Chapter 9: Issues Relevant to Statistical Process
Control by Variables
Learning Objectives

Discuss the importance of measurement processes to
the effective control of quality.

Differentiate between the concepts of accuracy and
precision, and describe how measurement processes
can be evaluated with respect to each.

Test for linearity in a measurement process and explain
why linearity is important.

Conduct a measurement study to compute the
components of measurement error, including bias,
repeatability, and reproducibility.
Learning Objectives

Conduct a measurement study using Minitab software.

Compute the signal-to-noise ratio (SNR) and the ratio
r and apply to the evaluation of measurement
variation.

Differentiate between resolution and discrimination and
compute a discrimination ratio to evaluate the suitability
of any measurement process.

Discuss the three fundamental reasons for collecting
data from industrial processes and how each reason
impacts quality improvement programs.
Learning Objectives

Describe how different sampling schemes can affect a
process analysis and conclusions concerning process
behavior.

Determine process capability for one-sided and twosided specifications for normally distributed data and
interpret the results.

Determine process capability for one-sided and twosided specifications for when data are not normally
distributed and interpret the results.
Learning Objectives

Use Minitab software to conduct capability analyses.

Use control charts to isolate and separate sources of
process variation.
Accuracy and Precision
The average measurement
corresponds to the true
value
Truth
Truth
.
..
.
.
.. . .
.
truth
time
Figure 9-1: An Accurate Measurement Process
.
.
.
.
.
Truth
..
.
.
.
truth
.
time
Imprecise Measurement Process
Truth
.
.
.
. ... .. . . ..
Precise Measurement Process
Figure 9-2: Measurement Precision
truth
time
Biconical Disc
Upper Die
Upper Heating Platen
Gate Tolerances
Sample 3 (large)
Die Cavity
Where Samples
Are Loaded
j = torque
Sample 2 (medium)
Lower Die
Lower Heating Platen
Oscillating Rotor Shaft
Output Measures for an Oscillating
Disc Rheometer:
Sample 1 (small)
time
t = 3 min
• Elastic Modulus
• Viscous Modulus
• Tangent Delta
• Cure Rate
Rheometer Plots for One Batch of Rubber
Figure 9-3: A Rheometer Test for Tire Rubber
10
Measuring Known Standard
1
458
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
456.71
453.29
455.24
453.84
455.45
455.25
455.66
455.36
455.40
453.74
456.42
454.91
455.21
457.86
456.59
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
453.85
454.74
455.67
453.18
454.82
457.69
455.41
454.35
454.28
455.63
456.31
455.94
454.75
454.62
453.81
1
UCL=457.452
457
456
Individual Value
Weights
Weights
of 454
of 454
Measurement
Measurement
gm
gm
Standard
Standard
455
_
X=454
454
453
452
451
LCL=450.548
450
1
4
7
10
13
16
19
Observation
22
25
28
31
Testing Whether Average =
True Value
Measuring Known Standard
459
UCL=458.608
458
Figure 9-4: A Test for
Accuracy
Individual Value
457
456
_
X=455.156
455
454
453
452
LCL=451.704
451
1
4
7
10
13
16
19
Observation
22
25
28
31
Actual Average of Process
Linearity
Bivariate Fit of Actual Average By Measured Standard
1250
Ac tual Av erage
1000
Plot of Average Measures
Against Standards
Measured
750
500
250
0
0
250
500
750
1000
1250
Meas ured Standar d
Bivariate Fit of Actual Average By Measured Standard
1250
Actual Average
1000
750
500
250
Figure 9-5: Test for Linearity
0
0
250
500
750
Measured Standard
Li near Fi t
1000
1250
How to Conduct a Measurement Study
Measured Value = True Mean of the Process + Process Variability
+ Measurement Bias + Operator Effect (Reproduceability)
+ Replication Error (Repeatability)
2
2
 x2   process
  measurement
where
2
 x2 = total observed variance =  observed

2
x
2
 process
= variance due to process
2
 measurement
= variance due to measurement
2
2
2
2
 measurement
  bias
  repeatability
  reproduceabil
ity
True Mean
Figure 9-6: Anatomy of a Measurement
Measurement Data; Jack and Jill Combined
Sample Size: n = 4
UCL=455.861
_
_
X=454.001
454
LCL=452.141
452
1
4
7
10
13
16
Sample
19
22
25
28
6
Sample Range
Sample Mean
456
UCL=5.824
4
_
R=2.553
2
0
LCL=0
1
4
7
10
13
16
Sample
19
22
25
28
2
 bias
0
2
2
 measurement
 R   2.553 2
2
  

1.2399
 1.537



d
2.059


 2
 measurement  1.537  1.240
Figure 9-7: Measurement Study
Measurement Data; Jack followed by Jane
Sample Size: n = 2
UCL=456.548
Sample Mean
456
_
_
X=454.001
454
452
LCL=451.454
1
1
7
13
19
25
37
43
49
55
1
1
4.5
Sample Range
31
Sample
UCL=4.425
Jane is more variable than Jack
3.0
_
R=1.354
1.5
0.0
LCL=0
1
7
13
19
25
31
Sample
37
43
49
55
Figure 9-8: Combining
Two Operators on
One Chart
Jack
Jane
Measurement Data; Jack's Data
Sample Size: n = 2
UCL=456.221
Sample Mean
456
_
_
X=453.923
454
452
LCL=451.626
1
4
7
10
13
16
Sample
19
22
25
28
4.5
Sample Range
UCL=3.992
Jack
3.0
 Jack 
R 1.222

 1.08
d 2 1.128
in control
_
R=1.222
1.5
0.0
LCL=0
1
4
7
10
13
16
Sample
19
22
25
28
Measurement Data; Jane's Data
Sample Size: n = 2
UCL=456.873
Sample Mean
456
_
_
X=454.079
454
452
LCL=451.285
1
1
4
7
10
13
16
Sample
19
22
25
28
R 1.486

 1.32
d 2 1.128
out of control
UCL=4.854
4.5
Sample Range
Jane
 Jane 
3.0
_
R=1.486
1.5
0.0
LCL=0
1
4
7
10
13
16
Sample
19
22
25
28
Figure 9-9: Independent Charts
for Measurement Data
SNR 
 observed
14.128

 11.39
 measurement 1.240
2
1.537
 1   measurement


 0.0077 


2
2
SNR



14.128
observed
2
0.77% of observed variation is due to measurement error
Production Data; Jack and Jane Combined
Sample Mean
Sample Size: n = 4
480
UCL=481.42
460
_
_
X=460.23
 observed 
R 29.09

 14.128
d 2 2.059
2
2
2
 observed
  process
  measurement
440
LCL=439.04
1
7
13
19
25
31
Sample
37
43
49
55
UCL=66.36
2
 process
 14.128   1.537  198.06
2
 process  198.06  14.07
Sample Range
60
40
_
R=29.09
20
0
LCL=0
1
7
13
19
25
31
Sample
37
43
49
55
Figure 9-10: Production Data; Jack and Jane Combined
Variables Control Chart
Variables Control Chart
Variables Control Chart
XBar of Nearest 1000
12.200
12.155
12.145
12.140
Avg=12.13950
12.135
Mean of Nearest 100
UCL=12.14810
LCL=12.13090
12.130
12.150
UCL=12.15020
12.145
12.140
Avg=12.14000
12.135
Mean of Nearest 10
12.150
Mean of Nearest 1000
XBar of Nearest 10
XBar of Nearest 100
12.175
UCL=12.17746
12.150
12.125
Avg=12.12500
12.100
12.075
12.130
LCL=12.12980
12.125
2 4 6 8 10 12 14 16 18 20 22 24 26 28
LCL=12.07254
12.050
2 4 6 8 10 12 14 16 18 20 22 24 26 28
2 4 6 8 10 12 14 16 18 20 22 24 26 28
Sample
Sample
Sample
Note: Sigma used for limits based on range.
Note: Sigma used for limits based on range.
Note: Sigma used for limits based on range.
R of Nearest 1000
0.035
0.020
0.015
Avg=0.01180
0.010
0.005
0.000
LCL=0.00000
-0.005
Range of Nearest 100
UCL=0.02693
0.025
UCL=0.03195
0.030
0.025
0.020
0.015
Avg=0.01400
0.010
0.005
Sample
A
Resolution to the
Nearest 1000th
of an Inch
13 Different Range
Values
0.10
Avg=0.0720
0.05
0.00
0.000
LCL=0.0000
LCL=0.00000
-0.005
2 4 6 8 10 12 14 16 18 20 22 24 26 28
UCL=0.1643
0.15
Range of Nearest 10
0.030
Range of Nearest 1000
R of Nearest 10
R of Nearest 100
2 4 6 8 10 12 14 16 18 20 22 24 26 28
2 4 6 8 10 12 14 16 18 20 22 24 26 28
Sample
Sample
B
Resolution to the
Nearest 100th
of an Inch
3 Different Range
Values
C
Resolution to the
Nearest 10th
of an Inch
2 Different Range
Values – 28% Zeroes
Figure 9-11: Charts Showing Different Levels of Discrimination
Figure 9-13: Pull Down Menu Sequence for Linearity and Bias Test
Enter Column Names
Then Click on “OK”
Figure 9-14: Linearity and Bias Dialogue Box
Gage Linearity and Bias Study for Process A
G age name:
D ate of study :
Reported by :
Tolerance:
M isc:
M easurement P rocess A
1.5
Regression
95% CI
Data
Avg Bias
1.0
Large p values mean that process
provides linear results over range of
reference values
P redictor
C onstant
S lope
S
G age Linearity
C oef
S E C oef
-0.2212
0.2077
0.0003536 0.0003424
0.756464
0.5
Bias
0.0
0
-0.5
-1.0
Reference
A v erage
100
300
500
700
1000
R-S q
G age Bias
Bias
-0.037351
-0.092200
-0.099472
-0.343129
0.206581
0.141467
P
0.292
0.307
2.2%
P
0.731
0.599
0.619
0.312
0.413
0.591
Large p values mean that bias is not
statistically significant
-1.5
-2.0
0
200
400
600
800
Refer ence V alue
1000
Figure 9-15: Linearity and Bias Analysis for Measurement Process A
Gage Linearity and Bias Study for Process B
G age name:
D ate of study :
Reported by :
Tolerance:
M isc:
M easurement P rocess B
3
Regression
95% CI
Data
Avg Bias
Small p values mean that process is not
linear over the range of reference
values
P redictor
C onstant
S lope
S
G age Linearity
C oef
S E C oef
-1.1966
0.2530
0.0025251 0.0004171
0.921425
R-S q
P
0.000
0.000
43.3%
2
Bias
1
0
0
Reference
A v erage
100
300
500
700
1000
G age Bias
Bias
0.11642
-1.03403
-0.25433
0.02079
0.48474
1.36491
P
0.394
0.001
0.302
0.957
0.204
0.002
-1
-2
0
200
400
600
800
Refer ence V alue
1000
Small p values mean that bias is
statistically significant for large and
small values
Figure 9-16: Linearity and Bias Analysis for Measurement Process B
Gauge R and R Study
Figure 9-17: Minitab Worksheet for Gauge R & R
Figure 9-18: Drop Down Menu Sequence for Gauge R & R Study
Figure 9-19: Gauge R & R Dialogue Box
Gage R&R (Xbar/R) for Meas
G age name:
D ate of study :
Reported by :
Tolerance:
M isc:
Jack and Jane M easurement P rocess
Components of Variation
Meas by Part
100
% Contribution
500
Percent
% Study Var
475
50
450
0
Gage R&R
Repeat
Reprod
1
Part-to-Part
2
3
4
5
6
R Chart by Opr
Sample Range
Jack
10 11
12 13
14 15
UCL=4.529
500
475
2
_
R=1.386
0
LCL=0
450
Jack
Jane
Opr
Xbar Chart by Opr
475
450
Jack
Jane
Opr * Part Interaction
500
_
_
UCL=466.49
X=463.89
LCL=461.28
Average
500
Sample Mean
8
9
Part
Meas by Opr
Jane
4
7
Opr
Jack
Jane
475
450
1
2
3
4
5
6
7
8 9 10 11 12 13 14 15
Part
Figure 9-20: Gauge R & R Minitab Report Form
1
4
Machine A
3
Machine B
2
Sampling Schemes
1.
1
Inspection
Station
2.
3.
Alternate between inspection stations 1 and 2 taking samples of size 4 from
Machine A, then Machine B, and so on.
Take samples of size 4 at inspection station 3 by selecting 2 items from Machine A
and 2 items from Machine B.
Take samples of size 4 at inspection station 4 by randomly selecting items from the
combined output of Machines A and B.
Figure 9-21: Various Sampling Schemes from a Two-Machine Production Process
Histogram (with Normal Curve) of Machine A
Mean
StDev
N
200
150
Frequency
 A  18.99
 A  2.021
18.99
2.021
2000
100
50
0
12
14
16
18
20
Measurement
22
Histogram (with Normal Curve) of Machine B
200
Mean
StDev
N
12.05
1.747
2000
24
 B  12.05
 B  1.747
Frequency
150
100
50
0
6.4
8.0
9.6
11.2
12.8
Measurement
LSL
= 10mm
14.4
Figure 9-22: Machine Distributions
16.0
Target
= 15mm
USL
= 20mm
Xbar-R Chart of Sampling Scheme 1
1
20.0
Sample M ean
Rules violations:
Rule 1 – 23 points
Rule 4 – 16 points
Rule 8 – 23 points
1
1
1
1
1
1
17.5
1
1
1
1
4
U C L=18.26
8
_
_
X=15.46
15.0
8
12.5
1
1
1
10.0
1
1
1
4
7
10
1
13
16
Sample
1
1
19
1
22
LC L=12.67
4
1
1
25
1
28
U C L=8.741
Sample Range
8
6
_
R=3.832
4
2
0
LC L=0
1

4
7
10
13
16
Sample
19
22
25
28
R 3.832

 1.86
d 2 2.059
Figure 9-23: Control Charts for Sampling Scheme 1
Xbar-R Chart of Sampling Scheme 2
Rules violations:
Rule 7 – 5 points
U C L=21.68
Sample M ean
20.0
17.5
7
7
15.0
7
12.5
7
_
_
X=15.47
7
10.0
LC L=9.25
1
4
7
10
13
16
Sample
19
22
25
28
Sample Range
20
U C L=19.46
15
10
_
R=8.53
5
0
LC L=0
1

4
7
10
13
16
Sample
19
22
25
28
R
8.53

 4.14
d 2 2.059
Figure 9-24: Control Charts for Sampling Scheme 2
Xbar-R Chart of Sampling Scheme 3
Rules violations:
None
U C L=21.51
Sample M ean
20.0
17.5
_
_
X=15.37
15.0
12.5
10.0
LC L=9.23
1
4
7
10
13
16
Sample
19
22
25
28
Sample Range
20
U C L=19.23
15
10
_
R=8.43
5
0
LC L=0
1

4
7
10
13
16
Sample
19
22
25
28
R
8.43

 4.09
d 2 2.059
Figure 9-25: Control Charts for Sampling Scheme 3
ET  USL  LSL
ET  USL  LSL
Capable
and
Meeting
Requirements
Not
Capable
of
Meeting
Requirements
NT  6 x
LSL
NT  6 x
USL
Capable
and
Not Meeting
Requirements
LSL
USL
A process is capable if
NT  ET
A process is not capable if
NT > ET
NT  6 x
Figure 9-26: Process Capability
Histogram of Trace Contaminants in PPM
30
Mean = 62.6
USL = 150
25
Target = 75
Frequency
20
15
s = 70.15
10
5
0
0
50
100
150
200
250
Parts Per Million
300
350
Figure 9-27: Histogram of Trace Contaminants
Histogram of Box-Cox Transformed Trace Contaminant Data With Lambda = 0
20
Mean = 3.46
Target = 4.32
Frequency
15
USL = 5.01
10
s = 1.434
5
0
-1.5
0.0
1.5
3.0
4.5
Transformed Trace Contaminant Density
6.0
Figure 9-28: Transformed Data for Trace Contaminants
Process Capability of Trace Contaminants
Using Box-Cox Transformation With Lambda = 0
Target*
U S L*
transformed data
P rocess Data
LS L
*
Target
75
USL
150
S ample M ean
62.5975
S ample N
100
S tD ev (Within)
64.6625
S tD ev (O v erall) 70.1488
Within
O v erall
P otential (Within) C apability
Cp
*
C PL
*
C P U 0.37
C pk 0.37
O v erall C apability
A fter Transformation
LS L*
Target*
U S L*
S ample M ean*
S tD ev (Within)*
S tD ev (O v erall)*
Pp
*
PPL
*
P P U 0.36
P pk 0.36
C pm 0.14
*
4.31749
5.01064
3.45993
1.41486
1.43437
-1.5
O bserv ed P erformance
% < LS L
*
% > U S L 8.00
% Total 8.00
0.0
E xp. Within P erformance
% < LS L*
*
% > U S L* 13.65
% Total
13.65
1.5
3.0
4.5
6.0
E xp. O v erall P erformance
% < LS L*
*
% > U S L* 13.98
% Total
13.98
Figure 9-29: Minitab Capability Analysis Using Box-Cox Transformation
Figure 9-30: Minitab Drop Down Sequence for Capability Analysis
If Data is Non-normal Click
on “Box-Cox”
Enter Desired l
Designate Where Data is
Found
Enter Specifications
To Enter a Target Value
Click on “Options”
Then Enter Target
Select Either PPM or %
Figure 9-31: Minitab Capability Dialogue
Boxes
Process Capability of Acme Fastener Dimension
LSL
Target
USL
P rocess D ata
LS L
11
Target
15.5
USL
20
S ample M ean
14.6492
S ample N
150
S tD ev (Within)
1.71459
S tD ev (O v erall) 1.71793
W ithin
Ov erall
P otential (Within) C apability
Cp
0.87
C P L 0.71
C P U 1.04
C pk
0.71
O v erall C apability
Pp
PPL
PPU
P pk
C pm
10.5
O bserv ed P erformance
P P M < LS L 13333.33
PPM > USL
0.00
P P M Total
13333.33
12.0
13.5
E xp. Within P erformance
P P M < LS L 16654.87
PPM > USL
901.95
P P M Total
17556.82
15.0
16.5
18.0
0.87
0.71
1.04
0.71
0.78
19.5
E xp. O v erall P erformance
P P M < LS L
16827.53
PPM > USL
920.76
P P M Total
17748.29
Figure 9-32: Minitab Display of Capability Results for Acme Fasteners
Figure 9-33: Dropdown Menu Sequence for Non-Normal Distributions
Trace Contaminant Data
Johnson Transformation with SB Distribution Type
2.461 + 0.874 * Ln( ( X + 5.772 ) / ( 791.123 - X ) )
Target*
U S L*
transformed data
P rocess D ata
LS L
*
Target
75
USL
150
S ample M ean 62.5975
S ample N
100
S tD ev
70.1488
S hape1
2.46109
S hape2
0.874114
Location
-5.77218
S cale
796.895
O v erall C apability
Pp
*
PPL
*
PPU
0.43
P pk
0.43
E xp. O v erall P erformance
% < LS L
*
% > U S L 9.67
% Total 9.67
A fter Transformation
LS L*
Target*
U S L*
S ample M ean*
S tD ev *
*
0.553585
1.22437
0.0124166
0.931791
O bserv ed P erformance
% < LS L
*
% > U S L 8.00
% Total 8.00
-1.6
-0.8
0.0
0.8
1.6
2.4
Figure 9-34: Capability Display Using Johnson Transformation
Trace Contaminant Data
Using Box-Cox Transformation With Lambda = 0
Xbar C har t
C apability H istogr am
Sample Mean
Target*
USL*
UCL=5.358
5
Specifications
_
_
X=3.460
3
1
3
5
7
9
11
13
15
17
19
-1.5
0.0
1.5
R C har t
Sample Range
5.01064
3.0
4.5
6.0
Nor mal P r ob P lot
8
AD: 1.865, P: < 0.005
1
UCL=6.959
_
R=3.291
4
0
LCL=0
1
3
5
7
9
11
13
15
17
19
0
Last 2 0 Subgr oups
5
2.5
0.0
5
10
Sample
10
T r ansfor med C apa P lot
Within
Within
5.0
Values
4.31749
USL*
LCL=1.562
1
15
StDev
1.41486
Cp
Cpk
*
0.37
Overall
O v erall
StDev
1.43437
Pp
Ppk
Cpm
*
0.36
0.14
S pecs
20
Trace Contaminant Data
Johnson Transformation with SB Distribution Type
2.461 + 0.874 * Ln( ( X + 5.772 ) / ( 791.123 - X ) )
Xbar C har t
Sample Mean
The Johnson
Transformation Shows
Better Stability Than the
Box-Cox Transformation
Target*
C apability H istogr am
USL*
UCL=1.198
1
Specifications
USL*
_
_
X=0.012
0
-1
LCL=-1.173
1
3
5
7
9
11
13
15
17
19
-1.6
-0.8
Sample Range
R C har t
0.0
0.8
1.6
2.4
Nor mal P r ob P lot
AD: 0.260, P: 0.704
UCL=4.345
4
_
R=2.055
2
0
LCL=0
1
3
5
7
9
11
13
15
17
19
-2
Last 2 0 Subgr oups
0
0
Pp
Ppk
-2
10
4
Overall
Overall
Location
Scale
5
2
T r ansfor med C apa P lot
2
Values
1.22437
Normal Probability Plot
Indicates That The Johnson
Transformation Fits The
Normal Distribution Better
Than the Box-Cox
Transformation
15
0.0124166
0.931791
*
0.43
Specs
20
Sample
Figure 9-35: Comparison of Box-Cox and Johnson Transformations for Trace Contaminant Data
Figure 9-36: Control Charts for Fill Volume – Combined Stream
Figure 9-37: Comparison of Control Charts by Line
Most Line 1 Points are Below the Centerline
Violates Length of Longest Run Rule
Line 1
Line 2
Figure 9-38: Line 1 Data Followed By Line 2 Data
Figure 9-39: Line 1 Data Collected By Filler Head
Histogram of Filler Head 1
Histogram of Filler Head 2
180
Mean
StDev
N
160
246.0
1.028
2000
140
160
Head 1
Frequency
80
249.0
0.9946
2000
80
60
40
40
20
20
243
244
245
246
247
Fill Volume
248
0
249
244
245
Histogram of Filler Head 3
246
247
Fill Volume
248
249
250
Histogram of Filler Head 4
180
Mean
StDev
N
160
248.0
0.9946
2000
140
180
160
140
Head 3
100
80
100
80
60
60
40
40
20
20
245
246
247
248
249
Fill Volume
250
251
Head 4
120
Frequency
120
Frequency
Mean
StDev
N
100
60
0
247.0
0.9752
2000
Head 2
120
100
0
Mean
StDev
N
140
120
Frequency
180
0
246
247
248
249
Fill Volume
250
Figure 9-40A: Histograms of Fill Volumes – Heads 1 - 4
251
252
Histogram of Filler Head 5
Histogram of Filler Head 6
180
Mean
StDev
N
160
250.0
1.007
2000
140
Head 5
160
Frequency
80
100
80
60
60
40
40
20
20
247
248
249
250
Fill Volume
251
252
0
253
248
249
Histogram of Filler Head 7
Mean
StDev
N
140
251
252
Fill Volume
253
254
255
252.0
1.003
2000
180
Mean
StDev
N
160
140
120
Head 7
Head 8
120
100
Frequency
Frequency
250
Histogram of Filler Head 8
160
80
60
100
80
60
40
40
20
20
0
251.0
1.019
2000
Head 6
120
100
0
Mean
StDev
N
140
120
Frequency
180
249.6
250.4
251.2
252.0 252.8
Fill Volume
253.6
254.4
255.2
0
250
251
252
253
Fill Volume
254
Figure 9-40B: Histograms of Fill Volumes – Heads 5 - 8
255
256
253.0
0.9744
2000