Quality Control
Statistical Process Control (SPC)
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Step 5 – Control
Statistical Process Control
(SPC)
Use data from the actual
process
Estimate distributions
Look at capability - is good
quality possible
Statistically monitor the process
over time
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Quality
Two types of variation
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Two Types of Variation
Common Cause Variation (low level)
Common Cause Variation (high level)
Assignable Cause Variation
• Need to measure and reduce common cause variation
• Identify assignable cause variation as soon as possible
• What
is common cause variation for one person might be
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assignable cause to the other
Detect Abnormal Variation in the Process:
Identifying Assignable Causes
Process
Parameter
Upper Control Limit (UCL)
• Track process parameter over time
- average weight of 5 bags
- control limits
- different from specification limits
Center Line
Lower Control Limit (LCL)
Time
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• Distinguish between
- common cause variation
(within control limits)
- assignable cause variation
(outside control limits)
Statistical Process Control
Capability
Analysis
Eliminate
Assignable Cause
Conformance
Analysis
Investigate for
Assignable Cause
Capability analysis
• What is the currently "inherent" capability of my process when it is "in control"?
Conformance analysis
• SPC charts identify when control has likely been lost and assignable cause
variation has occurred
Investigate for assignable cause
• Find “Root Cause(s)” of Potential Loss of Statistical Control
Eliminate or replicate assignable cause
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• Need Corrective Action To Move Forward
 Statistical process control (SPC) involves testing a random
sample of output from a process to determine whether the
process is producing items within a preselected range.
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Statistical
Process Normal Behavior
Control
(SPC) Charts
UCL
LCL
1
2
3
4
5
6
Samples
over time
UCL
Possible problem, investigate
LCL
1
2
3
4
5
6
Samples
over time
UCL
Possible problem, investigate
LCL
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1
2
3
4
5
6
Samples
over time
Control Limits are based on the Normal Curve
x
m
-3
-2
-1
Standard
deviation
units or “z”
units.
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0
1
2
3
z
Control Limits
Forming the Upper control limit (UCL) and the Lower
control limit (LCL):
UCL = Process Mean + 3 Standard Deviations
LCL = Process Mean – 3 Standard Deviations
UCL
+3σ
Process Average
- 3σ
LCL
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time
Control Chart Basics
Special Cause Variation:
Range of unexpected variability
UCL
Common Cause
Variation: range of
expected variability
+3σ
Process Mean
- 3σ
LCL
time
UCL = Process Mean + 3 Standard Deviations
LCL = Process Mean – 3 Standard Deviations
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Process Variability
Special Cause of Variation:
A measurement this far from the process average is
very unlikely if only expected variation is present
UCL
±3σ → 99.7% of
process values should
be in this range
Process Mean
LCL
time
UCL = Process Mean + 3 Standard Deviations
LCL = Process Mean – 3 Standard Deviations
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In-control Process
 A process is said to be in control when the control chart does not
indicate any out-of-control condition
 Contains only common causes of variation
 If the common causes of variation is small, then control chart can be used to
monitor the process
 If the common causes of variation is too large, you need to alter the process
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Process In Control
 Process in control: points are randomly distributed around
the center line and all points are within the control limits
UCL
Process Mean
LCL
time
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Process Not in Control
Out of control conditions:
 One or more points outside control limits
 8 or more points in a row on one side of the center line
 8 or more points in a row moving in the same direction
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Process Not in Control
One or more points
outside control limits
UCL
Process
Average
Process
Average
LCL
LCL
Eight or more points in a
row moving in the same
direction
UCL
Process
Average
LCL
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Eight or more points in a
row on one side of the
center line
UCL

Out-of-control
Processes
When the control chart indicates an out-of-control condition (a point
outside the control limits or exhibiting trend, for example)
 Contains both common causes of variation and assignable causes of
variation
 The assignable causes of variation must be identified
 If detrimental to the quality, assignable causes of variation must be removed
 If increases quality, assignable causes must be incorporated into the process
design
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Types of Statistical Sampling
 Attribute (Go or no-go information)

Defectives refers to the acceptability of product across a
range of characteristics.

Defects refers to the number of defects per unit which may be
higher than the number of defectives. (good or bad, function
or malfunction, 0 or 1)

p-chart application
 Variable (Continuous)

Usually measured by the mean and the standard deviation.

X-bar and R chart applications
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Example of Constructing a p-Chart:
Required Data
Sample
No. of
No.
Samples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
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100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
Number of
defects found
in each sample
4
2
5
3
6
4
3
7
1
2
3
2
2
8
3
Statistical Process Control Formulas:
Attribute Measurements (p-Chart)
Given:
sp =
T o ta l N u m b e r o f D e fe c tiv e s
p =
T o ta l N u m b e r o f O b s e rv a tio n s
p (1 - p)
n
UCL = p + z sp
LCL = p - z sp
Where p is the fraction defective, s p is the
standard deviation, n is the sample size, z is
the number of standard deviations for a
specific confidence. Typically, z  3 (99.7
percent confidence) or z  2.58 (99 percent
confidence) Compute control limits:
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Example of Constructing a p-chart: Step 1
1. Calculate the
sample proportions,
p (these are what
can be plotted on the
p-chart) for each
sample
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Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
n
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
Defectives
4
2
5
3
6
4
3
7
1
2
3
2
2
8
3
p
0.04
0.02
0.05
0.03
0.06
0.04
0.03
0.07
0.01
0.02
0.03
0.02
0.02
0.08
0.03
Example of Constructing a p-chart: Steps 2&3
2. Calculate the average of the sample proportions
55
p =
1500
= 0.036
3. Calculate the standard deviation of the
sample proportion
sp =
p (1 - p)
=
n
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.036(1- .036)
= .0188
100
Example of Constructing a p-chart: Step 4
4. Calculate the control limits
UCL = p + z sp
LCL = p - z sp
.036  3(.0188)
UCL = 0.0924
LCL = -0.0204 (or 0)
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Example of Constructing a p-Chart: Step 5
5. Plot the individual sample proportions, the average
of the proportions, and the control limits
0.1
0.09
0.08
0.07
0.06
p
UCL
LCL
0.05
0.04
0.03
0.02
0.01
0
1
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2
3
4
5
6
7
8
9
10 11 12 13 14 15
Some notes for p-charts
 The size of the sample must be large enough to allow counting of the
attribute. A rule of thumb when setting up a p chart is to make the
sample large enough to expect to count the attribute twice in each
sample.
 The assumption is that the sample size is fixed. If the sample size varies,
the standard deviation and upper and lower control limits should be
recalculated for each sample.
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Some notes for p-charts
T o ta l N u m b e r o f D e fe c tiv e s
p =
T o ta l N u m b e r o f O b s e rv a tio n s
UCL = p + z sp
LCL = p - z sp
nn
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sp =
p (1 - p)
n
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
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n
100
50
100
100
75
100
100
50
100
100
100
100
100
100
100
Defectives
4
2
5
3
6
4
3
7
1
2
3
2
2
8
3
p
0.04
0.04
0.05
0.03
0.08
0.04
0.03
0.14
0.01
0.02
0.03
0.02
0.02
0.08
0.03
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
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n
100
50
100
100
75
100
100
50
100
100
100
100
100
100
100
n-bar
91.66666667
UCL
LCL
Defectives
4
2
5
3
6
4
3
7
1
2
3
2
2
8
3
sum defects
55
0.101401806
-0.021401806
p
0.04
0.04
0.05
0.03
0.08
0.04
0.03
0.14
0.01
0.02
0.03
0.02
0.02
0.08
0.03
p-bar
0.04
sp
0.020467
0.16
0.14
0.12
0.1
p
UCL
LCL
0.08
0.06
0.04
0.02
0
1
2
3
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4
5
6
7
8
9
10
11
12
13
14
15
Process Control With Variable
Measurements: Using X-bar and R Charts
 Size of the samples (keep the sample size small, 4-5 is preferred)
 the sample needs to be taken within a reasonable length of time
 the larger the sample, the more it costs to take.
 Number of samples (25 or so samples is suggested to set up the
chart)
 Frequency of samples
 Control limits
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Example of x-bar and R Charts:
Required Data
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Obs 1
10.68
10.79
10.78
10.59
10.69
10.75
10.79
10.74
10.77
10.72
10.79
10.62
10.66
10.81
10.66
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Obs 2
10.689
10.86
10.667
10.727
10.708
10.714
10.713
10.779
10.773
10.671
10.821
10.802
10.822
10.749
10.681
Obs 3
10.776
10.601
10.838
10.812
10.79
10.738
10.689
10.11
10.641
10.708
10.764
10.818
10.893
10.859
10.644
Obs 4
10.798
10.746
10.785
10.775
10.758
10.719
10.877
10.737
10.644
10.85
10.658
10.872
10.544
10.801
10.747
Obs 5
10.714
10.779
10.723
10.73
10.671
10.606
10.603
10.75
10.725
10.712
10.708
10.727
10.75
10.701
10.728
Example of x-bar and R charts: Step 1. Calculate sample
means, sample ranges, mean of means, and mean of
ranges.
3
4
5
6
7
8
9
10
11
12
13
14
15
10.78
10.59
10.69
10.75
10.79
10.74
10.77
10.72
10.79
10.62
10.66
10.81
10.66
10.667
10.727
10.708
10.714
10.713
10.779
10.773
10.671
10.821
10.802
10.822
10.749
10.681
10.838
10.812
10.79
10.738
10.689
10.11
10.641
10.708
10.764
10.818
10.893
10.859
10.644
10.785
10.775
10.758
10.719
10.877
10.737
10.644
10.85
10.658
10.872
10.544
10.801
10.747
10.723
10.73
10.671
10.606
10.603
10.75
10.725
10.712
10.708
10.727
10.75
10.701
10.728
Averages
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10.759
10.727
10.724
10.705
10.735
10.624
10.710
10.732
10.748
10.768
10.733
10.783
10.692
0.171
0.221
0.119
0.143
0.274
0.669
0.132
0.179
0.163
0.250
0.349
0.158
0.103
10.728
0.220400
Example of x-bar and R charts: Step 2. Determine Control
Limit Formulas and Necessary Tabled Values
 E.L.Grant and R.Leavenworth computed a table, where
n is the number of observations in subgroup, A2 is the
factor for X-bar chart, D3 and D4 are factors for R chart.
x Chart Control Limits
UCL = x + A 2 R
LCL = x - A 2 R
R Chart Control Limits
UCL = D 4 R
LCL = D 3 R
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n
2
3
4
5
6
7
8
9
10
11
A2
1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.34
0.31
0.29
D3
0
0
0
0
0
0.08
0.14
0.18
0.22
0.26
D4
3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
Example of x-bar and R charts: Steps 3&4. Calculate x-bar
Chart and Plot Values
UCL = x + A 2 R  10.728 - .58(0.2204 ) = 10.856
LCL = x - A 2 R  10.728 - .58(0.2204 ) = 10.601
10.900
UCL
10.850
Means
10.800
10.750
10.700
Central
Line
LCL
10.650
10.600
10.550
1
2
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3
4
5
6
7
8
Sample
9
10
11
12
13
14
15
Example of x-bar and R charts: Steps 5&6. Calculate
R-chart and Plot Values
UCL = D 4 R  (2.11)( 0.2204 )  0.46504
LCL = D 3 R  (0)( 0.2204 )  0
0 .8 0 0
0 .7 0 0
0 .6 0 0
0 .5 0 0
R
UCL
0 .4 0 0
0 .3 0 0
0 .2 0 0
R
0 .1 0 0
LCL
0 .0 0 0
1
2
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3
4
5
6
7
8
S a m p le
9
10
11
12
13
14
15
The R Chart
 Monitors variability in a process
 The characteristic of interest is measured on a numerical scale
 Is a variables control chart
 Shows the sample range over time
 Range = difference between smallest and largest values in the subgroup
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The R Chart
1.
Find the mean of the subgroup ranges (the center line of the
R chart)
2.
Compute the upper and lower control limits for the R chart
3.
Use lines to show the center and control limits on the R chart
4.
Plot the successive subgroup ranges as a line chart
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The X Chart
 Shows the means of successive subgroups over time
 Monitors process average
 Must be preceded by examination of the R chart to
make sure that the variation in the process is in control
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The X Chart
 Compute the mean of the subgroup means (the center line of
the X chart)
 Compute the upper and lower control limits for the X chart
 Graph the subgroup means
 Add the center line and control limits to the graph
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