Description
the statistical quality control methods, types of control chart, control chart applications.
Statistical Quality Control Methods
Statistical Process Control (SPC)
• A methodology for monitoring a process to
identify special causes of variation and signal
the need to take corrective action when
appropriate
• SPC relies on control charts
SPC: Basic Forms of Variation
• Assignable variation is caused by factors
that can be clearly identified and
possibly managed
- Random variation is inherent in the
production process
Types of Control Charts
• Either attributes or variables can be
measured and compared to standards.
• Attributes are characteristics that are
classified into one of two categories, usually
defective (not meeting specifications) or non-
defective (meeting specifications).
• Variables are characteristics that can be
measured on a continuous scale (weight,
length, etc.).
Control Chart Applications
• Establish state of statistical control
• Monitor a process and signal when it goes out of
control
• Determine process capability
Commonly Used Control Charts
• Variables data
– x-bar and R-charts
– x-bar and s-charts
• Attribute data
– For “defectives” (p-chart, np-chart)
– For “defects” (c-chart, u-chart)
Developing Control Charts
1. Prepare
– Choose measurement
– Determine how to collect data, sample size, and
frequency of sampling
– Set up an initial control chart
2. Collect Data
– Record data
– Calculate appropriate statistics
– Plot statistics on chart
Next Steps
3. Determine trial control limits
– Center line (process average)
– Compute UCL, LCL
4. Analyze and interpret results
– Determine if in control
– Eliminate out-of-control points
– Re-compute control limits as necessary
Final Steps
5. Use as a problem-solving tool
– Continue to collect and plot data
– Take corrective action when necessary
6. Compute process capability
Control Limits
We establish the Upper Control Limits (UCL) and the Lower
Control Limits (LCL) with plus or minus 3 standard deviations
from some x-bar or mean value. Based on this we can
expect 99.7% of our sample observations to fall within these
limits.
x
LCL UCL
99.7%
Example of x-bar and R Charts:
Required Data
Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5
1 10.68 10.689 10.776 10.798 10.714
2 10.79 10.86 10.601 10.746 10.779
3 10.78 10.667 10.838 10.785 10.723
4 10.59 10.727 10.812 10.775 10.73
5 10.69 10.708 10.79 10.758 10.671
6 10.75 10.714 10.738 10.719 10.606
7 10.79 10.713 10.689 10.877 10.603
8 10.74 10.779 10.11 10.737 10.75
9 10.77 10.773 10.641 10.644 10.725
10 10.72 10.671 10.708 10.85 10.712
11 10.79 10.821 10.764 10.658 10.708
12 10.62 10.802 10.818 10.872 10.727
13 10.66 10.822 10.893 10.544 10.75
14 10.81 10.749 10.859 10.801 10.701
15 10.66 10.681 10.644 10.747 10.728
Example of x-bar and R charts: Step 1. Calculate sample means, sample ranges, mean of
means, and mean of ranges.
Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 Avg Range
1 10.68 10.689 10.776 10.798 10.714 10.732 0.116
2 10.79 10.86 10.601 10.746 10.779 10.755 0.259
3 10.78 10.667 10.838 10.785 10.723 10.759 0.171
4 10.59 10.727 10.812 10.775 10.73 10.727 0.221
5 10.69 10.708 10.79 10.758 10.671 10.724 0.119
6 10.75 10.714 10.738 10.719 10.606 10.705 0.143
7 10.79 10.713 10.689 10.877 10.603 10.735 0.274
8 10.74 10.779 10.11 10.737 10.75 10.624 0.669
9 10.77 10.773 10.641 10.644 10.725 10.710 0.132
10 10.72 10.671 10.708 10.85 10.712 10.732 0.179
11 10.79 10.821 10.764 10.658 10.708 10.748 0.163
12 10.62 10.802 10.818 10.872 10.727 10.768 0.250
13 10.66 10.822 10.893 10.544 10.75 10.733 0.349
14 10.81 10.749 10.859 10.801 10.701 10.783 0.158
15 10.66 10.681 10.644 10.747 10.728 10.692 0.103
Averages 10.728 0.220400
Example of x-bar and R charts: Step 2. Determine Control Limit Formulas and Necessary
Tabled Values
x Chart Control Limits
UCL = x + A R
LCL = x - A R
2
2
R Chart Control Limits
UCL = D R
LCL = D R
4
3
n A2 D3 D4
2 1.88 0 3.27
3 1.02 0 2.57
4 0.73 0 2.28
5 0.58 0 2.11
6 0.48 0 2.00
7 0.42 0.08 1.92
8 0.37 0.14 1.86
9 0.34 0.18 1.82
10 0.31 0.22 1.78
11 0.29 0.26 1.74
Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart and Plot Values
10.601
10.856
= ) .58(0.2204 - 10.728 R A - x = LCL
= ) .58(0.2204 - 10.728 R A + x = UCL
2
2
=
=
10.550
10.600
10.650
10.700
10.750
10.800
10.850
10.900
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Sample
M
e
a
n
s
UCL
LCL
Example of x-bar and R charts: Steps 5&6. Calculate R-chart and Plot Values
0
0.46504
= =
= =
) 2204 . 0 )( 0 ( R D = LCL
) 2204 . 0 )( 11 . 2 ( R D = UCL
3
4
0 . 0 0 0
0 . 1 0 0
0 . 2 0 0
0 . 3 0 0
0 . 4 0 0
0 . 5 0 0
0 . 6 0 0
0 . 7 0 0
0 . 8 0 0
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5
S a m p l e
R
UCL
LCL
For a process to be in control
• No points outside control limits
• The number of points above and below the
center line is about the same
• The points seem to fall randomly above and
below the center line
• Most points, but not all are near the center
line, and only a few are close to the control
limits
Typical Out-of-Control Patterns
• Point outside control limits
• Sudden shift in process average
• Cycles
• Trends
• Hugging the center line
• Hugging the control limits
• Instability
Population and Sampling
Distributions
f(x)
x
Population Distribution
Sampling Distribution
of Sample Means
Mean = µ
Std. Dev. = o
x
Mean = x = µ
Std. Error =
x
x
?
? =
n
f(x)
=
x
Capability Versus Control
Control
Capability
Capable
Not Capable
In Control Out of Control
IDEAL
Process Capability
• Process Capability is the inherent ability of a
process to produce similar items that can be
controlled for a sustained period of time, given
a certain set of conditions. The process limits
are compared to a set of given specification
limits.
• The purpose of process capability study is to
determine the limits within which a process
operates and thereby to reduce its variability
Process Capability Index
The process capability index, Cp is
defined as the ratio of the specification
width to the natural tolerance of the
process. Cp relates the natural variation
of the process with the design
specifications in a single, quantitative
measure.
Process Capability Index
C
p
=
UTL - LTL
6o
C
pl
, C
pu
}
UTL - µ
3o
C
pl
=
µ - LTL
3o
C
pk
= min{
C
pu
=
Process Capability
This is a “one-sided” Capability Index
Concentration on the side which is closest to the
specification - closest to being “bad”
)
`
¹
¹
´
¦ ÷ ÷
=
o o 3
;
3
X UTL LTL X
Min C
pk
Values of C
pk
• C
pk
>= 1.33 would imply that the process both
in control and is capable
• 1.00 =10
x and MR
no
yes
x and s
x and R
no
yes
defective defect
constant
sample
size?
p-chart with
variable sample
size
no
p or
np
yes
constant
sampling
unit?
c u
yes no
Acceptance sampling:
• Form of inspection applied to lots or batches of items
before or after a process, to judge conformance with
predetermined standards
Acceptance Sampling
• Purposes
– Determine quality level
– Ensure quality is within predetermined level
• Advantages
– Economy
– Less handling damage
– Fewer inspectors
– Upgrading of the inspection job
– Applicability to destructive testing
– Entire lot rejection (motivation for improvement)
Acceptance Sampling (Continued)
• Disadvantages
– Risks of accepting “bad” lots and rejecting
“good” lots
– Added planning and documentation
– Sample provides less information than 100-
percent inspection
• Sampling plans
–single
–double
–Multiple
SAMPLING ERRORS
Acceptance Sampling
Error, Risk, & Tolerance
– Error
• Sampling error: sample is biased or unrepresentative
• Inspection error: measurement is incorrect
– Risk
• Type I error (producer’s risk)
• Type II error (consumer’s risk)
– Tolerance levels
• Acceptable Quality Level (AQL)
• Lot Tolerance Percent Defective (LTPD)
Terminology
• Acceptable Quality Level (AQL)
– Max. acceptable percentage of defectives
defined by producer
• The o (Producer’s risk)
– The probability of rejecting a good lot
• Lot Tolerance Percent Defective (LTPD)
– Percentage of defectives that defines
consumer’s rejection point
• The | (Consumer’s risk)
– The probability of accepting a bad lot
Operating Characteristic Curve
n = 99
c = 4
AQL LTPD
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8 9 10 11 12
Percent defective
P
r
o
b
a
b
i
l
i
t
y
o
f
a
c
c
e
p
t
a
n
c
e
| =.10
(consumer’s risk)
o = .05 (producer’s risk)
The OCC brings the concepts of producer’s risk, consumer’s risk,
sample size, and maximum defects allowed together
The shape or
slope of the
curve is
dependent on
a particular
combination
of the four
parameters
doc_549616154.pptx
the statistical quality control methods, types of control chart, control chart applications.
Statistical Quality Control Methods
Statistical Process Control (SPC)
• A methodology for monitoring a process to
identify special causes of variation and signal
the need to take corrective action when
appropriate
• SPC relies on control charts
SPC: Basic Forms of Variation
• Assignable variation is caused by factors
that can be clearly identified and
possibly managed
- Random variation is inherent in the
production process
Types of Control Charts
• Either attributes or variables can be
measured and compared to standards.
• Attributes are characteristics that are
classified into one of two categories, usually
defective (not meeting specifications) or non-
defective (meeting specifications).
• Variables are characteristics that can be
measured on a continuous scale (weight,
length, etc.).
Control Chart Applications
• Establish state of statistical control
• Monitor a process and signal when it goes out of
control
• Determine process capability
Commonly Used Control Charts
• Variables data
– x-bar and R-charts
– x-bar and s-charts
• Attribute data
– For “defectives” (p-chart, np-chart)
– For “defects” (c-chart, u-chart)
Developing Control Charts
1. Prepare
– Choose measurement
– Determine how to collect data, sample size, and
frequency of sampling
– Set up an initial control chart
2. Collect Data
– Record data
– Calculate appropriate statistics
– Plot statistics on chart
Next Steps
3. Determine trial control limits
– Center line (process average)
– Compute UCL, LCL
4. Analyze and interpret results
– Determine if in control
– Eliminate out-of-control points
– Re-compute control limits as necessary
Final Steps
5. Use as a problem-solving tool
– Continue to collect and plot data
– Take corrective action when necessary
6. Compute process capability
Control Limits
We establish the Upper Control Limits (UCL) and the Lower
Control Limits (LCL) with plus or minus 3 standard deviations
from some x-bar or mean value. Based on this we can
expect 99.7% of our sample observations to fall within these
limits.
x
LCL UCL
99.7%
Example of x-bar and R Charts:
Required Data
Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5
1 10.68 10.689 10.776 10.798 10.714
2 10.79 10.86 10.601 10.746 10.779
3 10.78 10.667 10.838 10.785 10.723
4 10.59 10.727 10.812 10.775 10.73
5 10.69 10.708 10.79 10.758 10.671
6 10.75 10.714 10.738 10.719 10.606
7 10.79 10.713 10.689 10.877 10.603
8 10.74 10.779 10.11 10.737 10.75
9 10.77 10.773 10.641 10.644 10.725
10 10.72 10.671 10.708 10.85 10.712
11 10.79 10.821 10.764 10.658 10.708
12 10.62 10.802 10.818 10.872 10.727
13 10.66 10.822 10.893 10.544 10.75
14 10.81 10.749 10.859 10.801 10.701
15 10.66 10.681 10.644 10.747 10.728
Example of x-bar and R charts: Step 1. Calculate sample means, sample ranges, mean of
means, and mean of ranges.
Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 Avg Range
1 10.68 10.689 10.776 10.798 10.714 10.732 0.116
2 10.79 10.86 10.601 10.746 10.779 10.755 0.259
3 10.78 10.667 10.838 10.785 10.723 10.759 0.171
4 10.59 10.727 10.812 10.775 10.73 10.727 0.221
5 10.69 10.708 10.79 10.758 10.671 10.724 0.119
6 10.75 10.714 10.738 10.719 10.606 10.705 0.143
7 10.79 10.713 10.689 10.877 10.603 10.735 0.274
8 10.74 10.779 10.11 10.737 10.75 10.624 0.669
9 10.77 10.773 10.641 10.644 10.725 10.710 0.132
10 10.72 10.671 10.708 10.85 10.712 10.732 0.179
11 10.79 10.821 10.764 10.658 10.708 10.748 0.163
12 10.62 10.802 10.818 10.872 10.727 10.768 0.250
13 10.66 10.822 10.893 10.544 10.75 10.733 0.349
14 10.81 10.749 10.859 10.801 10.701 10.783 0.158
15 10.66 10.681 10.644 10.747 10.728 10.692 0.103
Averages 10.728 0.220400
Example of x-bar and R charts: Step 2. Determine Control Limit Formulas and Necessary
Tabled Values
x Chart Control Limits
UCL = x + A R
LCL = x - A R
2
2
R Chart Control Limits
UCL = D R
LCL = D R
4
3
n A2 D3 D4
2 1.88 0 3.27
3 1.02 0 2.57
4 0.73 0 2.28
5 0.58 0 2.11
6 0.48 0 2.00
7 0.42 0.08 1.92
8 0.37 0.14 1.86
9 0.34 0.18 1.82
10 0.31 0.22 1.78
11 0.29 0.26 1.74
Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart and Plot Values
10.601
10.856
= ) .58(0.2204 - 10.728 R A - x = LCL
= ) .58(0.2204 - 10.728 R A + x = UCL
2
2
=
=
10.550
10.600
10.650
10.700
10.750
10.800
10.850
10.900
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Sample
M
e
a
n
s
UCL
LCL
Example of x-bar and R charts: Steps 5&6. Calculate R-chart and Plot Values
0
0.46504
= =
= =
) 2204 . 0 )( 0 ( R D = LCL
) 2204 . 0 )( 11 . 2 ( R D = UCL
3
4
0 . 0 0 0
0 . 1 0 0
0 . 2 0 0
0 . 3 0 0
0 . 4 0 0
0 . 5 0 0
0 . 6 0 0
0 . 7 0 0
0 . 8 0 0
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5
S a m p l e
R
UCL
LCL
For a process to be in control
• No points outside control limits
• The number of points above and below the
center line is about the same
• The points seem to fall randomly above and
below the center line
• Most points, but not all are near the center
line, and only a few are close to the control
limits
Typical Out-of-Control Patterns
• Point outside control limits
• Sudden shift in process average
• Cycles
• Trends
• Hugging the center line
• Hugging the control limits
• Instability
Population and Sampling
Distributions
f(x)
x
Population Distribution
Sampling Distribution
of Sample Means
Mean = µ
Std. Dev. = o
x
Mean = x = µ
Std. Error =
x
x
?
? =
n
f(x)
=
x
Capability Versus Control
Control
Capability
Capable
Not Capable
In Control Out of Control
IDEAL
Process Capability
• Process Capability is the inherent ability of a
process to produce similar items that can be
controlled for a sustained period of time, given
a certain set of conditions. The process limits
are compared to a set of given specification
limits.
• The purpose of process capability study is to
determine the limits within which a process
operates and thereby to reduce its variability
Process Capability Index
The process capability index, Cp is
defined as the ratio of the specification
width to the natural tolerance of the
process. Cp relates the natural variation
of the process with the design
specifications in a single, quantitative
measure.
Process Capability Index
C
p
=
UTL - LTL
6o
C
pl
, C
pu
}
UTL - µ
3o
C
pl
=
µ - LTL
3o
C
pk
= min{
C
pu
=
Process Capability
This is a “one-sided” Capability Index
Concentration on the side which is closest to the
specification - closest to being “bad”
)
`
¹
¹
´
¦ ÷ ÷
=
o o 3
;
3
X UTL LTL X
Min C
pk
Values of C
pk
• C
pk
>= 1.33 would imply that the process both
in control and is capable
• 1.00 =10
x and MR
no
yes
x and s
x and R
no
yes
defective defect
constant
sample
size?
p-chart with
variable sample
size
no
p or
np
yes
constant
sampling
unit?
c u
yes no
Acceptance sampling:
• Form of inspection applied to lots or batches of items
before or after a process, to judge conformance with
predetermined standards
Acceptance Sampling
• Purposes
– Determine quality level
– Ensure quality is within predetermined level
• Advantages
– Economy
– Less handling damage
– Fewer inspectors
– Upgrading of the inspection job
– Applicability to destructive testing
– Entire lot rejection (motivation for improvement)
Acceptance Sampling (Continued)
• Disadvantages
– Risks of accepting “bad” lots and rejecting
“good” lots
– Added planning and documentation
– Sample provides less information than 100-
percent inspection
• Sampling plans
–single
–double
–Multiple
SAMPLING ERRORS
Acceptance Sampling
Error, Risk, & Tolerance
– Error
• Sampling error: sample is biased or unrepresentative
• Inspection error: measurement is incorrect
– Risk
• Type I error (producer’s risk)
• Type II error (consumer’s risk)
– Tolerance levels
• Acceptable Quality Level (AQL)
• Lot Tolerance Percent Defective (LTPD)
Terminology
• Acceptable Quality Level (AQL)
– Max. acceptable percentage of defectives
defined by producer
• The o (Producer’s risk)
– The probability of rejecting a good lot
• Lot Tolerance Percent Defective (LTPD)
– Percentage of defectives that defines
consumer’s rejection point
• The | (Consumer’s risk)
– The probability of accepting a bad lot
Operating Characteristic Curve
n = 99
c = 4
AQL LTPD
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8 9 10 11 12
Percent defective
P
r
o
b
a
b
i
l
i
t
y
o
f
a
c
c
e
p
t
a
n
c
e
| =.10
(consumer’s risk)
o = .05 (producer’s risk)
The OCC brings the concepts of producer’s risk, consumer’s risk,
sample size, and maximum defects allowed together
The shape or
slope of the
curve is
dependent on
a particular
combination
of the four
parameters
doc_549616154.pptx