BASIC MEASUREMENT ERROR ANALYSIS

The basic measurement error study was designed as a "one point in time" study. The study gives quick, easy, and practical insight into the magnitude and sources of measurement error. Because the study separates measurement error into the major domains of repeatability and reproducibility , some people refer to this study as a gauge repeatability and reproducibility (GRR) study. The procedures detailed in this chapter are appropriate for assessing the precision of nondestructive, variables data measurement systems. Advanced statistical techniques are needed to analyze the repeatability of measurement systems that make attribute decisions or are destructive.

In this chapter, we use very simple approaches to measurement. Readers interested in advanced techniques should consult some of the chapter's references and selected bibliography sources. This measurement study approach uses very limited data and should be used as an initial screen of measurement devices and procedures. The study should be continued and expanded for measurements systems that demonstrate acceptable levels of measurement precision. The expanded study should be used to verify the findings of the basic measurement error analysis (MEA) and provide a method for monitoring the long- term effects of measurement system stability.

GENERAL GUIDELINES FOR MEA DATA COLLECTION

1. Determine and refine the measurement technique . The measurement device should have a scale that has a resolution that is at least 1/10 of the characteristic tolerance. If the part has a total tolerance of 0.001 (USL - LSL = 0.001), the measuring equipment should provide measured values that have at least four significant decimals (0.0001). All members of the study team must agree on the methods for calibrating the instrument, manipulating the fixtures, and reading the measurement scale and on where to measure the part or test specimen. A formal set of measurement procedures should be used as a definition of the measurement process.

2. Develop specific study procedures. Establish the details of the measurement error analysis before beginning the study. Make certain that all who participate in the study understand the measurement techniques, study analysis, and interpretation of the study findings.

3. Number the parts or test specimens. Identify each part or test specimen with a numeric code. The code should be indelible and unobtrusive . This identification number should match the numbers used for the data collection form, be compatible with computer software needs, and aid in the recording and analysis of data. The part identification numbers should be placed on the parts so that they do not interfere with the performance of the measurement system.

4. Execute the study so that the findings may be generalized to the manufacturing process. Study the measurement system under conditions that closely approximate the regular operating conditions. The findings from the measurement error analysis must be generalized to the normal operating status of the measurement system. Do not make improvements to the instrumentation or fixtures until the measurement error analysis is completed. If improvements are later made to the measurement system, a postimprovement MEA may be completed as a means of confirming the effectiveness of the repairs . Select the parts or test specimens from the regular process. Do not study the measurement system with special parts or the gage master. Sometimes, a measurement device is very precise when it is used to measure laboratory standards or masters but cannot provide the same level of precision while measuring actual parts. Production parts are used during the MEA as a means of ensuring the external validity of the study. The parts should be prepared for the study as they are prepared by production personnel during the regular process. People who participate in the MEA should be selected from among those who normally use the measurement device.

5. Manipulate the measuring device, fixtures, and parts for each measurement. The entire measurement procedure should be repeated for each trial of the study. Position the part in the fixture, read the measurement scale, and remove the part from the device for each measurement. The repeated readings for each part should be independent of each other.

6. Do not allow the appraisers access to previous study data. While a person is participating in the study as an appraiser, do not allow that person to see the study data. People realize that they are a potential source of error and tend to bias their interpretation of the measurement scale when they are allowed to refer to the study data. The internal validity of the study is maintained only when the parts are measured in a random sequence, part numbers are unobtrusive, and data collection activities are orderly and confidential.

DATA COLLECTION PROCEDURES

1. Select 10 parts from the manufacturing process. The parts should be selected from the manufacturing process so that they are representative of those that are measured during the regular manufacturing process. The findings of the MEA should apply to the full operating spectrum of the manufacturing process. The parts may be randomly selected from the manufacturing stream, or they may be specified. So that the full operating range of the instrument is studied, some of the parts should be on the low side of the product specification, and some should be near the high side of the product specification. A broad and plausible range of part sizes is needed if the linearity of the measurement system is to be assessed.

2. Identify each part with a part number. The parts or test specimens should be prepared for measurement as is typical during the manufacturing process; for instance, burrs should be removed, chips and cutting oil should be wiped off, and so on. The part numbers should be readily identifiable to the study analyst but not to the appraisers. The part numbers should be permanent and should be placed in such a way that they do not interfere with the performance system.

3. Calibrate the measurement instrument. Calibrate the measurement device before any parts are measured. A standard calibration procedure should be used. If a standard calibration procedure does not exist, one must be developed and used as part of the measurement system.

4. The first appraiser measures the parts. The first appraiser (Appraiser A) should measure the ten parts for the first trial. The parts should be measured in a random sequence and recorded by the study analyst in Column 1 of the data collection sheet (see Figure 15.10). The values should be recorded as each part is measured by the appraiser. Each measured value must be placed in the appropriate row of the data collection sheet. Comparing the measured values for two different parts does not contribute to the assessment of the measurement system precision. After Appraiser A completes the measurement for the first trial, that person immediately remeasures the 10 parts. This is the second trial for Appraiser A. The data from this trial should be recorded in Column 2 of the data collection sheet. A second random sequence should be used for measuring the parts. The appraiser should not be able to refer to the values reported during an earlier trial. After the second trial is completed, the appraiser should immediately remeasure the parts for the third trial. These measured values should be recorded in Column 3. A third random sequence should be used, and reading from previous trials should not be divulged to the appraiser.

   
   Figure 15.10: R&R data collection sheet.

5. The second appraiser measures the parts. The second appraiser (Appraiser B) should follow the sequence of actions outlined above for the first appraiser. New random orders should be used for these trials, and the data should be recorded in Columns 5, 6, and 7.

6. The third appraiser measures the parts. The third appraiser (Appraiser C) should follow the sequence of actions outlined above for the first appraiser. New random orders should be used for these trials, and the data should be recorded in Columns 9, 10, and 11.

7. Calculate the range for each part and each appraiser. Each appraiser measured each of the parts three times. The range between the highest and lowest appraiser readings for each part is a measure of repeatability for that appraiser. Calculate the 10 ranges for each appraiser and record these values in Columns 4, 8, and 12. Because the range is a distance between two extreme numbers, all of the values recorded in Columns 4, 8, and 12 should be positive. This and other steps of the procedure have been completed on the worksheets shown in Figure 15.11. The circled numbers indicate the step number and the answer that was calculated for the example.

   
   Figure 15.11: A complete R&R data sheet.

8. Sum the values in all columns. The sum of each of the 12 columns is used during the analysis. Place the sum of each column in the row labeled "Totals" (see Figure 15.11).

9. Calculate the mean range for each trial. The mean range for each trial is calculated using the following formula: (record the mean range for each appraiser in the cell labeled Rbar):

   

where

	=	mean range for the i th appraiser
ˆ‘ R	=	sum of the ranges for the 10 parts measured by the i th appraiser
n	=	the number of parts used in the study ( n = 10)

10. Calculate the mean of the mean ranges for all appraisers. The mean ranges for all three appraisers are averaged into a single number. Calculate this average with the following formula:

    

where

	=	the mean of the mean ranges
ˆ‘ R bar	=	the sum of the mean ranges for all appraisers
k	=	the number of appraisers

11. Calculate the upper control limit for the ranges. If the same measurement procedures and techniques were used to measure all of the parts, the ranges (Columns 4, 8, and 12) should be estimates of the same population. They should be within the UCL for ranges. The UCL for the ranges is calculated with the following formula:

    

where

UCL R	=	UCL for the ranges
D 4	=	factor for calculating control limits for ranges
	=	the mean of the mean ranges

The most common factors for ranges are as follows :

Number of Trials	D 4
2	3.267
3	2.574
4	2.282
5	2.114

Circle any range that exceeds the UCL _R and investigate the source of unusually large measurement error. After the source of the unusual error is uncovered and resolved, the appraiser and part that are responsible for the unusual range should be remeasured.

Note that the values that yield the out-of-control range may also be discarded and not used in the analysis. The mean range and the mean of the mean ranges must be recalculated based on a revised sample size .

12. Calculate the mean for the individual measurements made by each appraiser. The mean for the individual measurements made by each appraiser is used to calculate the error due to reproducibility. The sums for Trial 1 and Trial 3 are placed below the sum of Trial 2 for each appraiser. Add the three sums and record the total in the cell labeled "Sums." The mean of each appraiser's measurements is calculated with the following formula:

    

where

Xbar i	=	the mean for the i th appraiser's measurements
n — k	=	the total number of measurements per appraiser
& pound ; X i	=	the sum of all the measurements made by the i th appraiser

13. Calculate the range of the appraisers' means. The range between the means (RXbar) is used to estimate error due to reproducibility. Place the highest and lowest means in the table at the right bottom corner of the data sheet (see Figure 15.11). Calculate the range by subtracting the smallest mean from the largest mean.

14. Transfer the summary repeatability and reproducibility ranges to the measurement error calculation sheet. The measurement error calculation sheet (see Figure 15.12) is a worksheet that was designed as an aid for quantifying and comparing the error due to repeatability and reproducibility. Copy the mean of the mean ranges (R double bar) and the range of the appraiser means (RXbar) to the measurement error calculation sheet (see Figure 15.13). The mean of the mean ranges will be converted into an estimate for repeatability error. The range of the appraiser means will be converted into an estimate for reproducibility.

    
    Figure 15.12: R&R report.

    
    Figure 15.13: R&R calculation sheet.

15. Estimate the variation due to repeatability error. Use the following formula to estimate the variation due to repeatability error:

    

where

RPT	=	estimate of variation for repeatability error
	=	mean of the mean ranges
K 1	=	factor for converting ranges into estimates of variation

16. Estimate the variability due to reproducibility. Use the following formula to calculate the variation due to reproducibility:

    RPD = (RXbar) — ( K ₂ )

where

RPD	=	estimate of variation for reproducibility error
RXbar	=	range of the appraiser's means
K 2	=	factor for converting ranges into estimates of variation

17. Calculate the spread due to repeatability and reproducibility error. The spread for measurement error is calculated as a combination of error due to repeatability and reproducibility. Use the following formula to calculate the standard deviation due to repeatability and reproducibility:

    

where

R&R	=	estimate of variation for repeatability and reproducibility
RPT	=	estimate of variation for repeatability
RPD	=	estimate of variation for reproducibility

18. Calculate the percentage of the tolerance used by repeatability error. Error due to repeatability is evaluated by comparing the amount of repeatability variation with the product tolerance. The percentage of tolerance used by repeatability error is calculated with the following formula:

    %RPT = 100 ((RPT) ² / {(R&R) — (Tolerance)})

where

%RPT	=	percentage of the product tolerance used by repeatability error
RPT	=	estimated variation due to repeatability error
R&R	=	estimate of variation for repeatability and reproducibility
Tolerance	=	range between the upper and lower product specification limits (USL - LSL)

19. Calculate the percentage of the tolerance used by reproducibility error. Error due to reproducibility is evaluated by comparing the amount of reproducibility variation with the product tolerance. The percentage of the tolerance used by reproducibility error is calculated with the following formula:

    %RPD = 100((RPD) ² / {(R&R) — (Tolerance)})

where

%RPD	=	percentage of the product tolerance used by reproducibility error
RPD	=	estimated variation due to reproducibility error
R&R	=	estimate of variation for repeatability and reproducibility
Tolerance	=	range between the upper and lower product specification limits (USL - LSL)

20. Calculate the percentage of the tolerance used by the measurement system error. Measurement error is divided into two major domains: repeatability and reproducibility. These two forms of measurement error are combined when the error due to the measurement system is evaluated. The error due to the measurement system is evaluated by comparing the sum of the estimates for error variation with the product tolerance. The percentage of product tolerance used by the measurement system is calculated with the following formula:

    %R&R = (%RPT) + (%RPD)

where

%R&R	=	percentage of the product tolerance used by measurement system error
%RPT	=	percentage of the product tolerance used by repeatability error
%RPD	=	percentage of the product tolerance used by reproducibility error

SHORT-METHOD R&R STUDY

Accuracy: Have one operator measure the same part ten times and measure a standard ten times. Average the readings from each. The difference between the averages is the inaccuracy, or bias.

R&R: The steps for a short study in measurement are

1. Randomly select and number five parts.

2. Two operators each measure the parts and record them in the row that corresponds to the part number. The form used is similar to that in Figure 15.14.

   
   Figure 15.14: Form used for short R&R.

3. The difference in reading is recorded as range (positive numbers ONLY).

4. Calculate the average range, Rbar, as follows:

   

5. RR Error = ( / d ₂ ) — 5.15 (This formula accounts for 99% of the normal distribution). The common constants used for the d ₂ are shown in Table 8.1.

6. Divide RR Error by the tolerance and multiply by 100%; therefore,

   

ATTRIBUTE GAGE STUDY

An attribute gage study is one that compares each part to a specific set of limits, accepts the part if the limits are satisfied, and rejects the part otherwise . Most such gages are set up to accept or reject a set of master parts. Unlike a variable gage, an attribute gage cannot indicate how good or how bad a part is but only that the part is good or bad.

The short study for the attribute analysis is conducted by selecting 20 parts and having two operators measure all parts twice in a manner so as to prevent operator bias. In selecting the 20 parts, it is desirable that some of the parts will be slightly below and others will be slightly above both specification limits.

The gage is acceptable if all measurement decisions (four per part) agree. If the measurement decisions do not agree, the gage must be improved and/or evaluated. If the gage cannot be improved, it is unacceptable, and an acceptable alternate measurement system should be found.

A typical form, used for the short attribute gage study, is shown in Figure 15.15.

Attribute Gage Study
Item XYZ
Go-No Go Plug Gage
	Operator A		Operator B
	1	2	1	2
1	g	G	g	G
2	g	G	G	G
3	Ng	G	G	G
4	ng	ng	ng	ng
5	G	G	G	G
6	G	G	G	G
7	ng	ng	ng	Ng
8	ng	G	ng	G
9	G	G	G	G
10	G	G	G	G
11	G	G	G	G
12	G	ng	G	G
13	G	G	G	G
14	G	G	G	G
15	G	G	G	G
16	G	G	G	G
17	G	G	G	G
18	G	G	G	G
19	G	G	G	G
20	G	G	G	G

Figure 15.15: Attribute gauge study.

INTERPRETATION OF BASIC MEASUREMENT ERROR FINDINGS

Measurement error studies are conducted as a means of quantifying and evaluating the magnitude of measurement system error. The percentage of the product tolerance used by measurement system error is calculated during Step 20 of the basic measurement error study. This value compares the spread of error inherent to the measurement system with the product tolerance. The variation due to the measurement system should be smaller than the product tolerances. The following decision rules are usually used to evaluate the magnitude of measurement error:

• Measurement system error under 10% of the product tolerance ”acceptable level of measurement error.

• Measurement system error between 10% and 30% of the product tolerance ”may be considered acceptable based on the importance of the measured characteristic, cost of incorrect machine adjustment decisions, cost of the gauge, or cost and/or time needed for gage repairs.

• Measurement system error greater than 30% of the product tolerance ”measurement system should be considered unacceptable, and every effort should be made to reduce the level of measurement error.

If the percentage of the product tolerance used by measurement system error is less than 10%, the study of the measurement system should be continued to provide further insight to the nature of measurement error. This is because the basic measurement error study is a study of only one point in time. It is a good screening study that will identify poor measuring systems. The basic measurement error study does not provide information concerning the stability of the measuring system. The study should be extended so that additional measurement trials are made. These additional trials are needed for the assessment of the measurement system stability.

Measurement systems that yield unacceptable levels of measurement error must be diagnosed and improved. If the percentage of the product tolerance is unacceptably large for the basic measurement error study, the system will demonstrate even greater levels of error over a longer period of time.

• If the error due to repeatability (%RPT error) is significantly larger than that of reproducibility, the following may be required:

  • Maintenance or repair of the measurement hardware (gage, scale, meter, and/or fixtures),

  • Training for all appraisers related to the measurement procedure, or

  • Change of the measurement strategy (replace an analog readout with a digital readout as a means to avoid rounding errors).

• If the error due to reproducibility (%RPD error) is significantly larger than that of repeatability, the following may be required:

  • Standardization of the calibration technique used by the appraisers,

  • Training for specific appraisers related to the standard measurement procedure, or

  • Standardization of where on the parts measurements are taken.