IENG 486 - Lecture 15
Gage Capability Studies
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IENG 486 Statistical Quality &
Process Control
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Bonus Points # 3
 In teams of 4 people, go to the Project Office and
perform a gage R & R study on the 7 parts.
 Half of the teams measured with the micrometer
 Half of the teams measured with the dial calipers
 Entire team works together to analyze the data –
for the 4 Operators, estimate:





σ2 total
σ2 repeatability
σ2 reproducability
σ2 product
P/T for gage, assuming USL – LSL = 0.005”
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Bonus Points # 4
 In teams of 4 people, go to the CIM Lab (CM 203) and set up a control
chart strategy for the “pipe-bomb” machine.
 Dr. Jensen will demonstrate the system, each team operates afterward.
 The team will collect data using the scale, and track the data using the
spreadsheet template, each control chart should have 30 samples.
 Entire team works together to collect and analyze the data for the
system, and to create and interpret x – and R – charts.
 For the lab exercise, briefly report:



What your control chart strategy is (what did you measure and why)
Turn in print out of your trial control charts, and describe how the limits were
developed
For each control chart, use your Trial Control Limits* on all 30 sample points, and
interpret each chart for control using the 4 Western Electric Rules:


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Convert your Trial Control Limit data to Standards
Circle Western Electric Rule violations, and describe what they show
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Gage Capability Studies
 Ensuring adequate gage and inspection system capability
 In any problem involving measurement the observed
variability in product is due to two sources:


Product variability - σ2product
Gage variability - σ2gage
i.e., measurement error
 Total observed variance in product:
σ2total = σ2product + σ2gage (system)
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e.g. Assessing Gage
Capability
 Following data were taken by one operator
during gage capability study.
Measurement
Part #
1
2
3
4
5
6
…
17
18
19
20
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1
21
24
20
27
19
23
…
20
19
25
19
2
20
23
21
27
18
21
…
20
21
26
19
x-bar R
20.5
1
23.5
1
20.5
1
27
0
18.5
1
22
2
…
…
20
0
20
2
25.5
1
19
0
x  22.3
R  1.0
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e.g. Assessing Gage
Capability Cont'd
 Estimate standard deviation of measurement error:
 gage  R d2  1.0 1.128  0.8865
 Dist. of measurement error is usually well approximated by the
Normal, therefore

Estimate gage capability:
6ˆ gage  6  0.8865  5.32

That is, individual measurements expected to vary as much as
3 gage  3  0.8865  2.67
owing to gage error.
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Precision-to-Tolerance
(P/T) Ratio
 Common practice to compare gage capability with the width
of the specifications
 In gage capability, the spec width is called the tolerance band

(not to be confused with natural tolerance limits, NTLs)
P T
6ˆ gage
USL  LSL
 Specs for above example: 32.5 ± 27.5
PT
 Rule of Thumb:

6ˆ gage
USL  LSL

6  0.8865
55
 0.0967
P/T  0.1  Adequate gage capability
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Estimating Variance Components
of Total Observed Variability
 Estimate total variance:
2
ˆ total
 S2
2
1 n
S 
xi  x 


n  1 i 1
2
1 n
2

 xi  22.3  10.05

39 i 1
 Compute an estimate of product variance
 Since : 2
2
2
 total   product   gage
2
2
2
ˆ product
 ˆ total
 ˆ gage
 10.05   0.8865  9.26
2
ˆ product  9.26  3.04
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Gage Std Dev Can Be Expressed
as % of Product Std Dev
 Gage standard deviation as percentage of product
standard deviation :
ˆ gage
0.8865
100% 
100%  29.2%
ˆ product
3.04
 This is often a more meaningful expression,
because it does not depend on the width of the
specification limits
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Using x and R-Charts for a
Gage Capability Study
 On x chart for measurements:


Expect to see many out-of-control points
x chart has different meaning than for process control

shows the ability of the gage to discriminate between units
(discriminating power of instrument)
 Why? Because estimate of σx used for control limits based
only on measurement error, i.e.:   
R d
x
gage
2
X-bar Chart for Measurements
X-bar
30
UCL = 24.18
28
CTR = 22.30
26
LCL = 20.42
24
22
20
18
0
4
8
12
16
20
Subgroup
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Using x and R-Charts for a
Gage Capability Study
 On R-chart for measurements:


R-chart directly shows magnitude of measurement error
Values represent differences between measurements made by same
operator on same unit using same instrument
Range Chart for Measurements
4
UCL = 3.27
CTR = 1.00
Range
3
LCL = 0.00
2
1
0
0
4
8
12
16
20
Subgroup
 Interpretation of chart:


In-control: operator has no difficulty making consistent measurements
Out-of-control: operator has difficulty making consistent measurements
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Repeatability & Reproducibility:
Gage R & R Study
 If more than one operator used in study then measurement
(gage) error has two components of variance:
σ2total = σ2product + σ2gage
σ2reproducibility + σ2repeatability
 Repeatability:

σ2repeatability - Variance due to measuring instrument
 Reproducibility:

σ2reproducibility - Variance due to different operators
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Ex. Gage R & R Study
 20 parts, 3 operators, each operator measures each part twice
Operator i
xi
Ri
1
22.30
1.00
2
22.28
1.25
3
22.10
1.20
 Estimate repeatability (measurement error):
R
1
3
R

1
3
1.00  1.25  1.20   1.15
1
ˆ repeatabiltiy

 R2  R3 
R
1.15


 1.0195
d 2 1.128
Use d2 for n = 2 since each range uses 2 repeat measures
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Ex. Gage R & R Study
Cont'd
 Estimate reproducibility:

Differences in xi  operator bias since all three operators measured
the same parts
xmax  max  x1 , x2 , x3   22.30
xmin  min  x1 , x2 , x3   22.10
Rx  xmax  xmin  0.20
ˆ reproducibility

Rx
0.20


 0.1181
d 2 1.693
Use d2 for n = 3 since Rx is from sample of size 3
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Ex. Gage R & R Study
Cont'd
 Total Gage variability:
2
2
2
ˆ gage
 ˆ repeatability
 ˆ reproducibility
2
ˆ gage
 1.0195   0.1181  1.0533
2
2
 Gage standard deviation (measurement error):
ˆ gage  1.0533  1.0263
 P/T Ratio:

Specs: USL = 60, LSL = 5

Would like P/T < 0.1
6ˆ gage
6 1.0263
P


 0.1120
T USL  LSL
60  5
 Note: P T  0.1120  0.1
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Comparison of Gage
Capability Examples
σ2 repeatability
σ2 reproducibility
Single
operator
Three
operators
1.0195
0.1181
σ2 product
P/T
0.8865
0.0967
1.0263
0.1120
 Gage capability is not as good when we account
for both reproducibility and repeatability


Train operators to reduce σ2reproducability = 0.1181
Since σ2repeatability = 1.0195 (largest component), direct effort
toward finding another inspection device.
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