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Measurement System Analysis for Process Improvement

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 (M).

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
0
0

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, M. 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 M.

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.

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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, C. The value of C may be estimated, for example, by C = 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 C is best obtained from other studies. If such studies are not available, as is the case here, C 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 C is taken to be sBar = 2.55 millivolts. Since the increment of measurement (1 millivolt) is less than C = 2.55 millivolts, the increment of measurement is satisfactory.

Precision: Standard Deviation of the Measurement System, M

When the range chart exhibits statistical control, its precision may be quantified by the standard deviation of the measurement system, M, which may be estimated by M =Rbar/d2 = Rbar/1.13.4 In this case, M = 1.625/1.13 = 1.44 volts. M must be evaluated to see if it is small enough to give acceptable precision. As a rule of thumb, it is recommended here that M be less than half of C. 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 (C) needs to be reduced.
1. If M is less than 25% of C, put the effort into improving the manufacturing process before any work is done on the measurement process.
2. If M is between 25% and 50% of C, work on both the manufacturing and the measurement process.
3. If M is greater than 50% of C, put the effort initially into improving the measurement process.

In this example M = 1.44 millivolts is 56% of C, so the initial process improvement efforts should be put toward decreasing the variation in the measurement processes.

A mathematical discussion in the Appendix compares C and M.

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 C and M. These differences are shown in Table 2. Under conditions typically met in industry, the addition of variances will hold, i.e.,
C2 = P2+M2
where P 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 = C = (combined) Rbar/d2 = M = (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.