p-Charts: Attribute Based Control Charts

1
p-ChartsAttribute Based Control Charts

• By James Patterson

2
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

3
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 http//www.sytsma.com/tqmtools/pchart.ht
ml
4
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 http//deming.eng.clemson.edu/pub/tutori
als/qctools/ccmain1.htm
5
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 http//deming.eng.clemson
.edu/pub/tutorials/qctools/ccmain1.htm
6
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

7
What is a p-Chart?

• A process control chart that measures a
  proportion of defective or nonconforming items
  within a sample or population

8
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

9
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

10
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.

11
p-Chart Example

12
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

13
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

14
Example dataset for a p-Chart (Equal Sample
Sizes)

Sample Nonconforming Subgroup Sample Size 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

• The proportion of defective or nonconforming
  items in each sample is calculated by dividing
  the number defective by the sample size

15
Example dataset for a p-Chart (Unequal Sample
Sizes)

Sample Nonconforming Subgroup Sample Size 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

• The proportion of defective or nonconforming
  items in each sample is calculated by dividing
  the number defective by the sample size

16
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

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)
17
Creating a p-Chart with equal sample sizes
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

Adapted From Business Statistics, 5th
Edition Groebner, et al, pp 56 (See Reference
Slide)
18
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 Mean subgroup
proportion n Common Sample Size
Adapted From Business Statistics, 5th
Edition Groebner, et al, pp 56 (See Reference
Slide)
19
Creating a p-Chart with equal sample sizes
Estimate of the sample error for subgroup
proportions where p Mean subgroup
proportion n Common Sample Size

• The standard error will be used to calculate the
  upper and lower control limits in the next step
• Applied Formula

Adapted From Business Statistics, 5th
Edition Groebner, et al, pp 56 (See Reference
Slide)
20
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

Adapted From Business Statistics, 5th
Edition Groebner, et al, pp 56 (See Reference
Slide)
21
Creating a p-Chart with equal sample sizes

• Upper Control Limit
• Lower Control Limit

Adapted From Business Statistics, 5th
Edition Groebner, et al, pp 56 (See Reference
Slide)
22
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 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)
23
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 p-bar (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

24
Creating a p-Chart with equal sample sizes
25
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

Adapted From Business Statistics, 5th
Edition Groebner, et al, pp 56 (See Reference
Slide)
26
Creating a p-Chart with unequal sample sizes

• Next, calculate the upper and lower control
  limits for each subgroup individually
• Formula

Adapted From Business Statistics, 5th
Edition Groebner, et al, pp 56 (See Reference
Slide)
27
Creating a p-Chart with unequal sample sizes

• Formula
• Applied

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!
Adapted From Business Statistics, 5th
Edition Groebner, et al, pp 56 (See Reference
Slide)
28
Creating a p-Chart with unequal sample sizes

• Formula
• Applied

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!
Adapted From Business Statistics, 5th
Edition Groebner, et al, pp 56 (See Reference
Slide)
29
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 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
30
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 p-bar (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

31
Creating a p-Chart with unequal sample sizes
32
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

From Statistics for Dummies Deborah Rumsey, pp
307 (See Reference Slide)
33
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
departments 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.
34
A Real World Example
The following is the data collected over a
period of 30 days by the emergency department
manager
35
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
36
A Real World Example
37
A Real World Example
38
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.
39
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
40
P-Chart Exercise
Calculate Mean Subgroup Proportion Standard
Error UCL / UCL Build a p-Chart Analyze the
chart Is the process in control?
41
P-Chart Exercise
Solutions Mean Subgroup Proportion 0.049 Stan
dard Error 0.021 Upper Control
Limit 0.115 Lower Control Limit 0.000 Act
ually calculated -0.016, but a negative number
is not a legitimate number of defects,
therefore 0.000 is used as a realistic
substitute
42
P-Chart Exercise
Solutions
43
P-Chart Exercise
Solutions
44
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

45
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.
46
p-ChartsAttribute Based Control Charts

• By James Patterson