Description
Business statistics is the science of good decision making in the face of uncertainty and is used in many disciplines such as financial analysis, econometrics, auditing, production and operations including services improvement, and marketing research. These sources feature regular repetitive publication of series of data. This makes the topic of time series especially important for business statistics. It is also a branch of applied statistics working mostly on data collected as a by-product of doing business or by government agencies.
Business Statistics: A Decision-Making Approach
Introduction to Quality and Statistical Process Control
Chapter Goals
After completing this chapter, you should be able to: Use the seven basic tools of quality Construct and interpret x-charts and R-charts Construct and interpret p-charts Construct and interpret c-charts
-2
Chapter Overview
Quality Management and Tools for Improvement Philosophy of Quality Deming's 14 Points Juran's 10 Steps to Quality Improvement
-3
Tools for Quality Improvement The Basic 7 Tools Control Charts x-bar/R-charts p-charts c-charts
Themes of Quality Management
Primary focus is on process improvement Most variations in process are due to systems Teamwork is integral to quality management Customer satisfaction is a primary goal Organization transformation is necessary It is important to remove fear Higher quality costs less
-4
Deming's 14 Points
1. Create a constancy of purpose toward improvement
become more competitive, stay in business, and provide jobs
2. Adopt the new philosophy
Better to improve now than to react to problems late
3. Stop depending on inspection to achieve quality -- build in quality from the start
Inspection to find defects at the end of production is too late
4. Stop awarding contracts on the basis of low bids
Better to build long-run purchaser/supplier relationships
-5
Deming's 14 Points
(continued)
5. Improve the system continuously to improve quality and thus constantly reduce costs 6. Institute training on the job
Workers and managers must know the difference between common cause and special cause variation
7. Institute leadership
Know the difference between leadership and supervision
8. Drive out fear so that everyone may work effectively. 9. Break down barriers between departments so that people can work as a team.
-6
Deming's 14 Points
(continued)
10. Eliminate slogans and targets for the workforce
They can create adversarial relationships
11. Eliminate quotas and management by objectives 12. Remove barriers to pride of workmanship 13. Institute a vigorous program of education and self-improvement 14. Make the transformation everyone's job
-7
Juran's 10 Steps to Quality Improvement
1. Build awareness of both the need for improvement and the opportunity for improvement 2. Set goals for improvement 3. Organize to meet the goals that have been set 4. Provide training 5. Implement projects aimed at solving problems
-8
Juran's 10 Steps to Quality Improvement
(continued)
6. Report progress 7. Give recognition 8. Communicate the results 9. Keep score 10. Maintain momentum by building improvement into the company's regular systems
-9
The Deming Cycle
Plan
Act
The Deming Cycle
Do
The key is a continuous cycle of improvement
Study
-10
The Basic 7 Tools
1. Process Flowcharts 2. Brainstorming 3. Fishbone Diagram 4. Histogram 5. Trend Charts 6. Scatter Plots 7. Statistical Process Control Charts
-11
The Basic 7 Tools
(continued)
1. 2. 3. 4. 5. 6. 7. Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts
Map out the process to better visualize and understand opportunities for improvement
-12
The Basic 7 Tools
(continued)
1. 2. 3. 4. 5. 6. 7. Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts
Fishbone (cause-and-effect) diagram:
Cause 1 Cause 2
Sub-causes
Problem
Sub-causes
Show patterns of variation
Cause 3
Cause 4
-13
The Basic 7 Tools
(continued)
1. 2. 3. 4. 5. 6. 7. Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts
Identify trend y
time
Examine relationships y
x
-14
The Basic 7 Tools
(continued)
1. 2. 3. 4. 5. 6. 7. Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts
Examine the performance of a process over time X
time
-15
Introduction to Control Charts
Control Charts are used to monitor variation in a measured value from a process
Exhibits trend Can make correction before process is out of control
A process is a repeatable series of steps leading to a specific goal Inherent variation refers to process variation that exists naturally. This variation can be reduced but not eliminated
-16
Process Variation
Total Process Variation Common Cause = Variation Special Cause + Variation
Variation is natural; inherent in the world around us No two products or service experiences are exactly the same With a fine enough gauge, all things can be seen to differ
-17
Sources of Variation
Total Process Variation Common Cause = Variation Special Cause + Variation
Variation is often due to differences in: People Machines Materials Methods Measurement Environment
-18
Common Cause Variation
Total Process Variation Common Cause = Variation Special Cause + Variation
Common cause variation naturally occurring and expected the result of normal variation in materials, tools, machines, operators, and the environment
-19
Special Cause Variation
Total Process Variation Common Cause = Variation Special Cause + Variation
Special cause variation abnormal or unexpected variation has an assignable cause variation beyond what is considered inherent to the process
-20
Statistical Process Control Charts
Show when changes in data are due to:
Special or assignable causes Fluctuations not inherent to a process Represents problems to be corrected Data outside control limits or trend Common causes or chance Inherent random variations Consist of numerous small causes of random variability
-21
Control Chart Basics
Special Cause Variation: Range of unexpected variability
UCL
Common Cause Variation: range of expected variability
+3? - 3?
Process Average LCL time
UCL = Process Average + 3 Standard Deviations LCL = Process Average - 3 Standard Deviations
-22
Process Variability
Special Cause of Variation: A measurement this far from the process average is very unlikely if only expected variation is present
±3?? 99.7% of process values should be in this range
UCL Process Average LCL time
UCL = Process Average + 3 Standard Deviations LCL = Process Average - 3 Standard Deviations
-23
Statistical Process Control Charts
Statistical Process Control Charts x-chart and R-chart
Used for measured numeric data
p-chart
c-chart
Used for proportions (attribute data)
Used for number of attributes per sampling unit
-24
x-chart and R-chart
Used for measured numeric data from a process Start with at least 20 subgroups of observed values Subgroups usually contain 3 to 6 observations each
-25
Steps to create an x-chart and an R-chart
Calculate subgroup means and ranges Compute the average of the subgroup means and the average range value Prepare graphs of the subgroup means and ranges as a line chart
-26
Steps to create an x-chart and an R-chart
(continued)
Compute the upper and lower control limits for the x-chart Compute the upper and lower control limits for the R-chart Use lines to show the control limits on the x-chart and R-chart
-27
Example: x-chart
Process measurements:
Subgroup measures
Subgroup Individual measurements number
Mean, x 14.5 13.0 19.0 ?
Average subgroup mean = x
Range, R 6 73 ?
Average subgroup range = R
1 2
15 12 17 4? ?
17 16 21 ?
15 9 18 ?
11 15 20 ?
-28
Average of Subgroup Means and Ranges
Average of subgroup means: Average of subgroup ranges:
x i x ?? k
where: where: xi = ith subgroup average k = number of subgroups
R ?
i
R k
Ri = ith subgroup range k = number of subgroups
-29
Computing Control Limits
The upper and lower control limits for an x-chart are generally defined as
UCL = Process Average + 3 Standard Deviations LCL = Process Average - 3 Standard Deviations
or
UCL? x? 3? LCL? x? 3?
-30
Computing Control Limits
(continued)
Since control charts were developed before it was easy to calculate ?, the interval was formed using R instead The value A2R is used to estimate 3? , where A2 is from Appendix Q The upper and lower control limits are
UCL? x? A2(R) LCL? x? A2(R)
-31
where A2 = Shewhart factor for subgroup size n from appendix Q
Example: R-chart
The upper and lower control limits for an R-chart are
UCL? D4(R) LCL? D3(R)
where: D4 and D3 are taken from the Shewhart table (appendix Q) for subgroup size = n
-32
x-chart and R-chart
UCL
x-chart
x
LCL time UCL
R-chart
R
LCL time
-33
Using Control Charts
Control Charts are used to check for process control H0: The process is in control
i.e., variation is only due to common causes
HA: The process is out of control
i.e., special cause variation exists
If the process is found to be out of control, steps should be taken to find and eliminate the special causes of variation
-34
Process In Control
Process in control: points are randomly distributed around the center line and all points are within the control limits
x
UCL
x
LCL
time
-35
Process Not in Control
Out of control conditions:
One or more points outside control limits Nine or more points in a row on one side of the center line Six or more points moving in the same direction 14 or more points alternating above and below the center line
-36
Process Not in Control
One or more points outside control limits UCL Nine or more points in a row on one side of the center line UCL
x
LCL Six or more points moving in the same direction UCL
x
LCL 14 or more points alternating above and below the center line UCL
x
LCL
-37
x
LCL
Out-of-control Processes
When the control chart indicates an out-ofcontrol 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
-38
p-Chart
Control chart for proportions
Is an attribute chart
Shows proportion of nonconforming items
Example -- Computer chips: Count the number of defective chips and divide by total chips inspected Chip is either defective or not defective Finding a defective chip can be classified a "success"
-39
p-Chart
(continued)
Used with equal or unequal sample sizes (subgroups) over time
Unequal sizes should not differ by more than ±25% from average sample sizes Easier to develop with equal sample sizes
Should have np ? 5 and n(1-p) ? 5
-40
Creating a p-Chart
Calculate subgroup proportions Compute the average of the subgroup proportions Prepare graphs of the subgroup proportions as a line chart Compute the upper and lower control limits Use lines to show the control limits on the p-chart
-41
p-Chart Example
Subgroup number Sample size Number of successes Proportion of successes, p
1 2 3 ?
150 150 150
15 12 17 ?
.1000 .0800 .1133 ?
Average subgroup proportion = p
-42
Average of Subgroup Proportions
The average of subgroup proportions = p
If equal sample sizes: If unequal sample sizes:
??
pi k
??
npi i n
i
where: pi = sample proportion for subgroup i k = number of subgroups of size n
-43
where: ni = number of items in sample i ni = total number of items sampled in k samples
Computing Control Limits
The upper and lower control limits for an p-chart are
UCL = Average Proportion + 3 Standard Deviations LCL = Average Proportion - 3 Standard Deviations
or
UCL? LCL?
? 3? ? ?
-44
Standard Deviation of Subgroup Proportions
The estimate of the standard deviation for the subgroup proportions is
If equal sample sizes: If unequal sample sizes: Generally, sp is computed separately for each different sample size
sp? ?
where:
(p)(1 n
p = mean subgroup proportion
n = common sample size
-45
Computing Control Limits
(continued)
The upper and lower control limits for the p-chart are
UCL? p? LCL? p? If sample sizes are equal, this becomes
p) p
)
Proportions are never negative, so if the calculated lower control limit is negative, set LCL = 0
UCL? p? LCL? p?
n n
-46
p-Chart Examples
For equal sample sizes
UCL p LCL
For unequal sample sizes
UCL p LCL
sp is constant since
n is the same for all subgroups
sp varies for each
subgroup since ni varies
-47
c-Chart
Control chart for number of nonconformities (occurrences) per sampling unit (an area of opportunity)
Also a type of attribute chart
Shows total number of nonconforming items per unit
examples: number of flaws per pane of glass number of errors per page of code
Assume that the size of each sampling unit remains constant
-48
Mean and Standard Deviation for a c-Chart
The mean for a c-chart is The standard deviation for a c-chart is
xi c ?? k
where: xi = number of successes per sampling unit k = number of sampling units
s
c
-49
c-Chart Control Limits
The control limits for a c-chart are
UCL? c? 3 c LCL? c? 3 c
-50
Process Control
Determine process control for p-chars and c-charts using the same rules as for x-bar and R-charts Out of control conditions:
One or more points outside control limits Nine or more points in a row on one side of the center line Six or more points moving in the same direction 14 or more points alternating above and below the center line
-51
c-Chart Example
A weaving machine makes cloth in a standard width. Random samples of 10 meters of cloth are examined for flaws. Is the process in control?
Sample number Flaws found
1 2
2 1
3 3
4 0
5 5
6 1
7 0
-52
Constructing the c-Chart
The mean and standard deviation are:
x i c ?? k ?2? 1 3? 0? 5? 1 0 7 c? 1.7143? 1.3093 1.7143
s?
The control limits are:
UCL? 5.642
c?
3 c?
1.7143?
3(1.3093)?
LCL?
c?
3 c?
1.7143?
3(1.3093)? ? 2.214
Note: LCL < 0 so set LCL = 0
-53
The completed c-Chart
6 5 4 3 2 1 0 1 2 3 4 5 6 7
UCL = 5.642
c = 1.714 LCL = 0 Sample number
The process is in control. Individual points are distributed around the center line without any pattern. Any improvement in the process must come from reduction in common-cause variation
-54
Chapter Summary
Reviewed the philosophy of quality management
Demings 14 points Juran's 10 steps
Described the seven basic tools of quality Discussed the theory of control charts
Common cause variation vs. special cause variation
Constructed and interpreted x-charts and Rcharts Constructed and interpreted p-charts Constructed and interpreted c-charts
-55
doc_554575455.docx
Business statistics is the science of good decision making in the face of uncertainty and is used in many disciplines such as financial analysis, econometrics, auditing, production and operations including services improvement, and marketing research. These sources feature regular repetitive publication of series of data. This makes the topic of time series especially important for business statistics. It is also a branch of applied statistics working mostly on data collected as a by-product of doing business or by government agencies.
Business Statistics: A Decision-Making Approach
Introduction to Quality and Statistical Process Control
Chapter Goals
After completing this chapter, you should be able to: Use the seven basic tools of quality Construct and interpret x-charts and R-charts Construct and interpret p-charts Construct and interpret c-charts
-2
Chapter Overview
Quality Management and Tools for Improvement Philosophy of Quality Deming's 14 Points Juran's 10 Steps to Quality Improvement
-3
Tools for Quality Improvement The Basic 7 Tools Control Charts x-bar/R-charts p-charts c-charts
Themes of Quality Management
Primary focus is on process improvement Most variations in process are due to systems Teamwork is integral to quality management Customer satisfaction is a primary goal Organization transformation is necessary It is important to remove fear Higher quality costs less
-4
Deming's 14 Points
1. Create a constancy of purpose toward improvement
become more competitive, stay in business, and provide jobs
2. Adopt the new philosophy
Better to improve now than to react to problems late
3. Stop depending on inspection to achieve quality -- build in quality from the start
Inspection to find defects at the end of production is too late
4. Stop awarding contracts on the basis of low bids
Better to build long-run purchaser/supplier relationships
-5
Deming's 14 Points
(continued)
5. Improve the system continuously to improve quality and thus constantly reduce costs 6. Institute training on the job
Workers and managers must know the difference between common cause and special cause variation
7. Institute leadership
Know the difference between leadership and supervision
8. Drive out fear so that everyone may work effectively. 9. Break down barriers between departments so that people can work as a team.
-6
Deming's 14 Points
(continued)
10. Eliminate slogans and targets for the workforce
They can create adversarial relationships
11. Eliminate quotas and management by objectives 12. Remove barriers to pride of workmanship 13. Institute a vigorous program of education and self-improvement 14. Make the transformation everyone's job
-7
Juran's 10 Steps to Quality Improvement
1. Build awareness of both the need for improvement and the opportunity for improvement 2. Set goals for improvement 3. Organize to meet the goals that have been set 4. Provide training 5. Implement projects aimed at solving problems
-8
Juran's 10 Steps to Quality Improvement
(continued)
6. Report progress 7. Give recognition 8. Communicate the results 9. Keep score 10. Maintain momentum by building improvement into the company's regular systems
-9
The Deming Cycle
Plan
Act
The Deming Cycle
Do
The key is a continuous cycle of improvement
Study
-10
The Basic 7 Tools
1. Process Flowcharts 2. Brainstorming 3. Fishbone Diagram 4. Histogram 5. Trend Charts 6. Scatter Plots 7. Statistical Process Control Charts
-11
The Basic 7 Tools
(continued)
1. 2. 3. 4. 5. 6. 7. Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts
Map out the process to better visualize and understand opportunities for improvement
-12
The Basic 7 Tools
(continued)
1. 2. 3. 4. 5. 6. 7. Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts
Fishbone (cause-and-effect) diagram:
Cause 1 Cause 2
Sub-causes
Problem
Sub-causes
Show patterns of variation
Cause 3
Cause 4
-13
The Basic 7 Tools
(continued)
1. 2. 3. 4. 5. 6. 7. Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts
Identify trend y
time
Examine relationships y
x
-14
The Basic 7 Tools
(continued)
1. 2. 3. 4. 5. 6. 7. Process Flowcharts Brainstorming Fishbone Diagram Histogram Trend Charts Scatter Plots Statistical Process Control Charts
Examine the performance of a process over time X
time
-15
Introduction to Control Charts
Control Charts are used to monitor variation in a measured value from a process
Exhibits trend Can make correction before process is out of control
A process is a repeatable series of steps leading to a specific goal Inherent variation refers to process variation that exists naturally. This variation can be reduced but not eliminated
-16
Process Variation
Total Process Variation Common Cause = Variation Special Cause + Variation
Variation is natural; inherent in the world around us No two products or service experiences are exactly the same With a fine enough gauge, all things can be seen to differ
-17
Sources of Variation
Total Process Variation Common Cause = Variation Special Cause + Variation
Variation is often due to differences in: People Machines Materials Methods Measurement Environment
-18
Common Cause Variation
Total Process Variation Common Cause = Variation Special Cause + Variation
Common cause variation naturally occurring and expected the result of normal variation in materials, tools, machines, operators, and the environment
-19
Special Cause Variation
Total Process Variation Common Cause = Variation Special Cause + Variation
Special cause variation abnormal or unexpected variation has an assignable cause variation beyond what is considered inherent to the process
-20
Statistical Process Control Charts
Show when changes in data are due to:
Special or assignable causes Fluctuations not inherent to a process Represents problems to be corrected Data outside control limits or trend Common causes or chance Inherent random variations Consist of numerous small causes of random variability
-21
Control Chart Basics
Special Cause Variation: Range of unexpected variability
UCL
Common Cause Variation: range of expected variability
+3? - 3?
Process Average LCL time
UCL = Process Average + 3 Standard Deviations LCL = Process Average - 3 Standard Deviations
-22
Process Variability
Special Cause of Variation: A measurement this far from the process average is very unlikely if only expected variation is present
±3?? 99.7% of process values should be in this range
UCL Process Average LCL time
UCL = Process Average + 3 Standard Deviations LCL = Process Average - 3 Standard Deviations
-23
Statistical Process Control Charts
Statistical Process Control Charts x-chart and R-chart
Used for measured numeric data
p-chart
c-chart
Used for proportions (attribute data)
Used for number of attributes per sampling unit
-24
x-chart and R-chart
Used for measured numeric data from a process Start with at least 20 subgroups of observed values Subgroups usually contain 3 to 6 observations each
-25
Steps to create an x-chart and an R-chart
Calculate subgroup means and ranges Compute the average of the subgroup means and the average range value Prepare graphs of the subgroup means and ranges as a line chart
-26
Steps to create an x-chart and an R-chart
(continued)
Compute the upper and lower control limits for the x-chart Compute the upper and lower control limits for the R-chart Use lines to show the control limits on the x-chart and R-chart
-27
Example: x-chart
Process measurements:
Subgroup measures
Subgroup Individual measurements number
Mean, x 14.5 13.0 19.0 ?
Average subgroup mean = x
Range, R 6 73 ?
Average subgroup range = R
1 2
15 12 17 4? ?
17 16 21 ?
15 9 18 ?
11 15 20 ?
-28
Average of Subgroup Means and Ranges
Average of subgroup means: Average of subgroup ranges:
x i x ?? k
where: where: xi = ith subgroup average k = number of subgroups
R ?
i
R k
Ri = ith subgroup range k = number of subgroups
-29
Computing Control Limits
The upper and lower control limits for an x-chart are generally defined as
UCL = Process Average + 3 Standard Deviations LCL = Process Average - 3 Standard Deviations
or
UCL? x? 3? LCL? x? 3?
-30
Computing Control Limits
(continued)
Since control charts were developed before it was easy to calculate ?, the interval was formed using R instead The value A2R is used to estimate 3? , where A2 is from Appendix Q The upper and lower control limits are
UCL? x? A2(R) LCL? x? A2(R)
-31
where A2 = Shewhart factor for subgroup size n from appendix Q
Example: R-chart
The upper and lower control limits for an R-chart are
UCL? D4(R) LCL? D3(R)
where: D4 and D3 are taken from the Shewhart table (appendix Q) for subgroup size = n
-32
x-chart and R-chart
UCL
x-chart
x
LCL time UCL
R-chart
R
LCL time
-33
Using Control Charts
Control Charts are used to check for process control H0: The process is in control
i.e., variation is only due to common causes
HA: The process is out of control
i.e., special cause variation exists
If the process is found to be out of control, steps should be taken to find and eliminate the special causes of variation
-34
Process In Control
Process in control: points are randomly distributed around the center line and all points are within the control limits
x
UCL
x
LCL
time
-35
Process Not in Control
Out of control conditions:
One or more points outside control limits Nine or more points in a row on one side of the center line Six or more points moving in the same direction 14 or more points alternating above and below the center line
-36
Process Not in Control
One or more points outside control limits UCL Nine or more points in a row on one side of the center line UCL
x
LCL Six or more points moving in the same direction UCL
x
LCL 14 or more points alternating above and below the center line UCL
x
LCL
-37
x
LCL
Out-of-control Processes
When the control chart indicates an out-ofcontrol 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
-38
p-Chart
Control chart for proportions
Is an attribute chart
Shows proportion of nonconforming items
Example -- Computer chips: Count the number of defective chips and divide by total chips inspected Chip is either defective or not defective Finding a defective chip can be classified a "success"
-39
p-Chart
(continued)
Used with equal or unequal sample sizes (subgroups) over time
Unequal sizes should not differ by more than ±25% from average sample sizes Easier to develop with equal sample sizes
Should have np ? 5 and n(1-p) ? 5
-40
Creating a p-Chart
Calculate subgroup proportions Compute the average of the subgroup proportions Prepare graphs of the subgroup proportions as a line chart Compute the upper and lower control limits Use lines to show the control limits on the p-chart
-41
p-Chart Example
Subgroup number Sample size Number of successes Proportion of successes, p
1 2 3 ?
150 150 150
15 12 17 ?
.1000 .0800 .1133 ?
Average subgroup proportion = p
-42
Average of Subgroup Proportions
The average of subgroup proportions = p
If equal sample sizes: If unequal sample sizes:
??
pi k
??
npi i n
i
where: pi = sample proportion for subgroup i k = number of subgroups of size n
-43
where: ni = number of items in sample i ni = total number of items sampled in k samples
Computing Control Limits
The upper and lower control limits for an p-chart are
UCL = Average Proportion + 3 Standard Deviations LCL = Average Proportion - 3 Standard Deviations
or
UCL? LCL?
? 3? ? ?
-44
Standard Deviation of Subgroup Proportions
The estimate of the standard deviation for the subgroup proportions is
If equal sample sizes: If unequal sample sizes: Generally, sp is computed separately for each different sample size
sp? ?
where:
(p)(1 n
p = mean subgroup proportion
n = common sample size
-45
Computing Control Limits
(continued)
The upper and lower control limits for the p-chart are
UCL? p? LCL? p? If sample sizes are equal, this becomes
p) p
)
Proportions are never negative, so if the calculated lower control limit is negative, set LCL = 0
UCL? p? LCL? p?
n n
-46
p-Chart Examples
For equal sample sizes
UCL p LCL
For unequal sample sizes
UCL p LCL
sp is constant since
n is the same for all subgroups
sp varies for each
subgroup since ni varies
-47
c-Chart
Control chart for number of nonconformities (occurrences) per sampling unit (an area of opportunity)
Also a type of attribute chart
Shows total number of nonconforming items per unit
examples: number of flaws per pane of glass number of errors per page of code
Assume that the size of each sampling unit remains constant
-48
Mean and Standard Deviation for a c-Chart
The mean for a c-chart is The standard deviation for a c-chart is
xi c ?? k
where: xi = number of successes per sampling unit k = number of sampling units
s
c
-49
c-Chart Control Limits
The control limits for a c-chart are
UCL? c? 3 c LCL? c? 3 c
-50
Process Control
Determine process control for p-chars and c-charts using the same rules as for x-bar and R-charts Out of control conditions:
One or more points outside control limits Nine or more points in a row on one side of the center line Six or more points moving in the same direction 14 or more points alternating above and below the center line
-51
c-Chart Example
A weaving machine makes cloth in a standard width. Random samples of 10 meters of cloth are examined for flaws. Is the process in control?
Sample number Flaws found
1 2
2 1
3 3
4 0
5 5
6 1
7 0
-52
Constructing the c-Chart
The mean and standard deviation are:
x i c ?? k ?2? 1 3? 0? 5? 1 0 7 c? 1.7143? 1.3093 1.7143
s?
The control limits are:
UCL? 5.642
c?
3 c?
1.7143?
3(1.3093)?
LCL?
c?
3 c?
1.7143?
3(1.3093)? ? 2.214
Note: LCL < 0 so set LCL = 0
-53
The completed c-Chart
6 5 4 3 2 1 0 1 2 3 4 5 6 7
UCL = 5.642
c = 1.714 LCL = 0 Sample number
The process is in control. Individual points are distributed around the center line without any pattern. Any improvement in the process must come from reduction in common-cause variation
-54
Chapter Summary
Reviewed the philosophy of quality management
Demings 14 points Juran's 10 steps
Described the seven basic tools of quality Discussed the theory of control charts
Common cause variation vs. special cause variation
Constructed and interpreted x-charts and Rcharts Constructed and interpreted p-charts Constructed and interpreted c-charts
-55
doc_554575455.docx