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Measurement System Analysis for Process Improvement
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Robert F. Hart, Ph.D.
Marilyn K. Hart, Ph.D.
Introduction
In a manufacturing process, the perceived variation
includes both the true but unknown product variation
and the variation (lack of repeatability) within
the measurement system. When the perceived variation
of a manufacturing process is too large, a measurement
system study is needed to determine whether
the improvement efforts should be made on the
manufacturing process or on the measurement
process.2
Preliminaries
An essential part of the evaluation of a measurement
system is the verification that the measurement
system is in a state of statistical control.
Only then is the measurement system stable and
predictable, and only then do the bias and the
precision of the system have meaning. The "precision"
of the measurement system refers to the spread
of a set of replicate independent measurements
made on the characteristic. Precision is often
referred to as the "repeatability"
of the measurement process.
The "bias" of a system is the average
of independent repeated measurements minus the
"true" value for that measurement.
It may be difficult to establish the "true"
value (called an "Accepted Reference Value"
or ARV). The ARVs should come from a "higher
authority," by using a better gage under
more controlled conditions. Details of establishing
an ARV and calculating bias are beyond the scope
of this paper. If the average of the measurements
equals the ARV, the measurement system is said
to be "unbiased." Following the American
Society for Testing Materials (ASTM)1,
the word "accurate" should be avoided.
It has multiple definitions which can lead to
confusion.
Criteria for a Satisfactory Measurement
System
To evaluate the precision of a measurement
system, three characteristics must be evaluated:
statistical control of the measurement system,
the increment of measurement, and the standard
deviation of the measurement system (σM).
Statistical Control of the Measurement System
It is essential for the measurement system
to be in statistical control. Following the
Western Electric4 procedure, two
independent measurements would be made by the
same operator on each of fifty pieces of a product.
The authors' experience has shown that forty
pieces is sufficient and that is the number
used here. The 80 observations in Table 12
are duplicate coded product output voltage measurements
in millivolts from 40 consecutive pieces of
product. R is the range of the two measurements
for that piece.
Table 1. Data: Coded Product Output Voltage,
Millivolts
|
| Measure- |
|
|
Piece |
|
|
|
|
|
|
|
| ment |
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
| 1 |
1
|
4
|
10
|
6
|
3
|
7
|
5
|
7
|
3
|
3
|
| 2 |
2
|
5
|
5
|
2
|
3
|
7
|
5
|
3
|
8
|
4
|
| R |
1
|
1
|
5
|
4
|
0
|
0
|
0
|
4
|
5
|
1
|
|
| |
11
|
12
|
13
|
14
|
15
|
16
|
17
|
18
|
19
|
20
|
| 1 |
5
|
6
|
8
|
9
|
6
|
7
|
8
|
8
|
2
|
6
|
| 2 |
6
|
4
|
7
|
7
|
3
|
7
|
8
|
9
|
1
|
4
|
| R |
1
|
2
|
1
|
2
|
3
|
0
|
0
|
1
|
1
|
2
|
|
| |
21
|
22
|
23
|
24
|
25
|
26
|
27
|
28
|
29
|
30
|
| 1 |
4
|
2
|
7
|
6
|
3
|
5
|
8
|
5
|
2
|
1
|
| 2 |
5
|
1
|
9
|
4
|
2
|
4
|
4
|
3
|
5
|
2
|
| R |
1
|
1
|
2
|
2
|
1
|
1
|
4
|
2
|
3
|
1
|
|
| |
31
|
32
|
33
|
34
|
35
|
36
|
37
|
38
|
39
|
40
|
| 1 |
2
|
8
|
7
|
2
|
4
|
-1
|
3
|
3
|
4
|
9
|
| 2 |
1
|
8
|
7
|
7
|
3
|
0
|
4
|
7
|
4
|
9
|
| R |
1
|
0
|
0
|
5
|
1
|
1
|
1
|
4
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0
|
0
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|
When the 80 measurements are complete they
are arrayed as 40 subgroups (pieces) of size
2 (duplicate measurements on each piece). The
usual practice has been to make a Shewhart Xbar
and R chart on the 40 subgroups, but the authors'
experience has shown that it is preferable to
make only the R chart. The ranges reflect only
the measurement variation and, if the R chart
is in control, provide the needed estimate of
the standard deviation of the measurement process,
σM. The deletion of the Xbar
chart is a significant improvement since it
is very difficult to interpret and the information
it provides can be found by the evaluation of
σM.
Before proceeding with the control chart analysis,
the data must be checked for any evidence of
within-subgroup patterns, i.e., the first observation
of a pair must not be systematically higher
(or lower) than the second. This step is essential.
A simple way to accomplish this check is to
circle or underline the higher of the two measurements
in each of the 40 pairs and verify that there
does not seem to be a disproportionate number
of the higher values in either of the two rows
of measurements after ties are eliminated.
In Table 1, the higher value in each subgroup
is underlined. Seventeen of the 31 unpaired
first measurements are seen to be "high".
The data appear to be free from within-subgroup
patterns, so the R chart may be made.
The R chart for the 40 subgroups of size two
is shown in Figure 1. "Three sigma"
control limits are calculated for the R chart
in the usual way for n=2:
UCL(R)
= 3.27Rbar
LCL(R) = 0
Where Rbar is the average of the ranges from
all the subgroups. In this case, Rbar = 1.625,
so UCL(R) = 5.3.
Figure 1. R Chart
Because ranges are quite badly skewed to the
right, occasional points above the three sigma
limits can be expected. With 40 subgroups the
measurement system may be accepted as in statistical
control unless three or more points on the range
chart exceed the three sigma limit. In Figure
1, no points on the range chart are above the
three sigma limit, so it is accepted that the
measurement process is "in control."
Increment of Measurement
The increment of measurement must be fine enough
for the purpose intended. (This is to avoid
the problem of measuring precision ball bearings
with a yardstick.) For process improvement,
the increment of measurement (1 millivolt here)
must be smaller than the standard deviation
of the combined variability of the manufacturing
and measurement process, σC. The value of σC
may be estimated, for example, by
σC = Rbar/d2
using a conventional Xbar and R chart that is
in statistical control. The value Rbar/d2 is
typically thought of as the variability of the
product, but really is the combined variability
of both the manufactured product and the measurement
system. An estimate of the value of σC is best
obtained from other studies. If such studies
are not available, as is the case here, σC may
be estimated from the measurement data, Table
1. An s chart may be made on that data taken
as two subgroups of size 40, the first subgroup
being the first set of 40 measurements and the
second subgroup the second set. The estimate
of σC is taken to be sBar = 2.55 millivolts.
Since the increment of measurement (1 millivolt)
is less than σC = 2.55 millivolts, the increment
of measurement is satisfactory.
Precision: Standard Deviation of the Measurement
System, σM
When the range chart exhibits statistical control,
its precision may be quantified by the standard
deviation of the measurement system, σM, which
may be estimated by σM =Rbar/d2 = Rbar/1.13.4 In this case, σM = 1.625/1.13 = 1.44 volts.
σM must be evaluated to see if it is small enough
to give acceptable precision. As a rule of thumb,
it is recommended here that σM be less than
half of σC. This requirement is more lenient
than what is often used in the literature, but
has proven to be both adequate and realistic
in a large number of applications.
The following guidelines are useful for allocating
process improvement efforts when the perceived
product variation (σC) needs to be reduced.
1. If σM is less than 25% of σC, put the effort
into improving the manufacturing process before
any work is done on the measurement process.
2. If σM is between 25% and 50% of σC, work
on both the manufacturing and the measurement
process.
3. If σM is greater than 50% of σC, put the
effort initially into improving the measurement
process.
In this example σM = 1.44 millivolts is 56%
of σC, so the initial process improvement efforts
should be put toward decreasing the variation
in the measurement processes.
A mathematical discussion in the Appendix compares
σC and σM.
Conclusion
The need for measurement system analysis arises
when the perceived product variation (i.e.,
as perceived using the measurement system) is
too great. The analysis method proposed here
give guidelines for determining whether the
efforts for improvement should be directed toward
the production process or toward the measurement
system.
Appendix
Note the differences between conventional R
charts and measurement error R charts and between
σC and σM. These differences are shown in Table
2. Under conditions typically met in industry,
the addition of variances will hold, i.e.,
σC2 = σP2+σM2
where σP is the standard deviation of pure product
(with zero measurement error).3
| Conventional |
Measurement Error |
| 1 observation/piece |
n observations/piece |
| k subgroups |
k subgroups |
| for example, k=25, n=4 |
(k=40, n=2 in this paper) |
| |
|
| If in control: |
If in control: |
| Rbar/d2 = σC = σ(combined) |
Rbar/d2 = σM = σ(measurement) |
References
| 1. |
American Society for Testing
Materials. Standard Practice for Use of
the Terms Precision and Bias in ASTM Test
Methods, ASTM E177-86. ASTM: Philadelphia,
PA, 1986. |
| 2. |
Hart, M., and Hart, R.
F. “The Evaluation of a Measurement
System” Production and Inventory Management
Journal, Fourth Quarter, 1994: 22-26. |
| 3. |
Hart, R. F., and Hart,
M. "Is That Measurement Valid?"
Chemical Processing, October 1988: 28-32. |
| 4. |
Western Electric. Statistical
Quality Control Handbook. Newark, N. J.:
Western Electric Company, Inc., 1956. |
For more information, contact Drs. Robert and
Marilyn Hart at robthart@aol.com.
If you would like additional information, please
call our Support staff at (541) 752-4100 or
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