutive points either in or beyond zone A
is an extremely unlikely occurrence when the process mean follows the normal
distribution. Thus, this criteria applies only to charts for examining the process mean.
X, Y, and Z are all examples of this phenomena.
Four of Five Consecutive Points in Zone B or Beyond
The probability of having four out of five consecutive points either in or beyond zone B
is also an extremely unlikely occurrence when the process mean follows the normal
distribution. Again this criteria should only be applied to an chart when analyzing a
process mean.
X, Y, and Z are all examples of this phenomena.
Runs Above or Below the Centerline
The probability of having long runs (8 or more consecutive points) either above or below
the centerline is also an extremely unlikely occurrence when the process follows the
normal distribution. This criteria can be applied to both and r charts.
Example X above shows a run below the center line.
Linear Trends
The probability of 6 or more consecutive points showing a continuous increase or
decrease is also an extremely unlikely occurrence when the process follows the normal
distribution. This criteria can be applied to both and r charts.
X and Y are both examples of trends. Note that the zones play no part in the
interpretation of this out of control condition.
Oscillatory Trend
The probability of having 14 or more consecutive points oscillating back and forth is also
an extremely unlikely occurrence when the process follows the normal distribution. It
also signals an out of control condition. This criteria can be applied to both and r
charts.
X is an example of this out of control condition. Note that the zones play no part in the
interpretation of this out of control condition.
Avoidance of Zone C
The probability of having 8 or more consecutive points occurring on either side of the
center line and do not enter Zone C is also an extremely unlikely occurrence when the
process follows the normal distribution and signals an out of control condition. This
criteria can be applied to charts only. This phenomena occurs when more than one
process is being charted on the same chart (probably by accident—e.g., samples from two
machines mixed and put on a single chart), the use of improper sampling techniques, or
perhaps the process is over controlled or the data is being falsified by someone in the
system.
X is an example of this out of control condition.
Run in Zone C
The probability of having 15 or more consecutive points occurring the Zone C is also an
extremely unlikely occurrence when the process follows the normal distribution and
signals an out of control condition. This criteria can be applied to charts only. This
condition can arise from improper sampling, falsification of data, or a decrease in process
variability that has not been accounted for when calculating control chart limits, UCL and
LCL.
X is an example of this out of control condition.
Quality Control Tools
Production environments that utilize modern quality control methods are
dependant upon statistical literacy. The tools used therein are called
the seven quality control tools. These include:







Check Sheet
Pareto Chart
Flow Chart
Cause and Effect Diagram
Histogram
Scatter Diagram
Control Chart
Checksheet
The function of a checksheet is to present information in an efficient, graphical format.
This may be accomplished with a simple listing of items. However, the utility of the
checksheet may be significantly enhanced, in some instances, by incorporating a
depiction of the system under analysis into the form.
A defect location checksheet is a very simple example of how to incorporate graphical
information into data collection.
Additional data collection checksheet examples demonstrate the utility of this tool. The
data collected will be used in subsequent examples to demonstrate how the individual
tools are often interconnected.
Pareto Chart
Pareto charts are extremely useful because they can be used to identify those factors that
have the greatest cumulative effect on the system, and thus screen out the less significant
factors in an analysis. Ideally, this allows the user to focus attention on a few important
factors in a process.
They are created by plotting the cumulative frequencies of the relative frequency data
(event count data), in decending order. When this is done, the most essential factors for
the analysis are graphically apparent, and in an orderly format.
From the Pareto Chart it is possible to see that the initial focus in quality improvement
should be on reducing edge flaws. Although the print quality is also of some concern,
such defects are substantially less numerous than the edge flaws.
Flowchart
Flowcharts are pictorial representations of a process. By breaking the process down into
its constituent steps, flowcharts can be useful in identifying where errors are likely to be
found in the system.
By breaking down the process into a series of steps, the flowchart simplifies the analysis
and gives some indication as to what event may be adversely impacting the process.
Cause and Effect Diagram
This diagram, also called an Ishikawa diagram (or fish bone diagram), is used to associate
multiple possible causes with a single effect. Thus, given a particular effect, the diagram
is constructed to identify and organize possible causes for it.
The primary branch represents the effect (the quality characteristic that is intended to be
improved and controlled) and is typically labelled on the right side of the diagram. Each
major branch of the diagram corresponds to a major cause (or class of causes) that
directly relates to the effect. Minor branches correspond to more detailed causal factors.
This type of diagram is useful in any analysis, as it illustrates the relationship between
cause and effect in a rational manner.
Having decided on which problem to focus on, a Cause and Effect diagram of the related
process is created to help the user see the entire process and all of its components.
In many instances, attempts to find key problem areas in a process can be a hit or miss
proposition. In this instance, it was decided to collect data on the curetimes of the
material.
Histogram
Histograms provide a simple, graphical view of accumulated data, including its
dispersion and central tendancy. In addition to the ease with which they can be
constructed, histograms provide the easiest way to evaluate the distribution of data.
Data for a test of curetimes was collected and analyzed using a histogram.
From this chart, the curetime distribution does not appear to be a normal distribution as
might be expected, but is bimodal instead. Deviations from a normal distribution in a
histogram suggest the involvement of additional influences in the process.
Curing Time Test Results
order curetime defects
1
31.6583
0
2
29.7833
0
3
31.8791
0
4
33.9125
0
order curetime defects
51 40.53732
52 41.69992
53 38.01712
54 42.23068
3
3
2
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
34.4643
25.1848
37.76689
39.21143
41.34268
39.54590
29.5571
32.5735
29.4731
25.3784
25.0438
24.0035
25.4671
34.8516
30.1915
31.6222
46.25184
34.71356
41.41277
44.63319
35.44750
38.83289
33.0886
31.6349
34.55143
33.8633
35.18869
42.31515
43.43549
37.36371
38.85718
39.25132
37.05298
42.47056
35.90282
38.21905
38.57292
39.06772
32.2209
33.202
27.0305
33.6397
26.6306
42.79176
38.38454
37.89885
0
0
1
2
3
2
0
0
0
1
1
2
1
0
0
0
5
0
3
4
0
2
0
0
0
0
0
3
4
1
2
2
1
4
0
2
2
2
0
0
1
0
2
4
2
1
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
40.16485
38.35171
44.17493
37.32931
41.04428
38.63444
34.5628
28.2506
32.5956
25.3439
29.2058
32.0702
30.6983
40.30540
35.55970
39.98265
39.70007
33.95910
38.77365
35.69885
38.43070
40.05451
43.13634
44.31927
39.84285
39.12542
39.00292
34.9124
33.9059
28.2279
32.4671
28.8737
34.3862
33.9296
33.0424
28.4006
32.5994
30.7381
31.7863
34.0398
35.7598
42.37100
30.206
34.5604
27.93
30.8174
2
2
4
1
3
2
0
1
0
2
0
0
0
3
0
2
2
0
1
0
2
3
4
5
2
2
2
0
0
0
0
1
0
0
0
1
0
0
0
0
0
3
0
0
1
0
Scatter Diagram
Scatter diagrams are graphical tools that attempt to depict the influence that one variable
has on another. A common diagram of this type usually displays points representing the
observed value of one variable corresponding to the value of another variable.
Applying curing time test data to create a scatterplot, it is possible to see that there are
very few defects in the range of approximately 29.5 to 37.0 minutes. Thus, it is possible
to conclude that by establishing a standard cure time within this range, some degree of
quality improvement is likely.
Control Chart
The control chart is the fundamental tool of statistical process control, as it indicates the
range of variability that is built into a system (known as common cause variation). Thus,
it helps determine whether or not a process is operating consistently or if a special cause
has occurred to change the process mean or variance.
The bounds of the control chart are marked by upper and lower control limits that are
calculated by applying statistical formulas to data from the process. Data points that fall
outside these bounds represent variations due to special causes, which can typically be
found and eliminated. On the other hand, improvements in common cause variation
require fundamental changes in the process.
Applying statistical formulas to the data from the curetime tests of base material, it was
possible to construct X-bar and R charts to assess its consistency. As a result, we can see
that the process is in a state of statistical control.
order
ct1
ct2
ct3
ct4
1 27.34667 27.50085 29.94412 28.21249
2 27.79695 26.15006 31.21295 31.33272
3 33.53255 29.32971 29.70460 31.05300
4 37.98409 32.26942 31.91741 29.44279
5 33.82722 30.32543 28.38117 33.70124
6 29.68356 29.56677 27.23077 34.00417
7 32.62640 26.32030 32.07892 36.17198
8 30.29575 30.52868 24.43315 26.85241
9 28.43856 30.48251 32.43083 30.76162
10 28.27790 33.94916 30.47406 28.87447
11 26.91885 27.66133 31.46936 29.66928
12 28.46547 28.29937 28.99441 31.14511
13 32.42677 26.10410 29.47718 37.20079
14 28.84273 30.51801 32.23614 30.47104
15 30.75136 32.99922 28.08452 26.19981
16 31.25754 24.29473 35.46477 28.41126
17 31.24921 28.57954 35.00865 31.23591
18 31.41554 35.80049 33.60909 27.82131
19 32.20230 32.02005 32.71018 29.37620
20 26.91603 29.77775 33.92696 33.78366
21 35.05322 32.93284 31.51641 27.73615
22 32.12483 29.32853 30.99709 31.39641
23 30.09172 32.43938 27.84725 30.70726
24 30.04835 27.23709 22.01801 28.69624
25 29.30273 30.83735 30.82735 31.90733
Summary
The tools listed above are ideally utilized in a particular methodology, which typically
involves either reducing the process variability or identifying specific problems in the
process. However, other methodologies may need to be developed to allow for sufficient
customization to a certain specific process. In any case, the tools should be utilized to
ensure that all attempts at process improvement include:






Discovery
Analysis
Improvement
Monitoring
Implementation
Verification
Furthermore, it is important to note that the mere use of the quality control tools does not
necessarily constitute a quality program. Thus, to achieve lasting improvements in
quality, it is essential to establish a system that will continuously promote quality in all
aspects of its operation.
Control Charts as a tool in SQC
(Statistical Quality Control)
Overview
This page has been designed to help in understanding and learning the use, design and
analysis of Control Charts, which is the most important tool of Statistical Quality
Control.
The information has been formatted in the form of a tutorial, which will guide you
through the process. It includes the history, background information, the uses, the types
with examples, analysis of patterns, related software and additional sources of
information about control charts.
History
Control charting is one of the tools of Statistical Quality Control(SQC) It is the most
technically sophisticated tool of SQC. It was developed in the 1920s by Dr. Walter A.
Shewhart of the Bell Telephone Labs.
Dr. Shewhart developed the control charts as an statistical approach to the study of
manufacturing process variation for the purpose of improving the economic effectiveness
of the process. These methods are based on continuous monitoring of process variation.
Background Information
A typical control chart is a graphical display of a quality characteristic that has been
measured or computed from a sample versus the sample number or time. The chart
contains a center line that represents the average value of the quality characteristic
corresponding to the in-control state. Two other horizontal lines, called the upper control
limit(UCL) and and the lower control limit(LCL) are also drawn. These control limits are
chosen so that if the process is in control, nearly all of the sample points will fall between
them. As long as the points plot within the control limits, the process is assumed to be in
control, and no action is necessary.
However, a point that plots outside of the control limits is interpreted as evidence that the
process is out of control, and investigation and corrective action is required to find and
eliminate the assignable causes responsible for this behavior. The control points are
connected with straight line segments for easy visualization.
Even if all the points plot inside the control limits, if they behave in a systematic or
nonrandom manner, then this is an indication that the process is out of control.
Uses of Control charts
Control chart is a device for describing in a precise manner what is meant by statistical
control. Its uses are
1.
2.
3.
4.
5.
It is a proven technique for improving productivity.
It is effective in defect prevention.
It prevents unnecessary process adjustments.
It provides diagnostic information.
It provides information about process capability.
Types of control charts
1. Control Charts for Attributes
Introduction
Many quality characteristics cannot be conveniently represented numerically. In such
cases, each item inspected is classified as either conforming or nonconforming to the
specifications on that quality characteristic. Quality characteristics of this type are called
attributes. Examples are nonfunctional semiconductorchips, warped connecting rods, etc,.
p charts
This chart shows the fraction of nonconforming or defective product produced by
a manufacturing process. It is also called the control chart for fraction
nonconforming.
c charts
This shows the number of defects or nonconformities produced by a
manufacturing process.
u charts
This chart shows the nonconformities per unit produced by a manufacturing
process.
2. Control Charts for Variables
Introduction
Many quality characteristics can be expressed in terms of a numerical measurement. A
single measurable quality characteristic, such as a dimension, weight, or volume, is called
a Variable. Control charts for variables are used extensively. They usually lead to more
efficient control procedures and provide more information about process performance
than attributes control charts.
When dealing with a quality characteristic that is a variable, it is a standard practice to
control both the mean value of the quality characteristic and its variability. Control of the
process average or mean quality level is usually with the control chart for means, or the x
bar chart. The control of the process range is done by using the control chart for range, or
the R chart.
X bar and R charts
The X bar chart is developed from the average of each subgroup data. The R chart is
developed from the ranges of each subgroup data, which is calculated by subtracting the
maximum and the minimum value in each subgroup. An example showing the
development of the charts is provided.
Example showing procedure for making X bar and R Control Charts
Construction of Control charts
Plastic Keychains are being produced in a company named Etcetra. The plastic material
is first molded and then trimmed to the required shape. The curetimes during the molding
process affect the edge quality of the keychains produced. The aim is to achieve
statistical control of the curetimes using X bar and R charts.
Curetime data of twenty-five samples, each of size four, have been taken when the
process is assumed to be in control. These are shown in table 1. Save the raw data for this
table and try to draw the control charts. Compare the results with those given here.
Table 1
---------------------------------------------------------------Sample
No.
Observations
means
range
---------------------------------------------------------------1
27.34667 27.50085 29.94412 28.21249 28.25103
2.59745
2
27.79695 26.15006 31.21295 31.33272 29.12317
5.18266
3
33.53255 29.32971 29.70460 31.05300 30.90497
4.20284
4
37.98409 32.26942 31.91741 29.44279 32.90343
8.54130
5
33.82722 30.32543 28.38117 33.70124 31.55877
5.44605
6
29.68356 29.56677 27.23077 34.00417 30.12132
6.77340
7
32.62640 26.32030 32.07892 36.17198 31.79940
9.85168
8
30.29575 30.52868 24.43315 26.85241 28.02750
6.09553
9
28.43856 30.48251 32.43083 30.76162 30.52838
3.99227
10
28.27790 33.94916 30.47406 28.87447 30.39390
5.67126
11
26.91885 27.66133 31.46936 29.66928 28.92971
4.55051
12
28.46547 28.29937 28.99441 31.14511 29.22609
2.84574
13
32.42677 26.10410 29.47718 37.20079 31.30221 11.09669
14
28.84273 30.51801 32.23614 30.47104 30.51698
3.39341
15
30.75136 32.99922 28.08452 26.19981 29.50873
6.79941
16
31.25754 24.29473 35.46477 28.41126 29.85708 11.17004
17
31.24921 28.57954 35.00865 31.23591 31.51833
6.42911
18
31.41554 35.80049 33.60909 27.82131 32.16161
7.97918
19
32.20230 32.02005 32.71018 29.37620 31.57718
3.33398
20
26.91603 29.77775 33.92696 33.78366 31.10110
7.01093
21
35.05322 32.93284 31.51641 27.73615 31.80966
7.31707
22
32.12483 29.32853 30.99709 31.39641 30.96172
2.79630
23
30.09172 32.43938 27.84725 30.70726 30.27140
4.59213
24
30.04835 27.23709 22.01801 28.69624 26.99992
8.03034
25
29.30273 30.83735 30.82735 31.90733 30.71869
2.60460
---------------------------------------------------------------Means
30.40289 5.932155
----------------------------------------------------------------
The means and the ranges in each sample are calculated. The mean of the sample means
and the sample ranges are also calculated. The mean for the ranges gives the center line
for the R chart. Using n=4 from the table for calculating the control limits, we get that D3
= 0 and D4=2.282. Therefore the control limits for the R chart are LCL = Rmean x D3 =
0
UCL = Rmean x D4 = 13.525
The R chart is drawn with the 25 sample ranges plotted on the chart. The Control limits
and the center line are also drawn. This is shown in the figure below. All the points are
within the control limits and no particular pattern can be observed. Therefore the process
variability is in control.
Since the R chart indicates the process variablity is in control, the X bar chart is now
constructed. The center line is the mean of the sample means. Using the same table, and
taking n=4 the control limits calculated are
UCL = xmean_of_means + A2 x Rmean = 34.733
LCL = xmean_of_means - A2 x Rmean = 26.072
The X bar chart is drawn with the 25 sample means plotted on the chart. The Control
limits and the centre line are drawn too. This is shown in the figure below. No indication
of out of control condition is observed from this figure.
Since both the X bar and the R chart exhibit control, the process can be taken to be in
control at the stated levels and the control limits can be adopted for use in on-line
statistical process control.
Continuation of the X bar and R charts
Twelve additional samples of curetime data from the molding process were collected
from an actual production run. The data from these new samples are shown in table 2.
Save the raw data for this table and try to draw the control charts. Compare the results
with those given here
Table 2
----------------------------------------------------------------------Sample
No.
Observations
means
range
----------------------------------------------------------------------1
31.65830 29.78330 31.87910 33.91250 31.80830
4.12920
2
34.46430 25.18480 37.76689 39.21143 34.15686
14.02663
3
41.34268 39.54590 29.55710 32.57350 35.75480
11.78558
4
29.47310 25.37840 25.04380 24.00350 25.97470
5.46960
5
25.46710 34.85160 30.19150 31.62220 30.53310
9.38450
6
46.25184 34.71356 41.41277 44.63319 41.75284
11.53828
7
35.44750 38.83289 33.08860 31.63490 34.75097
7.19799
8
34.55143 33.86330 35.18869 42.31515 36.47964
8.45185
9
43.43549 37.36371 38.85718 39.25132 39.72693
6.07178
10
37.05298 42.47056 35.90282 38.21905 38.41135
6.56774
11
38.57292 39.06772 32.22090 33.20200 35.76589
6.84682
12
27.03050 33.63970 26.63060 42.79176 32.52314
16.16116
-----------------------------------------------------------------------
The X bar and the R charts are drawn with the new data with the same control limits
established before. They are shown below
The observations from the control charts :
X bar chart



R chart
Six points fall above UCL.
One point falls below LCL.
One point falls on UCL.


Two points fall above UCL.
Eight consecutive points are above the center line.
Both the charts show that the process is out of control. The possible reasons for it can be
understood by looking at the Analysis of Patterns on Control charts
Analysis of Patterns on Control Charts
A control chart may indicate an out-of-control condition either when one or more points
fall beyond the control limits, or when the plotted points exhibit some nonrandom pattern
of behavior.
The process is out of control if any one or more of the criteria is met.
1. One or more points outside of the control limits. This pattern may indicate:
o A special cause of variance from a material, equipment, method, or
measurement system change.
o Mis-measurement of a part or parts.
o Miscalculated or mis-plotted data points.
o Miscalculated or mis-plotted control limits.
2. A run of eight points on one side of the center line. This pattern indicates a shift in
the process output from changes in the equipment, methods, or materials or a shift
in the measurement system.
3. Two of three consecutive points outside the 2-sigma warning limits but still inside
the control limits. This may be the result of a large shift in the process in the
equipment, methods, materials, or operator or a shift in the measurement system.
4. Four of five consecutive points beyond the 1-sigma limits.
5. An unusual or non-random pattern in the data.
a. A trend of seven points in a row upward or downward. This may show
 Gradual deterioration or wear in equipment.
 Improvement or deterioration in technique.
b. Cycling of data can indicate
 Temperature or other recurring changes in the environment.
 Differences between operators or operator techniques.
 Regular rotation of machines.
 Differences in measuring or testing devices that are being used in
order.
6. Several points near a warning or control limit.