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1. 4/26/2013
1
IME 674: Quality Assurance &
Reliability
Kettering University
Justin Young, PhD
1
EXAM & TERM PROJECT INFO
IME 674: Lecture 4, Module 1
2
Exam I
• The first exam will take place in approximately
one week
• Will cover all material including this lecture
– Lectures 1-4
– Chapters 1-6 of the textbook
• Check blackboard for details and review guide
– Also an email sent to all students
3
Term Project
• Two options:
– Quality analysis of real process data
– Report on current state of art of a SPC topic (3
journal articles)
• 10 pages or less
• Term Project guidelines posted on blackboard
4
End Lecture 4, Module 1
5
CONTROL CHARTS –
AN INTRODUCTION
IME 674: Lecture 4, Module 2
6

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Variation
• No two natural items in any category are the
same.
• Variation may be quite large or very small.
• If variation very small, it may appear that
items are identical, but precision instruments
will show differences.
7
3 Categories of variation
• Within-piece variation
– One portion of surface is rougher than another
portion.
• Apiece-to-piece variation
– Variation among pieces produced at the same time.
• Time-to-time variation
– Service given early would be different from that given
later in the day.
8
Source of variation
• Equipment
– Tool wear, machine vibration, …
• Material
– Raw material quality
• Environment
– Temperature, pressure, humadity
• Operator
– Operator performs- physical & emotional
9
Both mean and variability must be
controlled
10
Control Chart Viewpoint
 Variation due to
 Common or chance causes
 Assignable causes
 Control chart may be used to discover
assig a le auses
11
Control charts identify variation
• Chance causes - common cause
– inherent to the process or random and not
controllable
– if only common cause present, the process is
o sidered sta le or i o trol
• Assignable causes - special cause
– variation due to outside influences
– if prese t, the pro ess is out of o trol
12

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Chance and Assignable Causes of Quality
Variation
• The eventual goal of SPC is reduction or
elimination of variability in the process by
identification of assignable causes.
13
Control charts help us learn more about
processes
• Separate common and special causes of
variation
• Determine whether a process is in a state of
statistical control or out-of-control
• Estimate the process parameters (mean,
variation) and assess the performance of a
process or its capability
14
15
Process in Control
• When a process is in control, there occurs a
natural pattern of variation.
• Natural pattern has:
– About 34% of the plotted point in an imaginary
band between 1s on both side CL.
– About 13.5% in an imaginary band between 1s
and 2s on both side CL.
– About 2.5% of the plotted point in an imaginary
band between 2s and 3s on both side CL.
16
Common Causes
17
• 34.13% of data lie between  and 1s above the mean ().
• 34.13% between  and 1s below the mean.
• Approximately two-thirds (68.28 %) within 1s of the mean.
• 13.59% of the data lie between one and two standard deviations
• Finally, almost all of the data (99.74%) are within 3s of the mean.
18

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Process Out of Control
• The term out of control is a change in the
process due to an assignable cause.
• When a point (subgroup value) falls outside its
control limits, the process is out of control.
19
Assignable Causes
(a) Mean
Grams
Average
20
Assignable Causes
(b) Spread
Grams
Average
21
Assignable Causes
(c) Shape
Grams
Average
22
Statistical Basis of the Control Chart
23
Basic Principles
A typical control chart has control limits set at
values 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).
Control Charts
UCL
Nominal
LCL
Assignable
causes likely
1 2 3
Samples 24

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Relationship between hypothesis testing and control
charts
• Choosing the control limits is equivalent to
setting up the critical region for testing
hypothesis
H0:  = 1.5
H1:   1.5
25
Control Chart Method
26
How the Shewhart Control Chart Works
27
Control Limits
• 3-Sigma Control Limits
– Probability of type I error is 0.0027
• Probability Limits
– Type I error probability is chosen directly
– For example, 0.001 gives 3.09-sigma control
limits
• Warning Limits
– Typically selected as 2-sigma limits
28
Control charts to monitor processes
• To monitor output, we use a control chart
– we check things like the mean, range, standard
deviation
• To monitor a process, we typically use two
control charts
– mean (or some other central tendency measure)
– variation (typically using range or standard
deviation)
29
Control Chart Method
30

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Control Chart Method
• Control limits are NOT specification limits
• CL are permissible limits of a quality
characteristic
• Evaluate variations in quality subgroup to
subgroup
• Limits established at 3 standard dev. from
central line; for normal distribution – we
expect 99.73% of items would lie within the
limits
31
Objectives Of Variable Control Chart
• What are the objectives ?
• For quality improvement
• To determine process capability
• For decisions in product specifications
• Provide information on production processes for current
decisions
– PIC – leave alone
– POOC – investigate, solve, rectify, improve
• Make decisions on recently produced items - release
next process, customer or other disposition method,
sorting, rework, reject
32
Types of Data
• Variable data
Product characteristic that can be measured
• Length, size, weight, height, time, velocity
• Attribute data
Product characteristic evaluated with a discrete choice
• Good/bad, yes/no
33
Variables data
• Examples
– Length, width, height
– Weight
– Temperature
– Volume
34
Control charts for variables
• X-bar chart
– In this chart the sample means are plotted in order to control
the mean value of a variable (e.g., size of piston rings, strength
of materials, etc.).
• R chart
– In this chart, the sample ranges are plotted in order to control
the variability of a variable.
• S chart
– In this chart, the sample standard deviations are plotted in order
to control the variability of a variable.
• S2 chart
– In this chart, the sample variances are plotted in order to
control the variability of a variable.
35
Control charts for variables
• Moving average–moving range chart (also
called MA–MR chart)
• Target charts (also called difference charts,
deviation charts and nominal charts)
• CUSUM (cumulative sum chart)
• EWMA (exponentially weighted moving
average chart)
• Multivariate chart
36

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End Lecture 4, Module 2
37
X-BAR AND R CHARTS
IME 674: Lecture 4, Module 3
38
X-bar and R charts
• The X- bar chart is developed from the
average of each subgroup data.
– used to detect changes in the mean between
subgroups.
• The R- chart is developed from the ranges of
each subgroup data
– used to detect changes in variation within
subgroups
39
Control chart components
• Centerline
– shows where the process average is centered or
the central tendency of the data
• Upper control limit (UCL) and Lower control
limit (LCL)
– describes the process spread
40
How to develop a control
chart?
41
Steps
1. Select quality characteristic
2. Choose rational subgroup
3. Collect data
4. Determine trial central line and control limits
5. Establish revised central line and control limits
6. Achieve the objective
42

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1. Select quality characteristic
• Measurable data (basic units, length, mass, time,
etc.)
• Affecting performance, function of product
• From Pareto analysis – highest % rejects, high
production costs
• Impossible to control all characteristics - selective
or use attributes chart
43
Define the problem
• Use other quality tools to help determine the
ge eral pro le that’s o urri g a d the
pro ess that’s suspe ted of ausi g it.
Select a quality characteristic to be measured
• Identify a characteristic to study - for example,
part length or any other variable affecting
performance.
44
2. Choose rational subgroup
• Variation within the group due only to chance
causes and can detect between groups
changes
• Two ways selecting subgroup samples:
– Select subgroup samples at one instant of time or
as close as possible
– Select period of time products are produced
45
Choose a subgroup size to be sampled
• Choose homogeneous subgroups
– Homogeneous subgroups are produced under the
same conditions, by the same machine, the same
operator, the same mold, at approximately the
same time.
• Try to maximize chance to detect differences
between subgroups, while minimizing chance
for difference with a group.
46
Choosing Samples
• Rational subgroup from homogeneous lot : same
machine, same operator
• Decisions on size of sample empirical judgment +
relates to costs
– choose n = 4 or 5  use R-chart
– when n  10  use s-chart
• Frequency of taking subgroups often enough to
detect process changes
47
3. Collect the data
• Generally, collect 20-25 subgroups (100 total
samples) before calculating the control limits.
• Each time a subgroup of sample size n is
taken, an average is calculated for the
subgroup and plotted on the control chart.
48

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Subgroup Number
Measure 1 2 3 4 5 … ….. … 25
x1
x2
x3
x4
x5
35
40
32
37
34
34
40
38
35
38
x 35.6 37.0
R 8 6
49
Example
50
4. Determine trial central line and
control limits
• Need an estimate of population mean and standard
deviation  Often unknown
Notation for variables control charts
• n - size of the sample (sometimes called a subgroup)
chosen at a point in time
• m - number of samples selected
• = average of the observations in the ith sample
(where i = 1, 2, ..., m)
• = gra d a erage or a erage of the a erages this
value is used as the center line of the control chart)
51
i
x
x
Notation and values
• Ri = range of the values in the ith sample
Ri = xmax - xmin
• = average range for all m samples
•  is the true process mean
• s is the true process standard deviation
R
52
Determine trial centerline
• The centerline should be the population
mean, 
• Since it is unknown, we use X Double bar, or
the grand average of the subgroup averages.
m
m
i
i


 1
X
X
x
53
Determine trial control limit
• The process standard deviation can be
estimated using a function of the sample
average range.
• This is an unbiased estimator of s
x
2
ˆ
d
R

s
54

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• Now, if population parameters were
known then
UCL =
LCL =
n
X
s
3

n
X
s
3

Determine trial control limit
x
55
• Now, if population parameters were
known then
UCL =
LCL =
n
X
s
3

n
X
s
3

Determine trial control limit
x
Use
Estimator!
2
ˆ
d
R

s
56
Determine trial control limit
• So, can be written as:
• Let A2 =
• Then,
n
X
s
3

x
n
d
R
2
3
n
d2
3
R
A
n
d
R
2
2
3

57
Trial chart parameters
x
R
A
x
LCL
x
Line
Center
R
A
x
UCL
2
2





 A2 is found in Appendix VI for various values of n.
58
Control chart for R
We need an estimate of sR
• We will use relative range
W = R/s, R = Ws
• Let d3 be the standard deviation of W
StdDev [R] = StdDev [Ws]
sR  d3s
• Use R to estimate sR
59
Control chart for R
• Estimator:
• Then:
60

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11
Trial chart parameters
R
R
D
LCL
R
Line
Center
R
D
R
UCL
3
4




 D3 and D4 are found in Appendix VI for various values of n.
61
4. Determine trial central line and
control limits
62
5. Establish revised central line and
control limits
Trial Control Limits:
• The control limits obtained from equations (6-4)
and (6-5) should be treated as trial control limits.
• If this process is in control for the m samples
collected, then the system was in control in the
past.
• If all points plot inside the control limits and no
systematic behavior is identified, then the
process was in control in the past, and the trial
control limits are suitable for controlling current
or future production.
63
Revise the charts
• In certain cases, control limits are revised
because:
–out-of-control points were included in the
calculation of the control limits.
–the process is in-control but the within
subgroup variation significantly improves.
64
Revising the charts
• Interpret the original charts
• Isolate the causes
• Take corrective action
• Revise the chart
– Only remove points for which you can determine an
assignable cause
65
Example 6.1 The Hard Bake Process
66

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67 68
End Lecture 4, Module 3
69
INTERPRETATION OF X-BAR CHARTS
& PROCESS CAPABILITY
IME 674: Lecture 4, Module 4
70
6. Achieving objective
• Initiate control charts
results in quality
improvement
• Less variation in sub-group
averages
• Reduction in variation of
range
• Can reduce frequency of
inspection - monitoring
purpose – even once/mth.
71
Improvement
72

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6. Achieving objective
• Use of control chart for monitoring future
production, once a set of reliable limits are
established, is called phase II of control chart
usage (Figure 6.4)
• A run chart showing individuals observations
in each sample, called a tolerance chart or tier
diagram (Figure 6.5), may reveal patterns or
unusual observations in the data
73
R-Chart
• Always look at the Range chart first. The control limits on the X-
bar chart are derived from the average range, so if the Range
chart is out of control, then the control limits on the X-bar chart
are meaningless.
• Look for out of control points. If there are any, then the special
causes must be eliminated.
• There should be more than five distinct values plotted, and no
one value should appear more than 25% of the time. If there are
values repeated too often, then you have inadequate resolution
of your measurements, which will adversely affect your control
limit calculations. In this case, you'll have to look at how you
measure the variable, and try to measure it more precisely.
• Once the effect of the out of control points from the Range chart
is removed, look at the X-bar Chart.
74
75 76
77
Analysis of Patterns on Control Charts
Nonrandom patterns can indicate out-of-control conditions
• Patterns such as cycles, trends, are often of considerable diagnostic
value
• Look for ru s - this is a sequence of observations of the same type
(all above the center line, or all below the center line)
• Runs of say 8 observations or more could indicate an out-of-control
situation.
– Run up: a series of observations are increasing
– Run down: a series of observations are decreasing
78

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• Patterns of the plotted points will provide
useful diagnostic information on the process,
and this information can be used to make
process modifications that reduce variability.
– Cyclic Patterns
– Mixture
– Shift in process level
– Trend
– Stratification
79 80
Process Out Of Control
1. A point falls outside control limits
– assignable cause present
– process producing subgroup avg. not from stable process
– must be investigated, corrected
– frequency distribution of
81 82
3. For two zones 1.5s each
• 2 or more points beyond
1.5s
83
3. Recurring cycles
• wavy, periodic high & low points
• seasonal effects of mtl.
• Recurring effects of temp.,
humidity (morning vs evening)
4. Two populations (mixture)
• many points near or outside
limits
• due to
• large difference in material
quality
• 2 or more machines
• different test method
• mtls from different supplier
84

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85 86
5.3.6 Discussion of the Sensitizing Rules
87
Warning Limits on Control Charts
• Warning limits (if used) are typically set at 2 standard
deviations from the mean.
• 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.
Caution!
88
Process In Control
• Individual parts will be more uniform – less
variation and fewer rejects
• Cost of inspection will decrease
• Process capability easily attained
• Trouble can be anticipated before it occurs
• Percentage of parts fall between two values can be
predicted with highest degree of accuracy, e.g.
filling machines
• X-R charts can be used as statistical evidence for
process control
89
Chart zones
• Based on our knowledge of the normal curve, a
control chart exhibits a state of control when:
♥ Two thirds of all points are near the center value.
♥ The points appear to float back and forth across
the centerline.
♥ The points are balanced on both sides of the
centerline.
♥ No points beyond the control limits.
♥ No patterns or trends.
90

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• Subgroup averages forms frequency distribution which is
normal distribution and limits – established at 3s from center
line.
• Choice of 3s is economic decision with respect to 2 types of
error
• Type I - occurs when looking for assignable cause but in reality
chance cause present > FALSE ALARM
• When limits set  3s Type I error probability = 0.27% or
3/1000
• Say point our control  due to assignable but 3/1000 of the
time can be due to chance cause
• Type II - assume chance cause present, but in fact assignable
cause present > TRUE ALARM
• Records indicate 3s limits balance between 2 errors.
91
Consequences of misinterpreting the
process
• Blaming people for problems that they cannot control
• Spending time and money looking for problems that do
not exist
• Spending time and money on unnecessary process
adjustments
• Taking action where no action is warranted
• Asking for worker-related improvements when process
improvements are needed first
92
Control vs. Specification Limits
• Control limits are derived
from natural process
variability, or the natural
tolerance limits of a process
• Specification limits are
determined externally, for
example by customers or
designers
• There is no mathematical or
statistical relationship
between the control limits
and the specification limits
93
Control Limits, Specification Limits, and Natural
Tolerance Limits
• There is no mathematical relationship between
control limits and specification limits.
• Do not plot specification limits on the charts
– Causes confusion between control and capability
– If individual observations are plotted, then
specification limits may be plotted on the chart.
94
Process Capability
• Tolerances
– design specifications reflecting product
requirements
• Process capability
– range of natural variability in a process what we
measure with control charts
95
Estimating Process Capability
• The x-bar and R charts give information about the
capability of the process relative to its specification limits.
• Assumes a stable process.
• We can estimate the fraction of nonconforming items for
any process where specification limits are involved.
• Assume the process is normally distributed, and x is
normally distributed, the fraction nonconforming can be
found by solving:
P(x < LSL) + P(x > USL)
96

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Process Capability
(b) Design specifications
and natural variation the
same; process is capable
of meeting specifications
most of the time.
Design
Specifications
Process
(a) Natural variation
exceeds design
specifications; process
is not capable of
meeting specifications
all the time.
Design
Specifications
Process
97
Process Capability (cont.)
(c) Design specifications
greater than natural
variation; process is
capable of always
conforming to
specifications.
Design
Specifications
Process
(d) Specifications greater
than natural variation,
but process off center;
capable but some output
will not meet upper
specification.
Design
Specifications
Process
98
Process Capability Measures
Process Capability Ratio
Cp =
=
tolerance range
process range
upper specification limit -
lower specification limit
6s
99 100
s


6
LSL
USL
Cp
Process Capability
• If Cp > 1, then a low # of nonconforming items will be
produced.
• If Cp = 1, (assume norm. dist) then we are producing about
0.27% nonconforming.
• If Cp < 1, then a large number of nonconforming items are
being produced.
101
Process-Capability Ratios (Cp)
 The percentage of the specification band that the
process uses up is denoted by
**The Cp statistic assumes that the process mean is
centered at the midpoint of the specification band
– it measures potential capability.
%
100
C
1
P̂
p









102

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18
Process Capability Measures
Process Capability Index
Cpk = minimum
x - lower specification limit
3s
=
upper specification limit - x
3s
=
,
103
End Lecture 4, Module 4
104
SAMPLE SIZE, OC CURVES, & ARL
IME 674: Lecture 4, Module 5
105
Changing Sample Size in and R Charts
• In some situations, it may be of interest to know the effect
of changing the sample size on the x-bar and R charts.
Needed information:
• = average range for the old sample size
• = average range for the new sample size
• nold = old sample size
• nnew = new sample size
• d2(old) = factor d2 for the old sample size
• d2(new) = factor d2 for the new sample size
x
old
R
new
R
106
Changing Sample Size in and R Charts
• Control Limits
x
old
2
2
2
old
2
2
2
R
)
old
(
d
)
new
(
d
A
x
LCL
R
)
old
(
d
)
new
(
d
A
x
UCL
chart
x














































old
2
2
3
old
2
2
new
old
2
2
4
R
)
old
(
d
)
new
(
d
D
,
0
max
UCL
R
)
old
(
d
)
new
(
d
R
CL
R
)
old
(
d
)
new
(
d
D
UCL
chart
R
107
Allocating Sampling Effort
• Guidelines for the Design of the Control Chart
– Choose a larger sample size and sample less
frequently? or, Choose a smaller sample size and
sample more frequently?
– The method to use will depend on the situation.
In general, small frequent samples are more
desirable.
108

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19
Rational Subgroups
• 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.
109
Selection of Rational Subgroups
• Select consecutive units of production.
– Pro ides a s apshot of the pro ess.
– Effective at detecting process shifts.
• Select a random sample over the entire sampling
interval.
– Can be effective at detecting if the mean has
wandered out-of-control and then back in-control.
110
Guidelines for Design of Control Charts
• Specify sample size, control limit width, and
frequency of sampling
• if the main purpose of the x-bar chart is to detect
moderate to large process shifts, then small
sample sizes are sufficient (n = 4, 5, or 6)
• if the main purpose of the x-bar chart is to detect
small process shifts, larger sample sizes are
eeded as u h as 5 to 5 … hi h is ofte
i pra ti al…alter ati e types of o trol harts
are a aila le for this situatio …see Chapter 8
111
• If increasing the sample size is not an option, then
sensitizing procedures (such as warning limits)
a e used to dete t s all shifts… ut this a
result in increased false alarms.
• R chart is insensitive to shifts in process standard
deviation.(the range method becomes less
effective as the sample size increases) may want
to use S or S2 chart.
• The OC curve can be helpful in determining an
appropriate sample size.
112
The Operating Characteristic Function
• How well the and R charts can detect
process shifts is described by operating
characteristic (OC) curves.
• Consider a process whose mean has shifted
from an in-control value by k standard
deviations. If the next sample after the shift
plots in-control, then you will not detect the
shift in the mean. The probability of this
occurring is called the β-risk.
x
113
Control Limits and Errors
LCL
Process
average
UCL
(a) Three-sigma limits
Type I error:
Probability of searching for
a cause when none exists
114

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20
Control Limits and Errors
Type I error:
Probability of searching for
a cause when none exists
UCL
LCL
Process
average
(b) Two-sigma limits
115
Type II error:
Probability of concluding
that nothing has changed
Control Limits and Errors
UCL
Shift in process
average
LCL
Process
average
(a) Three-sigma limits
116
Type II error:
Probability of concluding
that nothing has changed
Control Limits and Errors
UCL
Shift in process
average
LCL
Process
average
(b) Two-sigma limits
117
• If  is the probability of not detecting the
shift on the next sample, then 1 -  is the
probability of correctly detecting the shift
on the next sample.
118
The OC Function
• The probability of not detecting a shift in the
process mean on the first sample is b
– = P{LCL < Xbar < UCL m = m1 = m0 + ks}
• L= multiple of standard error in the control
limits
• k = shift in process mean (# of standard
deviations).
)
n
k
L
(
)
n
k
L
( 







119
Example
• We are using a 3s limit Xbar chart with
sample size equal to 5
– L = 3
– n = 5
• Determine the probability of detecting a
shift to 1 = 0 + 2s on the first sample
following the shift
– k = 2
120

21. 4/26/2013
21
Example, continued
So, =[3–2 SQRT(5)]-[-3–2 SQRT(5)]
= (-1.47) – (-7.37) = .0708
P(not detecting on first sample) = .0708
P(detecting on first sample) =1-
.0708=.9292
121
• The operating characteristic curves are
plots of the value  against k for various
sample sizes
If the shift is 1.0σ and the
sample size is n = 5, then
β = 0.75.
122
OC curve for the R chart
Use l  s1/s0 ( the ratio of new to old
process standard deviation ) in Fig. 6.14
R chart is insensitive when n is small
But, when n is large, W = R/s
loses efficiency, and S chart is better
123 124
125 126

22. 4/26/2013
22
• An assumption in performance properties is that the underlying
distribution of quality characteristic is normal
– If underlying distribution is not normal, sampling distributions can
be derived and exact probability limits obtained
• Burr (1967) notes the usual normal theory control limits are very
robust to normality assumption
• Schilling and Nelson (1976) indicate that in most cases, samples
of size 4 or 5 are sufficient to ensure reasonable robustness to
normality assumption for chart
• Sampling distribution of R is not symmetric, thus symmetric 3-
sigma limits are an approximation and -risk is not 0.0027. R
chart is more sensitive to departures from normality than
chart.
• Assumptions of normality and independence are not a primary
concern in phase I
The Effect of Non-normality
x
x
127
Which chart to use?
• X-bar / Range charts are used when you can rationally collect
measurements in groups (subgroups) of between two and ten
observations. Each subgroup represents a "snapshot" of the process at a
given point in time.
• For subgroup sizes greater than ten, use X-bar / Sigma charts, since the
range statistic is a poor estimator of process sigma for large subgroups. In
fact, the subgroup sigma is always a better estimate of subgroup variation
than subgroup range. The popularity of the Range chart is only due to its
ease of calculation, dating to its use before the advent of computers.
• For subgroup sizes equal to one, an Individual-X / Moving Range chart can
be used, as well as EWMA or CuSum charts.
– X-bar Charts are efficient at detecting relatively large shifts in the process
average, typically shifts of +-1.5 sigma or larger. The larger the subgroup, the
more sensitive the chart will be to shifts, providing a Rational Subgroup can be
formed. For more sensitivity to smaller process shifts, use an EWMA or CuSum
chart.
128
End Lecture 4, Module 5
129
X-BAR AND S CHARTS
IME 674: Lecture 4, Module 6
130
Construction and of and S Charts
• First, S2 is a u iased esti ator of s2
• Second, S is NOT an unbiased estimator of s
• S is an unbiased estimator of c4 s
where c4 is a constant
• The standard deviation of S is
x
2
4
c
1
s
131
• If a standard s is given the control limits for
the S chart are:
• B5, B6, and c4 are found in the Appendix for
various values of n.
s

s

s

5
4
6
B
LCL
c
CL
B
UCL
132

23. 4/26/2013
23
No Standard Given
• If s is unknown, we can use an average
sample standard deviation,



m
1
i
i
S
m
1
S
S
B
LCL
S
CL
S
B
UCL
3
4



133
Chart when Using S
The upper and lower control limits for the chart
are given as
where A3 is found in the Appendix
x
x
UCL x A S
CL x
LCL x A S
 

 
3
3
x
Construction and of and S Charts
134
Estimating Process Standard Deviation
• The process standard deviation, s can be
estimated by

s 
S
c4
135
The and S Control Charts
with Variable Sample Size
• The and S charts can be adjusted to
account for samples of various sizes.
• A eighted a erage is used i the
calculations of the statistics.
m = the number of samples selected.
ni = size of the ith sample
x
x
136
• The grand average can be estimated as:
• The average sample standard deviation is:





m
1
i
i
m
1
i
i
i
n
x
n
x
 
 



m
1
i
i
m
1
i
2
i
i
m
n
S
)
1
n
(
S
137
The and S Control Charts
with Variable Sample Size
• Control Limits
• If the sample sizes are not equivalent for each sample,
then
– there can be control limits for each point (control limits
may differ for each point plotted)
x
S
A
x
LCL
x
CL
S
A
x
UCL
3
3





S
B
LCL
S
CL
S
B
UCL
3
4



138

24. 4/26/2013
24
139
The S2 Control Chart
• There may be situations where the process variance
itself is monitored. An S2 chart is
where and are points found from the
chi-square distribution.
2
1
n
),
2
/
(
1
2
2
2
1
n
,
2
/
2
1
n
S
LCL
S
CL
1
n
S
UCL












2
1
n
,
2
/ 

 2
1
n
),
2
/
(
1 



140
End Lecture 4, Module 6
141
INDEPENDENT X -
MOVING RANGE CHARTS
IME 674: Lecture 4, Module 7
142
The Shewhart Control Chart
for Individual Measurements
• What if you could not get a sample size greater than 1 (n
=1)? Examples include
– Automated inspection and measurement technology is
used, and every unit manufactured is analyzed.
– The production rate is very slow, and it is inconvenient
to allow samples sizes of N > 1 to accumulate before
analysis
– Repeat measurements on the process differ only
because of laboratory or analysis error, as in many
chemical processes.
• The X and MR charts are useful for samples of sizes
n = 1.
143 144

25. 4/26/2013
25
The Shewhart Control Chart
for Individual Measurements
Moving Range Chart
• The moving range (MR) is defined as the
absolute difference between two
successive observations:
MRi = |xi - xi-1|
which will indicate possible shifts or
changes in the process from one
observation to the next.
145
X and Moving Range Charts
• The control limits on the moving range chart are:
where
2
2
d
MR
3
x
LCL
x
CL
d
MR
3
x
UCL





m
MR
MR
m
1
i
i

 
146
X and Moving Range Charts
• The X chart is the plot of the individual
observations. The control limits are
0
LCL
MR
CL
MR
D
UCL 4



147
The Shewhart Control Chart
for Individual Measurements
Interpretation of the Charts
• X Charts can be interpreted similar to charts. MR charts
cannot be interpreted the same as or R charts.
• “i e the MR hart plots data that are orrelated ith
one another, then looking for patterns on the chart does
not make sense.
• MR chart cannot really supply useful information about
process variability.
• More emphasis should be placed on interpretation of the
X chart.
x
x
148
The Shewhart Control Chart
for Individual Measurements
• The normality
assumption is often
taken for granted.
• When using the
individuals chart, the
normality assumption
is very important to
chart performance. P-Value: 0.063
A-Squared: 0.648
Anderson-Darling Normality Test
N: 10
StDev: 2.54733
Average: 52.4
58
57
56
55
54
53
52
51
50
.999
.99
.95
.80
.50
.20
.05
.01
.001
Probability
hardness
Normal ProbabilityPlot
149 150

26. 4/26/2013
26
151
End Lecture 4, Module 7
152