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# Measurement System Analysis For Attribute Measuring Processes

by Joseph D. Conklin

To rephrase an old management proverb, “What gets measured can be improved.” Six Sigma practitioners quickly come to appreciate the critical role of good measurement systems in initiating and sustaining process improvement. A good measurement system consists of many things, including accuracy (measuring the true or standard value), precision (low measurement variation), stability (predictable results under similar conditions over time) and discrimination (ability to distinguish true differences).1

This column will explain how to quantify contributions to variation when the measurement process yields attribute data.2 But first, let’s look at how this can be done when the measurement data are variable.3

When a measurement system analysis is set up for the variables case, it is common to measure the same set of units over multiple periods with multiple operators.4 The analysis produces variance component estimates for time periods, operators and any other factors included.5 These variance component estimates can be reported and arranged in a way that shows how each contributes to the variation of measurement.

To illustrate, consider this analysis for a system that measures the dimensions of a part in millimeters. Assume the analysis is performed using data from four parts measured over two periods by three operators.6 Each part in this example is measured twice by each operator in each period. The repeated measurements by each operator in each period are necessary to estimate the variation contributed by the measuring instrument itself. The data are reported in Table 1. The variance components associated with these factors are given in Table 2.7

To quantify the effect of the factors, add the variance components.8 The percentage contribution of each factor’s variance component to the total represents the best estimate of that factor’s impact on the total variation in the measuring system.9 The factors contributing to measurement variation are operators, periods and the measuring instrument itself. The part numbers also generate a variance component, but this is part of the normal process variation outside the measuring system

The first question to ask is whether the measurement variation is excessive. The answer depends on the requirements the process and the final product must meet.10 If the measurement variation is considered excessive, the corrective action depends on which factors dominate the variance components.

If periods dominate, you might consider a change in calibration procedures or in how you protect the measuring instrument from temperature or humidity factors that change with time. If operators dominate, you might consider standardizing procedures or improving operator training. If the instrument itself dominates, your most practical alternatives to reduce variation are redesign or replacement.

Because the variance component results in Table 2 show equal contributions by period and by instrument, the next direction for corrective action—if you conclude the overall variation is excessive—depends on which of the two factors is the easier or less expensive to address.

### Arc Sine Square Root Transformation

When the measuring system yields attribute data, variance components can be computed if the data are converted via the arc sine square root transformation.11 To begin showing how this works, take the data from Table 1, convert it to an attribute example, and keep the same factors. Instead of the parts being measured with a micrometer or caliper, let’s say they are run in lots of 10,000 through an automated go-no-go gage. To estimate the contribution of the attribute measuring device by itself, each operator in each period should run the same lot of 10,000 parts through twice. The repeated runs of the same lot should be run back to back or as close together in time as possible.

Table 3 (p. 51) displays the results. The variable for measurement system analysis is in the “fraction passing” column. This column is the ratio of the “number passing” and “lot size” columns. To apply the arc sine square root transformation, create a new column called “transformed data” with values equal to [2 x sin-1((fraction passing)1/2)]. Use the arc sine root transformation to convert data in the form of fractions or percentages. The converted data tend to be better aligned with the assumptions of traditional variance component analysis. The converted data for the attribute example are displayed in Table 4.

To obtain the variance component estimates from the converted data, you need the quantities from Table 5.12 The steps from the mean squares to the variance components are laid out in Table 6. Table 7 shows the contributions of the factors to the variation of the attribute measuring system.13 Because the data from the variables example were recycled, the results closely parallel those in Table 2 (p. 50). The measurement variation is dominated by joint and equal contributions of periods and the instrument.

You are often faced with the same decision as in the variables example case. If the measurement variation is not judged excessive, you need to make sure you maintain this desirable state. If you decide the measurement variation must be reduced, then you need to bring Six Sigma process improvement techniques to the project.

### REFERENCES AND NOTES

1. For a more in-depth discussion, see Evaluat-ing the Measurement Process, second edition, by Donald J. Wheeler and Richard W. Leyday (Addison-Wesley Publishing, 1990).
2. Examples of attribute data are number of defects or number of defective units. The result is the outcome of some counting process. The values are restricted to whole numbers.
3. Variables data are measured in units such as feet, degrees or volts. The results are not restricted to whole numbers; they may be fractional.
4. For illustrations of measurement system studies for variables data, see Measurement Systems Analysis, third edition (Automotive Industry Action Group, 2003). This reference does not exhaust all the possible ways to perform a measurement system analysis, but it is a good introduction to the subject.
5. Variance components are an instance of a statistical quantity called variance. Variance is a statistical index of variation in a process. For a more general treatment of the role of variance in understanding process variation, see Statistical Quality Control, seventh edition, by Eugene L. Grant and Richard S. Leavenworth (McGraw-Hill, 1996).
6. When performing an actual measurement system analysis, a different number of parts, time periods and operators may be used depending on the situation’s requirements. The standard designs in Measurement Systems Analysis (see reference 4) are widely applicable. When these are not adequate, the number of parts and operators can be determined by statistical formulas.
7. The variance component estimates are in the square of the units of the raw data, which, in this case, is square millimeters.
8. The complete details of how the analysis produces the variance component estimates are beyond the scope of this article. They involve the concepts of experimental designs with completely random factors. A useful reference on this subject is Design and Analysis of Experiments, fifth edition, by Douglas C. Montgomery (John Wiley, 2001). Some details are mentioned in this article when explaining the variance component estimates for the attribute data case.
9. Statistical software, such as that by the SAS Institute, exists for generating variance components. For more information, see SAS/STAT User’s Guide, Version 6, Vol. 2 (SAS Institute, 1989).
10. For a discussion of this issue, (see reference 4).
11. For a discussion of the arc sine square root transformation, see Statistical Principles in Experimental Design, second edition, by B.J. Winer (McGraw-Hill, 1971, p. 400).
12. More complete information can be found in Design and Analysis of Experiments (see reference 8).
13. The units for the variance components in Table 7 are the square of the units in Table 4, the fraction passing after applying the arc sine square root transformation. The units are not mentioned in Table 7 because they are not needed to interpret it.

JOSEPH D. CONKLIN is a mathematical statistician at the U.S. Department of Energy in Washington, D.C. He earned a master’s degree in statistics from Virginia Tech and is a Senior Member of ASQ. Conklin is also an ASQ certified quality manager, quality engineer, quality auditor and reliability engineer.

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