p-Charts:
Attribute Based
Control Charts
By James Patterson
6/29/2016
1
Topics of Discussion
 What is a Control Chart?
 What is a p-Chart?
 What information does a
p-Chart convey?
 How are p-Charts developed?
 An example from the real world
 A sample exercise
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What is a Control Chart?
A Control Chart is a graphical display
of process information which
compares item attributes or
quantitative values against a standard
or reference value, within a series of
upper and lower constraint values
Adapted From the World Wide Web, 10/02/04:
6/29/2016
http://www.sytsma.com/tqmtools/pchart.html
3
What is a Control Chart?
• Why are control charts used?
– To determine if the rate of production of
nonconforming products is stable
– To detect when a deviation from process
stability has occurred
Adapted From the World Wide Web, 10/02/04:
6/29/2016
http://deming.eng.clemson.edu/pub/tutorials/qctools/ccmain1.htm
4
What is a Control Chart?
• Control charts are good for:
–
–
–
–
–
Improving Productivity
Preventing Defects
Preventing Unnecessary Process Adjustments
Provide Diagnostic Information
Provide Information About Process Capability
From the World Wide Web:
6/29/2016
http://deming.eng.clemson.edu/pub/tutorials/qctools/ccmain1.htm
5
What are the features of
a control chart?
• A graphical representation of a range
of acceptable values that suggest
whether or not a process is in control
• Contains a reference or optimum
target value, an upper control limit,
and a lower control limit
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What is a p-Chart?
• A process control chart that measures a
proportion of defective or nonconforming
items within a sample or population
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What information does a
p-Chart convey?
• An element or item under inspection may have
one or more definable attributes… (an attribute is
an intrinsic property of a given item that either
does or does not exist)
• If any one of the inspected attributes is
nonconforming, the entire item is counted as
nonconforming
• The number of items in the sample that are
determined to be nonconforming are summed and
a proportion of the total is evaluated
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What information does a
p-Chart convey?
• The p-Chart is a graph of the proportion of
nonconforming items in each sample or
population
• The graph is then used to determine whether or
not a process is stable
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Rationale for a p-Chart
• What is the statistical basis for p-Charts?
• The Binomial Distribution
– Binomial probability distributions exist when the element
in question can have only two possible values, each of
which is mutually exclusive of the other.
– For example: Is the item defective? Yes or No? It cannot
be both Yes AND No.
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p-Chart Example
Control Chart
0.400
UCL (0.368)
0.350
Proportion
0.300
0.250
0.200
0.150
0.100
0.050
LCL (0.030)
0.000
1
3
5
7
9
11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Observations (Sample Number)
Proportion
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p-bar (0.199)
UCL (0.368)
LCL (0.030)
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Collecting a dataset for
a p-Chart
• The data required for a p-Chart should
meet the following criteria:
– Subgroup Sample Size (n) ≥ 50
• Sample size may be up to 100 or more, but
between 50 and 100 is adequate
– Number of subgroups (or samples
taken) ≥ 25
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Collecting a dataset for
a p-Chart
• The data required for a p-Chart should
meet the following criteria:
– When gathering data in the subgroup samples,
it is preferable (but not mandatory) that the
sample sizes be the same
– If sample sizes are not the same, a different
calculation will be required
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Example dataset for a
p-Chart
• The proportion of
defective or
nonconforming
items in each
sample is
calculated by
dividing the
number defective
by the sample
size
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(Equal Sample Sizes)
Sample
Subgroup
Sample Size
Nonconforming
Proportion
1
10
50
0.200
2
11
50
0.220
3
10
50
0.200
4
9
50
0.180
5
8
50
0.160
6
11
50
0.220
7
10
50
0.200
8
9
50
0.180
9
10
50
0.200
10
9
50
0.180
11
11
50
0.220
12
13
50
0.260
13
9
50
0.180
14
8
50
0.160
15
9
50
0.180
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Example dataset for a
p-Chart
• The proportion of
defective or
nonconforming
items in each
sample is
calculated by
dividing the
number defective
by the sample
size
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(Unequal Sample Sizes)
Sample
Subgroup
Sample Size
Nonconforming
Proportion
1
10
50
0.200
2
11
51
0.216
3
10
48
0.208
4
9
47
0.191
5
8
50
0.160
6
11
55
0.200
7
10
54
0.185
8
9
51
0.176
9
10
56
0.179
10
9
43
0.209
11
11
44
0.250
12
13
51
0.255
13
9
49
0.184
14
8
49
0.163
15
7
53
0.132
15
Creating a p-Chart with
equal sample sizes
• With equal sample sizes, the first step requires
calculating the mean subgroup proportion. This is
accomplished by averaging all of the proportions
calculated from each sample set
• Formula:
k
p
P
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i 1
k
i
Mean Subgroup Proportion (Equal Sample Sizes)
where: Pi =
Sample proportion for subgroup i
k =
Number of samples of size n
Adapted From:
Business Statistics, 5th Edition
Groebner, et al, pp 56 (See Reference Slide)
16
Creating a p-Chart with
equal sample sizes
k
p
 Pi
i 1
k
Mean Subgroup Proportion (Equal Sample Sizes)
where: Pi =
Sample proportion for subgroup i
k =
Number of samples of size n
• For this example, there are 25 subgroups (k) (only 15
shown on previous slides…)
• Applied Formula:
0.200  0.220  0.200  0.180  ...
p
 0.192
50
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Adapted From:
Business Statistics, 5th Edition
Groebner, et al, pp 56 (See Reference Slide)
17
Creating a p-Chart with
equal sample sizes
• Once the Mean Subgroup Proportion has been
determined, it is used to determine the standard error
for the subgroup proportions
• Formula:
Estimate of the sample error for subgroup
proportions where:
( p)(1  p)
sp 
n
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Adapted From:
p =
n =
Mean subgroup proportion
Common Sample Size
Business Statistics, 5th Edition
Groebner, et al, pp 56 (See Reference Slide)
18
Creating a p-Chart with
equal sample sizes
sp 
( p)(1  p)
n
Estimate of the sample error for subgroup
proportions where:
p =
n =
Mean subgroup proportion
Common Sample Size
• The standard error will be used to calculate the upper
and lower control limits in the next step
• Applied Formula:
(0.192)(1  0.192)
sp 
 0.056
50
6/29/2016
Adapted From:
Business Statistics, 5th Edition
Groebner, et al, pp 56 (See Reference Slide)
19
Creating a p-Chart with
equal sample sizes
• Use the sample error of the subgroup proportions to
calculate the upper and lower control limits for the chart
• Formulas:
UCL  p  3s p
LCL  p  3s p
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Adapted From:
Business Statistics, 5th Edition
Groebner, et al, pp 56 (See Reference Slide)
20
Creating a p-Chart with
equal sample sizes
UCL  p  3s p
• Upper Control Limit:
UCL  0.192  3(0.056)  0.359
• Lower Control Limit:
LCL  p  3s p
LCL  0.192  3(0.056)  0.025
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Adapted From:
Business Statistics, 5th Edition
Groebner, et al, pp 56 (See Reference Slide)
21
Creating a p-Chart with
equal sample sizes
• With the Mean Subgroup Proportion, standard error,
and upper / lower control limits determined, fill out the
table with the calculated data:
Sample
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Nonconforming
Sample Size
Proportion
UCL (0.359)
p-bar (0.192)
LCL (0.025)
1
10
50
0.200
0.359
0.192
0.025
2
11
50
0.220
0.359
0.192
0.025
3
10
50
0.200
0.359
0.192
0.025
4
9
50
0.180
0.359
0.192
0.025
5
8
50
0.160
0.359
0.192
0.025
6
11
50
0.220
0.359
0.192
0.025
Adapted From:
Business Statistics, 5th Edition
Groebner, et al, pp 56 (See Reference Slide)
22
Creating a p-Chart with
equal sample sizes
• The data table has been completed, and all of the
information necessary to construct the p-Chart is
compiled.
• The upper and lower control limits, as well as the “pbar” (Mean Subgroup Proportion) lines are fitted to the
graph. These should be equally spaced horizontal lines,
plotted as a line graph / chart
• Plot the subgroup proportions on the line graph…
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23
Creating a p-Chart with
equal sample sizes
p-Chart
0.400
UCL (0.359)
0.350
Proportion
0.300
0.250
0.200
0.150
0.100
0.050
LCL (0.025)
0.000
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Observations (Sample Number)
Proportion
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UCL (0.359)
p-bar (0.192)
LCL (0.025)
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Creating a p-Chart with
unequal sample sizes
• If the subgroup sample sizes are not equal, a
slightly different approach is required for
calculating the upper and lower control limits.
• First, begin by calculating the mean subgroup
proportion, using the same method as was
done in the equal sample size example
6/29/2016
Adapted From:
Business Statistics, 5th Edition
Groebner, et al, pp 56 (See Reference Slide)
25
Creating a p-Chart with
unequal sample sizes
• Next, calculate the upper and lower control
limits for each subgroup individually
• Formula:
 ( p )(1  p ) 

UCL  p  3

nk


 ( p )(1  p ) 

UCL  p  3

nk


6/29/2016
Adapted From:
Business Statistics, 5th Edition
Groebner, et al, pp 56 (See Reference Slide)
26
Creating a p-Chart with
unequal sample sizes
• Formula:
 ( p )(1  p ) 

UCL  p  3

nk


• Applied:
 (0.192)(1  0.192) 
  0.362
UCL  0.192  3

50


Note: The denominator is the sample size for the specific
subgroup for which the control limit is being calculated… it
is variable, not fixed as in the previous example!
6/29/2016
Adapted From:
Business Statistics, 5th Edition
Groebner, et al, pp 56 (See Reference Slide)
27
Creating a p-Chart with
unequal sample sizes
• Formula:
 ( p )(1  p ) 

LCL  p  3

nk


• Applied:
 (0.192)(1  0.192) 
  0.022
LCL  0.192  3

50


Note: The denominator is the sample size for the specific
subgroup for which the control limit is being calculated… it
is variable, not fixed as in the previous example!
6/29/2016
Adapted From:
Business Statistics, 5th Edition
Groebner, et al, pp 56 (See Reference Slide)
28
Creating a p-Chart with
unequal sample sizes
• With the Mean Subgroup Proportion, and upper / lower
control limits determined, fill out the table with the
calculated data (note the UCL / LCL will not graph as
straight lines):
Sample
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Nonconforming
Sample Size
Proportion
p-bar
UCL
LCL
1
10
50
0.200
0.192
0.362
0.022
2
11
51
0.216
0.192
0.365
0.019
3
10
48
0.208
0.192
0.368
0.016
4
9
47
0.191
0.192
0.364
0.020
5
8
50
0.160
0.192
0.347
0.036
6
11
52
0.212
0.192
0.362
0.022
7
10
51
0.196
0.192
0.359
0.025
8
9
50
0.180
0.192
0.355
0.029
9
10
49
0.204
0.192
0.365
0.019
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Creating a p-Chart with
unequal sample sizes
• The data table has been completed, and all of the
information necessary to construct the p-Chart is
compiled.
• The upper and lower control limits, as well as the “pbar” (Mean Subgroup Proportion) lines are fitted to the
graph. Note that the upper and lower control limits will
not be straight lines, and should be mirror images of
one another
• Plot the subgroup proportions on the line graph…
6/29/2016
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Creating a p-Chart with
unequal sample sizes
p-Chart
0.450
0.400
UCL
0.350
Proportion
0.300
0.250
0.200
0.150
0.100
LCL
0.050
0.000
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Observations (Sample Number)
Proportion
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p-bar
UCL
LCL
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Evaluating the p-Chart
• Four conditions or trends which warrant immediate
attention:
– Five sample means in a row above or below the target or
reference line
– Six sample means in a row that are steadily increasing or
decreasing (trending in one direction)
– Fourteen sample means in a row alternating above and
below the target or reference line
– Fifteen sample means in a row within 1 standard error of
the target or reference line
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From:
Statistics for Dummies
Deborah Rumsey, pp 307 (See Reference Slide)
32
A Real World Example
A local hospital emergency department manager keeps
track of whether or not patients that are awaiting
treatment are interviewed by the triage nurse within a
standard time, established by the department’s medical
director.
The medical staff requests that the patients be
interviewed within 10 minutes of arrival to the
emergency department waiting room. Each day, 50
charts are reviewed, and the triage time is compared
with the administration desk sign in time. If the time
elapsed is greater than 10 minutes, the chart is counted
as “nonconforming.”
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A Real World Example
The following is the data collected over a period of 30
days by the emergency department manager
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A Real World Example
The manager calculated the mean subgroup proportion,
standard error, and upper and lower control limits; and
added these to the table
Note that the lower control limit was calculated at 0.030; however, since it is not physically possible to
have a negative number of nonconforming charts, the
lower control limit is set to 0.00
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A Real World Example
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A Real World Example
Emergency Department Triage Time Conformance
Proportion of Nonconforming Charts
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
Observations
Proportion
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Mean Subgroup Proportion
UCL (0.20)
LCL (0.00)
37
A Real World Example
Interpretation of the chart:
The department manager was concerned with several
aspects of the stability of the triage process. It was
obvious that patients were not consistently being seen
within the 10 minute requested time, but there appeared
to be a pattern to it.
When the department manager compared the numerous
peaks to the calendar, he noted that this was
consistently occurring on weekends, when patient
volume was highest. He decided to adjust staffing
levels to see if this would rectify the problem.
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P-Chart Exercise
As the quality assurance manager for a small, contract
manufacturing company, you have been notified by a
customer that several recent orders have been rejected
due to nonconforming defects that were unacceptable. The
customer identified three separate defect categories;
however, any one defect would cause the whole part to be
rejected.
You have decided to evaluate the process by running
several batches through production and then counting the
number of parts that fail inspection for any reason. The
data you collect is on the following page
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P-Chart Exercise
Calculate:
Mean Subgroup
Proportion
Standard Error
UCL / UCL
Build: a p-Chart
Analyze the chart: Is
the process in
control?
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P-Chart Exercise
Solutions:
Mean Subgroup Proportion:
0.049
Standard Error:
0.021
Upper Control Limit:
0.115
Lower Control Limit:
0.000*
*Actually calculated -0.016, but a negative number
is not a legitimate number of defects, therefore
0.000 is used as a realistic substitute
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P-Chart Exercise
Solutions:
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P-Chart Exercise
Solutions:
Process Control Chart (Proportion Defective)
0.14
0.12
Proportion
0.1
0.08
0.06
0.04
0.02
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Observations
Proportion
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Mean Subgroup Proportion
UCL (0.115)
LCL (0.000)
43
P-Chart Exercise
Conclusion:
The process is trending out of control:
- Five sample means in a row, above the reference line
- More than six sample means on an increasing trend, albeit
with some alternation; however, the trend is clearly increasing
at the end
Recommend: Shut down the production line and evaluate
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References
Rumsey, Deborah (2003). Statistics for Dummies.
Hoboken, NJ: Wiley Publishing, Inc.
Jaising, Lloyd (2000). Statistics for the Utterly Confused.
New York, NY: McGraw-Hill
Groebner, David F., Shannon, Patrick W., Fry, Phillip C.,
Smith, Kent D. (2001). Business Statistics: A Decision
Making Approach, 5th Edition.
Upper Saddle River, NJ: Prentice Hall, Inc.
Foster, S. Thomas (2004). Managing Quality: An Integrative
Approach. Upper Saddle River, NJ: Prentice Hall, Inc.
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p-Charts:
Attribute Based
Control Charts
By James Patterson
6/29/2016
46