Statistical Quality Control
Six Sigma and R&R Project
Group A, SQC, May 2, 2005
Eric Klopp
Six Sigma: Is It Really Different
The article “Six Sigma: Is It Really Different” discusses the importance of Six
Sigma, especially in the enhancement of old quality improvement programs. Two of
these old programs that are mentioned in the article are DOE and SPC. DOE, or Design
of Experiments, is a great statistical tool to pinpoint a certain factor that may be
disrupting a process. SPC, or statistical process control, is a good problem solving
technique to achieve stability in a process and create a consistent product. One relatively
new program that is discussed in the article is DMAIC. DMAIC, which is an acronym for
Design – Measure – Analyze – Improve – Control, is a means of exploiting old
techniques such as control charts, DOE, and measurement capability studies into a
process that can be performed with ease. It first addresses or defines a problem, then
analyzes the data collected to improve a process. The last part, Control, is there to make
sure that this quality improvement process is sustained and does not fall apart like many
old programs.
There are several aspects of Six Sigma that differentiate it between previous
quality programs. Part of Six Sigma is convincing management that it must fully support
the program. Six Sigma sets apart leaders, known as Black Belts, to not only motivate an
individual, but also the organization as a whole. The training involved with becoming a
Black Belt is much more exhaustive than any other quality improvement program.
Computers are employed in Six Sigma programs to solve complex data problems. But the
main focus of Six Sigma is not to direct someone as to what needs to be done, but how to
achieve the goal and maintain it so that quality products can be manufactured for years to
come.
A Six Sigma program can drastically change the outlook for a company.
Implementing this program can provide a basis for any problem that may arise in a
company. If variability is increasing in a process, measurement data combined with
design of experiments can be used to locate the source of variability. By putting Six
Sigma into practice, efforts can be taken to make sure that source of variability will never
arise again. One company that is obviously credited with Six Sigma is Motorola. They
applied these procedures as a response to the demand for their products. Other successful
companies that have applied a Six Sigma program are Kodak, IBM, and Texas
Instruments. Six Sigma has the advantage of utilizing many of the same methods of
quality control that have been applied in the past, however its main advantage is that it
ties all of these programs together to become more effective than ever.
Reliability of Measurement System Using R&R Study and Characterization
Introduction
The problem has been given to us that a large manufacturer of washers was
having trouble controlling the internal diameter of their blanking process. Design of
Experiments was recommended to be used as a means of finding the variability of the
process, but ended unsuccessfully after three months of gathering data. A random sample
of data has been provided from four different operators using a vernier caliper to measure
the internal diameter of the washers. My objective, as a Six Sigma team member, is to
create x and R charts, use a tabular R&R procedure study on the vernier caliper, also use
analysis of variance to conduct an R&R study, and design a new system to implement to
reduce the variability caused by reproducibility and repeatability. Also, I will compare
the tabular study to the ANOVA study for R&R.
Experimental Results
Measurement Data Collected (internal diameter: mm):
sample
operator #1
operator #2
operator #3
operator #4
1
9.67
9.66
9.69
9.63
9.64
9.63
9.64
9.64
9.62
9.58
9.64
9.61
9.6
9.59
9.67
9.63
2
9.66
9.63
9.61
9.66
9.63
9.64
9.65
9.64
9.46
9.58
9.65
9.63
9.7
9.58
9.57
9.58
3
9.65
9.69
9.71
9.67
9.65
9.63
9.64
9.65
9.52
9.62
9.48
9.61
9.6
9.67
9.6
9.61
4
9.67
9.66
9.51
9.54
9.64
9.65
9.63
9.62
9.64
9.62
9.58
9.65
9.6
9.66
9.64
9.63
5
9.68
9.69
9.68
9.6
9.62
9.4
9.63
9.61
9.61
9.65
9.63
9.65
9.6
9.65
9.66
9.64
6
9.51
9.55
9.58
9.54
9.6
9.63
9.57
9.63
9.63
9.59
9.59
9.61
9.6
9.66
9.62
9.62
7
9.67
9.58
9.57
9.51
9.64
9.63
9.64
9.61
9.63
9.63
9.58
9.6
9.6
9.67
9.66
9.67
8
9.61
9.64
9.63
9.53
9.63
9.63
9.61
9.62
9.52
9.59
9.59
9.55
9.7
9.6
9.63
9.66
9
9.62
9.66
9.57
9.61
9.64
9.63
9.63
9.63
9.61
9.63
9.62
9.61
9.7
9.66
9.65
9.66
10
9.61
9.65
9.64
9.57
9.64
9.6
9.62
9.57
9.58
9.64
9.58
9.63
9.7
9.68
9.68
9.61
11
9.64
9.52
9.6
9.6
9.59
9.63
9.67
9.62
9.46
9.41
9.63
9.59
9.6
9.62
9.67
9.65
12
9.64
9.67
9.65
9.59
9.65
9.65
9.66
9.64
9.59
9.47
9.63
9.62
9.7
9.7
9.7
9.63
13
9.62
9.66
9.54
9.64
9.59
9.62
9.63
9.64
9.62
9.58
9.59
9.56
9.6
9.67
9.67
9.66
14
9.63
9.64
9.65
9.58
9.65
9.62
9.62
9.63
9.52
9.6
9.61
9.57
9.6
9.63
9.64
9.63
15
9.5
9.61
9.48
9.5
9.6
9.64
9.6
9.6
9.5
9.53
9.58
9.6
9.6
9.59
9.62
9.65
16
9.48
9.54
9.47
9.56
9.64
9.63
9.64
9.61
9.61
9.64
9.63
9.59
9.6
9.64
9.61
9.64
17
9.52
9.53
9.58
9.55
9.61
9.61
9.63
9.62
9.51
9.53
9.61
9.59
9.6
9.66
9.67
9.65
18
9.53
9.45
9.58
9.44
9.65
9.63
9.63
9.64
9.59
9.59
9.66
9.63
9.6
9.6
9.68
9.63
19
9.44
9.62
9.55
9.62
9.63
9.66
9.61
9.63
9.51
9.58
9.53
9.61
9.7
9.64
9.64
9.67
20
9.61
9.63
9.65
9.64
9.65
9.65
9.64
9.61
9.6
9.6
9.6
9.6
9.7
9.66
9.67
9.63
X bar
9.598
9.614
9.597
9.579
9.6295
9.6205
9.6295
9.623
9.5665
9.583
9.6005
9.6055
9.635
9.6415
9.6475
9.6375
The provided data was processed using the MINITAB computer software.
The data provided appears to follow
a normal distribution, with a mean
of 9.613 and a standard deviation of
0.05199.
Normal Distribution of Measurement Data
Normal - 95% CI
99.9
Mean
StDev
N
AD
P-Value
99
95
90
9.613
0.05199
320
8.844
<0.005
Percent
80
70
60
50
40
30
20
10
5
1
0.1
9.4
9.5
9.6
measurement
9.7
9.8
Figure 1- Normal Distribution of Data
The following results were displayed when an x and R charts were calculated:
X bar and R chart for Operator 1
9.70
1
Sample M ean
U C L=9.6694
9.65
_
_
X=9.597
9.60
9.55
1
9.50
1
3
5
7
9
11
Sample
13
15
LC L=9.5246
1
1
17
19
U C L=0.2268
Sample Range
0.20
0.15
_
R=0.0994
0.10
0.05
0.00
The X bar and R chart for
Operator #1 shows a few points
that appear to be out of control.
The X bar chart depicts a
downward trend. However, the R
chart is quite consistent. This may
be an indication that the operator is
using the measuring device
correctly, but it is not taking
accurate measurements.
LC L=0
1
3
5
7
9
11
Sample
13
15
17
19
Figure 2- X bar and R for Operator #1
X bar and R chart for Operator 2
U C L=9.6707
Sample M ean
9.65
_
_
X=9.6256
9.60
LC L=9.5805
1
9.55
1
3
5
7
9
11
Sample
13
15
17
19
1
0.20
Sample Range
Besides point 5, this operator’s
measurements seem to be a very
stable and in-control. Operator #2
seems to be taking very accurate
and precise data with his/her
vernier calipers. Point 5 lies well
about the upper control limit on
the R chart and perhaps it should
be investigated why this point lies
so far off.
0.15
U C L=0.1413
0.10
_
R=0.0619
0.05
0.00
LC L=0
1
3
5
7
9
11
Sample
13
Figure 3- X bar and R for Operator #2
15
17
19
X bar and R chart for Operator 3
U C L=9.6594
Sample M ean
9.65
_
_
X=9.5889
9.60
9.55
LC L=9.5184
9.50
1
3
5
7
9
11
Sample
13
15
17
19
U C L=0.2208
Sample Range
0.20
0.15
_
R=0.0968
0.10
0.05
0.00
LC L=0
1
3
5
7
9
11
Sample
13
15
17
The data from Operator #3 appears
to have some trends in variability,
but mainly in-control. Point 11
seems out of the ordinary and
perhaps actions should be taken to
see the cause of this widespread
variability. The X bar chart shows
a slight cyclic pattern to it. This
may be due to factors such as
changes in temperature or operator
fatigue.
19
Figure 4- X bar and R for Operator #3
X bar and R chart for Operator 4
U C L=9.6903
Sample M ean
9.68
9.66
_
_
X=9.6404
9.64
9.62
9.60
LC L=9.5904
1
3
5
7
9
11
Sample
13
15
17
19
0.16
Sample Range
The X bar chart for Operator #4
depicts an upward shift in the
process level. The R chart, on the
other hand, is quite consistent.
This may lead us to believe that
the operator is using the calipers
correctly, but something may be
changing with the caliper that is
causing the measurements to
change.
U C L=0.1565
0.12
0.08
_
R=0.0686
0.04
0.00
LC L=0
1
3
5
7
9
11
Sample
13
Figure 5- X bar and R for Operator #4
15
17
19
Gage R&R (Xbar/R) for measurement
Reported by :
Tolerance:
M isc:
G age name:
D ate of study :
Components of Variation
measurement by product
100
% Contribution
9.75
Percent
% Study Var
9.60
50
9.45
0
Gage R&R
Repeat
Reprod
1
Part-to-Part
2
3
4
5
Sample Range
2
3
7
4
9.75
UCL=0.1691
0.1
_
R=0.0741
0.0
LCL=0
9.60
9.45
1
2
3
4
operator * product Interaction
4
UCL=9.6670
_
X=9.6130
9.6
LCL=9.5590
9.7
Average
Sample Mean
2
3
operator
Xbar Chart by operator
1
9 10 11 12 13 14 15 16 17 18 19 20
measurement by operator
0.2
9.7
8
product
R Chart by operator
1
6
operator
1
2
9.6
3
4
9.5
9.5
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 1 9 2 0
product
Figure 6- Gage R&R for measurement
Gage R&R Study - XBar/R Method
Source
Total Gage R&R
Repeatability
Reproducibility
Part-To-Part
Total Variation
VarComp
0.0018084
0.0012960
0.0005124
0.0002744
0.0020829
%Contribution
(of VarComp)
86.82
62.22
24.60
13.18
100.00
Source
Total Gage R&R
Repeatability
Reproducibility
Part-To-Part
Total Variation
StdDev (SD)
0.0425256
0.0360005
0.0226360
0.0165663
0.0456384
Study Var %Study Var
(6 * SD)
(%SV)
0.255153
93.18
0.216003
78.88
0.135816
49.60
0.099398
36.30
0.273830
100.00
The graphs shown in Figure 6 depict the data for all four operators. A close look
at the X double bar chart for all operators shows that operator 1 had a lot of out of control
data near the end of his run compared to the other three operators. This may be due to
operator fatigue or perhaps machine tool wear. From this X double bar chart, it appears
that there is quite a bit of variability in the measurements taken between operators. From
the output data for our study, 62.22% of the variability was from Repeatability, whereas
only 24.60% was from reproducibility. What this indicates is that the variability caused
by the gauge itself is greater than variability due to different operators using the gauge.
This is a very good indication that a new measuring device is desperately needed.
The following is the output for the ANOVA based R&R study:
ANOVA: measurement versus product, operator
Factor
product
operator
Type
random
random
Levels
20
4
Factor
Values
product
1, 2, 3,
18, 19, 20
operator 1, 2, 3, 4
4,
5,
6,
7,
8,
9, 10, 11, 12, 13, 14, 15, 16, 17,
Analysis of Variance for measurement
Source
product
operator
product*operator
Error
Total
S = 0.0403152
1
2
3
4
DF
19
3
57
240
319
SS
0.091186
0.139743
0.241075
0.390075
0.862080
R-Sq = 54.75%
Source
product
operator
product*operator
Error
Variance
component
0.00004
0.00053
0.00065
0.00163
MS
0.004799
0.046581
0.004229
0.001625
F
1.13
11.01
2.60
P
0.344
0.000
0.000
R-Sq(adj) = 39.86%
Error
term
3
3
4
Expected Mean Square
for Each Term (using
restricted model)
(4) + 4 (3) + 16 (1)
(4) + 4 (3) + 80 (2)
(4) + 4 (3)
(4)
The variance due to reproducibility is the sum of the operator variance and the
product*operator variance (0.00053 + 0.00065). The repeatability variance is listed as the
Error variability (0.00163). This ANOVA output displays that the repeatability variance
is higher than the reproducibility variance. This again indicates that the gauge is causing
more variance in our process than the operator.
Both of these forms of analysis have lead us to the same conclusion, that the
variance caused by the measuring device, the vernier caliper, has been greater than the
variance caused by the operators themselves. The tabular method is based on finding the
reproducibility and repeatability variances by R double bar or R sub X double bar divided
by d2, which based on the number of observations per part per operator. With the
ANOVA method, variance components may be estimated by a process of calculating
mean squares and sum of squares. It is hard to say that one method is better than the
other, being that they have different ways of telling us the same common result. Using
the computer software MINITAB, both methods are easy to perform.
One complication that I found when using MINITAB was introducing error when
having the type in the raw data given. If this system were to be implemented in a factory
setting, it would be ideal for all data to be directly recorded onto a computer. Once it is in
the computer, it is easier to transfer this data without any typing error introduced. An
example of this that I have seen first hand is at Andersen Window Corporation. They
have a computer terminal at almost all operator workstations so that defective or culled
material data can be directly input into the main computer system for further analysis.
Another Problem with MINITAB is that when dealing with all these different forms of
analysis, several windows open up on your screen and it is very easy to get lost.
However, MINITAB is a great statistical tool that can save many, many hours of
calculations.
If I were to design a measurement system that would reduce reproducibility and
repeatability variability, it would be a more accurate measuring device. The problem with
measuring with a vernier caliper is that there is a great chance of an operator incorrectly
using it. I propose having a shaft, perhaps one or two feet long, that is tapered so that one
end is 25% larger than the specified diameter and the other end is 25% smaller. This shaft
would be initially calibrated to a specific diameter with increments along the taper. This
would allow you to simply slide the washer onto the shaft and read the value wherever it
stopped. It would be a fast procedure, allowing the operator to spend less time away from
production, and it would reduce gauge variability by having essentially no moving parts
to influence error. The shaft could be painted in such a way that the tolerance band could
be green for “go” and red for “no go.” This would allow the operator to quickly see
whether his parts are acceptable or not without having to look at the exact increments.