Average and Range Method :: SAS/QC(R) 12.1 User's Guide

Average and Range Method

The average and range method is widely used in industry because its calculations can be done by hand. It measures both repeatability and reproducibility for a measurement system.

All calculations described here are based upon a specified multiple of $\sigma$ , where the multiple $\nu$ can be 4, 5.15, or 6. Figure A.23 shows a sample gage report created with the GAGE application using the average and range method.

The measure of repeatability (or equipment variation), denoted by , is calculated as

$EV = \overline{R} \times K_1$

where $\overline{R}$ is the average range and is the adjustment factor

$K_1 = \frac{\nu }{d_2}$

Figure A.23: Average and Range Method Sample Report

                   Average and Range Method

    Test ID: Gasket         Performed By:
    Date:    04/17/02       John Smith

    Part No. & Name:   Gasket
    Characteristics:
    Specification:     0.6-1.0 mm

    Gage Name: Thickness
    Gage No.:  X-2034
    Gage Type: 0-10 mm


      MEASUREMENT UNIT ANALYSIS       % PROCESS VARIATION
    Repeatability
        EV  =      0.1747               % EV   =  18.70 %
    Reproducibility
        AV  =      0.1570               % AV   =  16.80 %
    Gage R&R
        R&R =      0.2349               % R&R  =  25.14 %
    Part Variation
        PV  =      0.9042               % PV   =  96.79 %
    Total Variation
        TV  =      0.9342


         Results are based upon predicting 5.15 sigma.
    (99.0% of the area under the normal distribution curve)

The quantity (Duncan 1974, Table M) depends on the number of trials used to calculate a single range. In the GAGE application, the number of trials can vary from 2 to 4. Use of is valid when $\# operators \times \# parts \ge 16$ ; otherwise, the GAGE application uses (Duncan 1974, Table D3), which is based on the number of ranges calculated from $\# operators \times \# parts$ and on the number of trials.

The measure of reproducibility (or appraiser variation), denoted by , is calculated as

$AV = \sqrt {(\overline{X}_{diff} \times K_2)^2 - \frac{(EV)^2}{nr}}$

where $\overline{X}_{diff}$ is the difference between the maximum operator average and the minimum operator average, is the adjustment factor

$K_2 = \frac{\nu }{d_2^*}$

n is the number of parts, and r is the number of trials. Reproducibility is contaminated by gage error and is adjusted by subtracting ${(EV)^2}/{nr}$ . The quantity (Duncan 1974, Table D3) depends on the number of operators used to calculate a single range. In the GAGE application, the number of operators can vary from 1 to 4. When there is only one operator, reproducibility is set to zero.

The measure of repeatability and reproducibility, denoted by $R\& R$ , is calculated as

$R\& R = \sqrt {(EV)^2 + (AV)^2}$

Part-to-part variation, denoted by , is calculated as

$PV = R_ p \times K_3$

where is the range of part averages and is the adjustment factor

$K_3 = \frac{\nu }{d_2^*}$

Here the quantity (Duncan 1974, Table D3) depends on the number of parts used to calculate a single range. In the GAGE application, the number of parts can vary from 2 to 15.

Total variation, denoted by , is based on gage R&R and part-to-part variation.

$TV = \sqrt {(R\& R)^2 + (PV)^2}$

The measures of repeatability, reproducibility, gage R&R, part variation, and total variation are shown in Figure A.23 under the heading “MEASUREMENT UNIT ANALYSIS.” The right-hand side of the report shows the “% PROCESS VARIATION” analysis, which compares the gage factors to total variation. The percent of total variation accounted for by each factor is calculated as follows:

$\displaystyle \% EV$	$\displaystyle =$	$\displaystyle 100 \left[\frac{EV}{TV}\right]$
$\displaystyle \% AV$	$\displaystyle =$	$\displaystyle 100 \left[\frac{AV}{TV}\right]$
$\displaystyle \% R\& R$	$\displaystyle =$	$\displaystyle 100 \left[\frac{R\& R}{TV}\right]$
$\displaystyle \% PV$	$\displaystyle =$	$\displaystyle 100 \left[\frac{PV}{TV}\right]$

Note that the sum of these percentages does not equal 100%. You can use these percentages to determine whether the measurement system is acceptable for its intended application.

Instead of percent of process variation, your analysis may be based on percent of tolerance. For this you must specify a tolerance value. Then $\% EV$ , $\% AV$ , $\% R\& R$ , and $\% PV$ are calculated by substituting the tolerance value for (the denominator) in the preceding formulas. A sample report with “% TOLERANCE ANALYSIS” is shown in Figure A.24.

What is considered acceptable for $\% R\& R$ ? Barrentine (1991) gives the following guidelines:

10% or less	excellent
11% to 20%	adequate
21% to 30%	marginally acceptable
over 30%	unacceptable

In general, interpretation may be guided by local standards.

Figure A.24: Average and Range Method with % Tolerance Analysis

                   Average and Range Method

    Test ID: Gasket         Performed By:
    Date:    04/17/02       John Smith

    Part No. & Name:   Gasket
    Characteristics:
    Specification:     0.6-1.0 mm

    Gage Name: Thickness
    Gage No.:  X-2034
    Gage Type: 0-10 mm                  Tolerance: 0.4


      MEASUREMENT UNIT ANALYSIS       % TOLERANCE ANALYSIS
    Repeatability
        EV  =      0.1747               % EV   =  43.68 %
    Reproducibility
        AV  =      0.1570               % AV   =  39.25 %
    Gage R&R
        R&R =      0.2349               % R&R  =  58.72 %
    Part Variation
        PV  =      0.9042               % PV   = 226.05 %
    Total Variation
        TV  =      0.9342


         Results are based upon predicting 5.15 sigma.
    (99.0% of the area under the normal distribution curve)