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p-ChartsAttribute Based Control Charts

- By James Patterson

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

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

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

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

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

What is a p-Chart?

- A process control chart that measures a

proportion of defective or nonconforming items

within a sample or population

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

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

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.

p-Chart Example

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

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

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

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

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)

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)

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)

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)

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)

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)

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)

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

Creating a p-Chart with equal sample sizes

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)

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)

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)

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)

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

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

Creating a p-Chart with unequal sample sizes

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)

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.

A Real World Example

The following is the data collected over a

period of 30 days by the emergency department

manager

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

A Real World Example

A Real World Example

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.

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

P-Chart Exercise

Calculate Mean Subgroup Proportion Standard

Error UCL / UCL Build a p-Chart Analyze the

chart Is the process in control?

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

P-Chart Exercise

Solutions

P-Chart Exercise

Solutions

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

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.

p-ChartsAttribute Based Control Charts

- By James Patterson