- Seven tools; - Process variability; - Important use of the control chart; - Statistical basis of the control chart: > Basic principles and type of control chart; > Choice of control limits; > Sampling size and sampling frequency; > Average run length; > Rational subgroups; > Analysis of patterns on control charts; > Sensitizing rules for control charts; > Phase I and Phase II of control chart.

1. Instructor:
M. Mujiya Ulkhaq
Department of Industrial Engineering
Methods and Philosophy of SPC
Statistical Process Control

2. • Seven tools;
• Process variability;
• Important use of the control chart;
• Statistical basis of the control chart:
– Basic principles and type of control chart;
– Choice of control limits;
– Sampling size and sampling frequency;
– Average run length;
– Rational subgroups;
– Analysis of patterns on control charts;
– Sensitizing rules for control charts;
– Phase I and Phase II of control chart.
2-2

3. SPC is a powerful collection of problem-solving tools useful in achieving process
stability and improving capability through the reduction of variability.
The magnificent seven or the seven tools:
Histogram Check Sheet
Cause-and-Effect
Diagram
Pareto Chart
Defect Concentration
Diagram
Scatter Diagram
Control Chart 2-3

4. • “Chance” or “common” causes
of variability represent the
inherent, natural variability of
a process; its background noise.
• Variation resulting from
“assignable” or “special” caus-
es represents generally large,
unsatisfactory disturbances to
the usual process performance.
A process that is operating with only chance causes of
variation present is said to be in statistical control.
A process that is operating in the presence of assignable
causes is said to be out of control.
Figure 2.1. Chance and assignable causes of variation
Montgomery (2013), p. 190
2-4

5. a) Stationary behavior, uncorrelated data (white noise);
b) Stationary behavior, autocorrelated data;
c) Nonstationary behavior.
Figure 2.2. Data from three different processes.
Montgomery (2013), p. 196
Shewhart control charts are most effective
when the in-control process data is
stationary and uncorrelated
2-5*The concept of autocorrelation will be discussed later.

6. • Most processes do not operate in a state of statistical control.
• Consequently, the routine and attentive use of control charts will identify assign-
nable causes. If these causes can be eliminated from the process, variability will
be reduced and the process will be improved.
• The control chart only detects assignable causes.
– Management, operator, and engineering
action will be necessary to eliminate the
assignable causes.
– In identifying and eliminating assignable
causes, it is important to find the root cause
of the problem and to attack it.
Figure 2.3. Process improvement using the control chart
Montgomery (2013), p. 194
The most important use of a control
chart is to improve the process
2-6

7. Out-of-control action plans (OCAPs) is a
flowchart or text-based description of the
sequence of activities that must take place
following the occurrence of an activating
event.
The OCAP consists of:
• checkpoints, which are potential assign-
nable causes, and
• terminators, which are actions taken to
resolve the out-of-control condition, prefe-
rably by eliminating the assignable cause.
Figure 2.4. An example of OCAP for the hard-bake process.
Montgomery (2013), p. 195
Control charts without an OCAP
are not likely to be useful as a
process improvement tool
2-7

8. The control chart can be used as an estimating device.
– Certain process parameters can be estimated, such as the mean, standard deviation,
fraction nonconforming or fallout, and so forth.
These estimates may then be used to determine the
capability of the process to produce acceptable
products.
Process-capability studies* have considerable
impact on many management decision problems
that occur over the product cycle, including make
or buy decisions, plant and process improvements
that reduce process variability, and contractual
agreements with customers or vendors regarding
product quality.
*This topic will be discussed later. 2-8

9. Control charts have had a long history of use in U.S. industries and in many
offshore industries as well.
There are at least five reasons for their popularity:
1. Control charts are proven technique for improving productivity;
2. Control charts are effective in defect prevention;
3. Control charts prevent unnecessary process adjustment;
4. Control charts provide diagnostic
information;
5. Control charts provide information
about process capability.
2-9

10. The 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.
• Center line: the target value.
• Control limits: values of such that if
the process is in control, nearly all
points will lie between the upper
control limit (UCL) and the
lower control limit (LCL).
Only chance causes are present
Figure 2.5. A typical control chart
Montgomery (2013), p. 190
2-10

11. As long as the points plot within the control limits, the
process is assumed to be in control, and no
action is necessary.
Out-of-control situations:
• If at least one point plots beyond the control limits;
• If the points behave in a systematic or nonrandom
manner.
assignable or special causes of variation have
been removed; characteristic parameters like
the mean, standard deviation, and probability
distribution are constant; and process
behavior is predictable.
2-11

12. • A point plotting within the control limits is equivalent to failing to
reject the hypothesis of statistical control;
• A point plotting outside the control limits is equivalent to rejecting
the hypothesis of statistical control.
H0: μ = μ0
H1: μ ≠ μ0
process mean is in control;
failing to reject the null hypothesis
process mean is out of control;
reject the null hypothesis
2-12

13. The hypothesis testing framework is useful in many ways, but there are some
differences in viewpoint between control charts and hypothesis tests*.
• The validity of assumptions is usually checked when testing the statistical
hypotheses; whereas control charts are used to detect departures from an assumed
state of statistical control.
• Only the sustained shift fits nicely within the usual statistical hypothesis
testing model.
we should not worry too much about assumptions such as the form of
the distribution or independence when we are applying control charts
to a process to reduce variability and achieve statistical control.
Type of mean shifts:
• Sustained shift;
• Steady drift or trend shift;
• Abrupt or sudden shift.
*Woodall, W. H. (2000). “Controversies and contradictions in statistical process
control,” Journal of Quality Technology, Oct., 32(4), pp. 341–350. 2-13

14. Control charts may be classified into two general types:
• Variables Control Charts
– These charts are applied to data that follow a continuous distribution
(measurement data).
– chart is the most widely used chart for controlling central tendency;
– R chart or s charts are used to control process variability.
• Attributes Control Charts
– These charts are applied to data that follow a discrete distribution.
– In these cases, we may judge each unit of product as either conforming or
nonconforming on the basis of whether or not it possesses certain attributes,
or we may count the number of nonconformities (defects) appearing on a
unit of product.
x
2-14

15. General model of a control chart:
w is a sample statistic that measures some
quality characteristic of interest;
μw is the mean of w;
σw is the standard deviation of w; and
L is “distance” of the control limits from the
center line.
Specifying the control limits is one of the critical decisions that must be made in
designing a control chart.
• Widening the control limits will decrease the risk of type I error but increase the
risk of type II error.
• Contrarily, narrowing the control limits will increase the risk of type I error but
decrease the risk of type II error.
UCL = μw + Lσw
Center Line = μw
LCL = μw – Lσw
2-15

16. 3-Sigma Limits
• The use of 3-sigma limits generally gives good results in practice;
• These limits are often referred to as action limits.
If the data is assumed to be normally
distributed, than the probability of
type I error (α) is 0.0027. That is,
an incorrect out-of-control signal
or false alarm will be generated in
only 27 out of 10,000 points.
Furthermore, the probability that a
point taken when the process is in
control will exceed the three-sigma
limits in one direction only is
0.00135.
2-16

17. Probability Limits
Instead of specifying the control limit as a multiple of the standard deviation (or
constant = L), we could have directly chosen the type I error probability and
calculated the corresponding control limit, called the probability limits.
For example, if we specified a 0.001 type I error probability in one direction, then
the appropriate multiple of the standard deviation would be 3.09. The control limit
for the chart would then be:
UCL = μw + 3.09σw
Center Line = μw
• In the UK and parts of Western Europe, the probability limits are often used
• In the US, the constant (L) is used as a standard practice.
2-17

18. Warning Limits
• Warning limits (if used) are typically set at 2 standard deviations
from the mean (L = 2);
• If one or more points fall between the warning limits and the
control limits, or close to the warning limits the process may not be
operating properly.
Good thing: warning limits often increase the sensitivity of
the control chart.
Bad thing: warning limits could result in an increased risk
of false alarms.
2-18

19. • In general, larger samples will make it easier to detect
small shifts in the process.
• If the process shift is relatively large, then the smaller
sample sizes can be used.
The most desirable situation from the point of view of detecting shifts would be to
take large samples very frequently; however, this is usually not economically
feasible.
Allocating sampling effort: take small samples at short intervals
or larger samples at longer intervals.
Current industry practice tends to favor
smaller, more frequent samples 2-19

20. Another way to evaluate the decisions regarding sample size and sampling frequency
is through the average run length (ARL) of the control chart.
ARL as a performance measurement of the control chart.
When the process is in-control (no mean shift) the value of ARL0 has to be
large; When there is a mean shift in the process, the value of out-of-control
ARL has to be as small as possible.
The in-control ARL (ARL0) is the average number of points that
must be plotted before a point indicates an out-of-control condition;
The out-of-control ARL (ARL1) is the average number of points to
detect the process shift.
2-20

21. Sometimes it is more appropriate to express the performance of the control chart in
terms of the average time to signal (ATS).
If samples are taken at fixed intervals of time that are h hours apart, then
h ARLATS
Consider a problem with the in-control
average run length of 370. Suppose
we are sampling every hour, so we
will have a false alarm about every
370 hours on the average.
Consider a problem with the out-of-
control average run length of 2.86.
Since the time interval between
samples is 1 hour, the average time
required to detect this shift is 2.86
hours.
Acceptable?
• Sample every half hour
ATS = 1.43 hours
• Use larger sample size
If n = 10, then β = 0.9
and ARL1 = 1.11
ATS = 1.11 hours
2-21

22. Subgroups or samples should be selected so that if assignable causes are present, the
chance for differences between subgroups will be maximized, while the
chance for differences due to these assignable causes within a subgroup will be
minimized.
Constructing rational subgroups:
1. Select consecutive units of production.
– Provides a “snapshot” of the process.
– Provides a better estimate of the standard deviation
of the process in the case of variables control charts
– Effective at detecting process shifts.
Figure 2.6. The snapshot approach to rational subgroups.
(a) Behavior of the process mean; (b) Corresponding x-bar and R charts
Montgomery (2013), p. 202 2-22

23. Subgroups or samples should be selected so that if assignable causes are present, the
chance for differences between subgroups will be maximized, while the
chance for differences due to these assignable causes within a subgroup will be
minimized.
Constructing rational subgroups:
2. Select a random sample over the entire
sampling interval.
– It is used to make decisions about the
acceptance of all units of product that have been
produced since the last sample.
– Can be effective at detecting if the mean
has wandered out-of-control and
then back in-control. Figure 2.7. The random sample approach to rational subgroups.
(a) Behavior of the process mean; (b) Corresponding x-bar and R charts
Montgomery (2013), p. 202 2-23

24. Western Electronic Rules*:
1. One point plots outside the 3-sigma
control limits.
2. Two out of three consecutive points plot
beyond the 2-sigma warning limits.
3. Four out of five consecutive points plot
at a distance of 1-sigma or beyond from
the center line.
4. Eight consecutive points plot on one side
of the center line.Figure 2.8. Western Electric or zone rules, with the last four
points showing a violation of rule 3
Montgomery (2013), p. 204
*Western Electric (1956). Statistical Quality Control Handbook, Western Electric
Corporation, Indianapolis, IN. 2-24

25. Another Sensitizing Rules for Shewhart Control Charts:
5. Six points in a row steadily increasing or decreasing (“runs”)
– Run up: a series of observations are increasing;
– Run down: a series of observations are decreasing.
6. Fifteen points in a row in zone C (both above and below the center line).
7. Fourteen points in a row alternating up and down.
8. Eight points in a row on both sides of the center line with none in zone C.
9. An unusual or nonrandom pattern in the data.
– Patterns such as cycles, trends, are often of considerable diagnostic value.
10. One or more points near a warning or control limit.
It should be used carefully because of
the increased risk of false alarms
2-25

26. Suppose that the analyst uses k decision rules and that criterion i has type I error
probability αi, then the overall type I error or false alarm probability for the
decision based on all k tests is (provided that all k decision rules are independent):
*Champ, C. W., and W. H. Woodall (1987). “Exact results for Shewhart control
charts with supplementary runs rules,” Technometrics, 29(4), pp. 393–399.
  

k
i
i
1
11 
When Western electronic rules are used, these rules does improve the ability
of the control chart to detect smaller shifts, but the ARL0 can be substantially
degraded. For example, assuming independent process data and using a
Shewhart control chart, the value of ARL0 is 91.25, in contrast to 370 for the
Shewhart control chart alone*.
αi
2-26

27. • In phase I, a set of process data is gathered and analyzed all at once in a
retrospective analysis, constructing trial control limits to determine if the
process has been in control over the period of time during which the data were
collected, and to see if reliable control limits can be established to monitor future
production.
– Typically m = 20 or 25 subgroups are used in phase I.
– The ARL is not usually a reasonable performance measure for phase I; analyst is more
interested in the probability that an assignable cause will be detected.
• In phase II, we use the control chart to monitor the process by comparing
the sample statistic for each successive sample as it is drawn from the process to
the control limits.
– The ARL is a valid basis for evaluating the performance of a control chart in phase II.
2-27

28. 1. Management leadership;
2. A team approach, focusing on
project-oriented applications;
3. Education of employees at all
levels;
4. Emphasis on reducing variability;
5. Measuring success in quantitative (economic) terms;
6. A mechanism for communicating successful results through
out the organization.
2-28

29. Instructor:
M. Mujiya Ulkhaq
Department of Industrial Engineering
Statistical Process Control
Thank You for Your Attention