9.4 Variable Gage R&R: Repeatability and Reproducibility

9.4 Variable Gage R&R: Repeatability and Reproducibility

Understanding Precision: Repeatability (EV) and Reproducibility (AV)

While accuracy (bias, linearity, stability) assesses whether a measurement system is centered on the true reference standard, precision assesses the spread or dispersion of repeated measurements. In industrial quality control, precision is evaluated through a Gage Repeatability and Reproducibility (Gage R&R) study.

Total measurement system precision error is mathematically partitioned into two distinct physical components:

PRECISION ERROR DECOMPOSITION:

                    TOTAL GAGE R&R VARIATION (sigma_GRR)
                                    |
                   +----------------+----------------+
                   |                                 |
             REPEATABILITY                     REPRODUCIBILITY
       Equipment Variation (EV)            Appraiser Variation (AV)
                   |                                 |
       - Same appraiser                  - Different appraisers
       - Same gage                       - Same gage
       - Same part / characteristic      - Same parts
       - Short time intervals            - Different techniques / clamping
       - Inherent gage noise / play      - Human interface differences

1. Repeatability (Equipment Variation, EV)

Repeatability is the variation observed when one appraiser uses the same gage to measure the same characteristic on the same identical part multiple times under identical operating conditions.

• What it reflects: The inherent mechanical, structural, and electrical noise of the measurement device itself.
• Root causes of high EV: Mechanical backlash in gear teeth, loose spindle bearings, sloppy slideways, electronic sensor drift, lack of gage rigidity, elastic fixture flexing, or friction in indicator dials.

2. Reproducibility (Appraiser Variation, AV)

Reproducibility is the variation observed in the average of measurements made by different appraisers using the same gage when measuring identical characteristics on the same set of parts.

• What it reflects: The variability introduced by the human beings operating the measurement equipment.
• Root causes of high AV: Inconsistent clamping force, differences in operator hand squeeze, varying sight angles causing parallax, improper part seating against datums, differing interpretations of zero, or poor operational training.

Combining EV and AV into Total Gage R&R ($GRR$)

Because repeatability and reproducibility are independent sources of variance, their variances sum directly: $\sigma^2_{GRR} = \sigma^2_{\text{Repeatability}} + \sigma^2_{\text{Reproducibility}} = \sigma^2_{EV} + \sigma^2_{AV}$ $\mathbf{GRR = \sqrt{EV^2 + AV^2}}$

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Variable Gage R&R Study Architecture and Design Rules

To perform a statistically rigorous variable Gage R&R study, the quality technician must follow standardized AIAG design protocols:

The Standard Study Structure: 10-3-3 Architecture

• 10 Parts ($n = 10$): Representative parts selected from the manufacturing process.
• 3 Appraisers ($k = 3$): Operators who normally inspect the product on production shifts.
• 3 Trials ($r = 3$): Each operator measures each part three separate times.
• Total Data Points: $10 \text{ parts} \times 3 \text{ operators} \times 3 \text{ trials} = \mathbf{90 \text{ individual measurements}}$. (Note: For coarse screening or smaller operations, a $10 \times 2 \times 2 = 40$-point study is sometimes permitted, but the 10-3-3 structure is the industrial benchmark).

Three Inflexible Execution Rules

1. Parts Must Span Process Variation: The 10 parts must not be chosen consecutively from one hour of production. They must be selected randomly across days or shifts to encompass the full range of historical manufacturing variation (from the low end to the high end of typical production). Selecting 10 nearly identical parts artificially deflates Part Variation ($PV$), causing calculated $%GRR$ to appear disastrously inflated!
2. Blind Numbering: The 10 parts must be coded or labeled so that the appraisers cannot identify which part they are measuring. If an operator sees "Part 3" and remembers they measured it at $1.002"$ earlier, they will subconsciously bias their second reading to $1.002"$, destroying the study's validity.
3. Completely Randomized Order: In Trial 1, Appraiser A measures all 10 parts in randomized order (e.g., 7, 2, 9, 1, ...). Then Appraiser B measures them in a newly randomized order, followed by Appraiser C. In Trials 2 and 3, the parts are re-randomized. Technicians must never allow an appraiser to measure the same part 3 times consecutively!

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Mathematical Methodologies: Average and Range ($\bar{X}-R$) vs. ANOVA

Two mathematical techniques are recognized by AIAG for calculating Gage R&R:

1. The Average and Range ($\bar{X}-R$) Method

The Average and Range method is the traditional calculation technique historically favored on shop floors because it can be computed manually using range statistics and tabulations of $d_2^*$ constants.

Step-by-Step Computational Formulas:

1. Range per Part Trial: For each appraiser on each part, calculate the range across trials: $R_{ij} = X_{\max} - X_{\min}$.
2. Average Range per Appraiser: Calculate the average range for Appraiser A ($\bar{R}_A$), Appraiser B ($\bar{R}_B$), and Appraiser C ($\bar{R}_C$).
3. Grand Average Range ($\bar{\bar{R}}$): $\bar{\bar{R}} = \frac{\bar{R}_A + \bar{R}_B + \bar{R}_C}{k}$
4. Equipment Variation (Repeatability, $EV$): $EV = \bar{\bar{R}} \times K_1 = \frac{\bar{\bar{R}}}{d_2^*(m, g)}$ Where $K_1 = 1 / d_2^$. For $r = 3$ trials, $K_1 = 0.5908$ ($d_2^ \approx 1.693$).
5. Appraiser Averages: Calculate the overall grand average measurement for each appraiser ($\bar{X}_A, \bar{X}_B, \bar{X}_C$).
6. Appraiser Difference ($\bar{X}_{\text{diff}}$): $\bar{X}_{\text{diff}} = \bar{X}_{\max} - \bar{X}_{\min}$
7. Appraiser Variation (Reproducibility, $AV$): $AV = \sqrt{(\bar{X}_{\text{diff}} \times K_2)^2 - \frac{EV^2}{n \times r}}$ Where $n = 10$ parts, $r = 3$ trials, and $K_2 = 1 / d_2^*$ for $k$ appraisers (for $k = 3$ appraisers, $K_2 = 0.5231$). (Important rule: If the value under the square root is negative, set $AV = 0$).
8. Total Gage R&R ($GRR$): $GRR = \sqrt{EV^2 + AV^2}$
9. Part Variation ($PV$): Calculate the average for each part across all appraisers and trials ($\bar{X}{P1}, \dots, \bar{X}{P10}$). $R_p = \bar{X}_{P,\max} - \bar{X}_{P,\min}$ $PV = R_p \times K_3$ Where $K_3 = 1 / d_2^*$ for $n$ parts (for $n = 10$ parts, $K_3 = 0.3146$).
10. Total Variation ($TV$): $TV = \sqrt{GRR^2 + PV^2}$

2. The ANOVA (Analysis of Variance) Method

The ANOVA method is the globally preferred modern standard for Gage R&R. While the range method only estimates main effects, ANOVA decomposes the total sum of squares into four statistical components:

1. Parts Variance ($\sigma^2_{\text{Part}}$)
2. Appraisers Variance ($\sigma^2_{\text{Appraiser}}$)
3. Appraiser-by-Part Interaction Variance ($\sigma^2_{\text{Appraiser} \times \text{Part}}$)
4. Equipment Error / Repeatability ($\sigma^2_{\text{Residual / Error}}$)

[!IMPORTANT] The Decisive Advantage of ANOVA: ANOVA explicitly measures the Operator-by-Part Interaction. If the interaction $p$-value is statistically significant ($p < 0.05$), it indicates that certain appraisers obtain systematically different readings on certain specific part geometries (for example, Appraiser A exerts excessive clamping force on thin-walled parts, but measures thick parts accurately). The Average and Range method cannot detect interaction effects!

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AIAG Acceptance Benchmarks and Tolerance vs. Process Variation

Once total $GRR$ is calculated, it must be evaluated as a percentage against an engineering baseline. In modern MSA, two distinct baselines are evaluated:

1. $%GRR$ Relative to Total Variation ($\sigma_{\text{Process}}$)

Evaluates whether the measurement system is capable of monitoring process shifts for Statistical Process Control (SPC): $\%GRR_{TV} = \frac{GRR}{TV} \times 100\%$

2. $%GRR$ Relative to Specification Tolerance ($\text{Tolerance}$)

Evaluates whether the measurement system is capable of inspecting and sorting parts against blueprint specifications: $\%GRR_{\text{Tol}} = \frac{6 \times GRR}{\text{USL} - \text{LSL}} \times 100\% \quad \left(\text{or } \frac{5.15 \times GRR}{\text{Tolerance}} \times 100\% \text{ in older manuals}\right)$ (Note: AIAG 4th Edition standardizes on $6\sigma$, representing $99.73%$ coverage, whereas older 3rd Edition manuals utilized $5.15\sigma$ for $99.0%$ coverage).

The AIAG Acceptance Criteria Matrix

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Number of Distinct Categories ($ndc$) and Process Discrimination

A gage may possess high sensitivity, yet still be incapable of dividing process variation into meaningful sub-groups. To quantify practical measurement discrimination, the AIAG standard defines the Number of Distinct Categories ($ndc$):

[!IMPORTANT] Number of Distinct Categories ($ndc$): Represents the number of non-overlapping $97%$ confidence intervals that span the expected product variation. It defines the gage's ability to detect process variation. $\mathbf{ndc = 1.41 \times \left(\frac{PV}{GRR}\right) = \sqrt{2} \times \left(\frac{PV}{GRR}\right)}$ Note: Always truncate (round down) the calculated value to the nearest integer!

NUMBER OF DISTINCT CATEGORIES (ndc) INTERPRETATION:

   ndc = 1:  [=============================================]  (UNACCEPTABLE: Gage sees parts as one chunk!)
             Cannot distinguish parts; completely obscured by gage noise.
             
   ndc = 3:  [ Low Group ]   [ Mid Group ]   [ High Group ]   (COARSE SCREENING ONLY)
             Can only sort into coarse buckets; invalid for SPC control charts.
             
   ndc >= 5: [1]   [2]   [3]   [4]   [5]   [6]   [7]   [8]   (ACCEPTABLE FOR SPC)
             Gage cleanly separates natural process variation into distinct categories.

AIAG Criteria for $ndc$:

• $ndc \ge 5$: Acceptable. The measurement system possesses adequate discrimination to support Statistical Process Control (SPC) control charts and process capability analysis ($C_p, C_{pk}$).
• $2 \le ndc < 5$: Conditionally Acceptable for Coarse Sorting Only. The system can divide parts into high, medium, and low groups, but is too coarse to detect process trends or maintain variables control charts.
• $ndc < 2$ (i.e., $ndc = 1$): Unacceptable. Measurement noise completely swamps part-to-part variation. The system cannot distinguish one part from another.

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Attribute Agreement Analysis: Fleiss' Kappa and Concordance

When quality inspections produce binary or nominal classifications (Pass/Fail, Conforming/Nonconforming, Good/Bad visual defect sorting) rather than continuous variables data, technicians must conduct an Attribute Agreement Analysis (Attribute Gage R&R).

Study Setup

• Select 30 to 50 parts with known reference master status (determined by engineering experts or laboratory metrology).
• The sample must include conforming parts, nonconforming parts, and critically: borderline / marginal parts.
• Have 2 to 3 appraisers inspect the parts across 2 to 3 trials in blind, randomized order.

Key Metrics Evaluated

1. Within-Appraiser Agreement (Repeatability): How consistently does Operator A classify the same part across repeated trials?
2. Between-Appraiser Agreement (Reproducibility): How frequently do Appraiser A, Appraiser B, and Appraiser C agree with each other on the same part?
3. Agreement with Reference Standard (Accuracy / Effectiveness): How frequently do appraiser classifications match the true known master status?

Fleiss' Kappa ($\kappa$) Statistic

Raw percent agreement is deceptive because appraisers can agree purely by random guessing. To eliminate chance agreement, MSA calculates Cohen's Kappa (for 2 raters) or Fleiss' Kappa (for $> 2$ raters): $\kappa = \frac{P_{\text{observed}} - P_{\text{expected}}}{1 - P_{\text{expected}}}$ Where $P_{\text{observed}}$ is the proportion of observed agreements, and $P_{\text{expected}}$ is the proportion of agreements expected purely by random chance.

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Step-by-Step Worked Numerical Example: Full Gage R&R Calculation

Scenario Data

A quality technician conducts a variable Gage R&R study on a CNC lathe turning diameter. The study employs $n = 10$ parts, $k = 3$ operators, and $r = 3$ trials ($90$ total data points). The engineering print specification is $1.2500" \pm 0.0100"$ (Total Tolerance $= 0.0200"$).

The preliminary data analysis yields the following calculated summary statistics:

• Grand Average Range across all operators and parts: $\bar{\bar{R}} = 0.00085"$
• Operator Grand Averages: $\bar{X}_A = 1.25040"$, $\bar{X}_B = 1.25012"$, $\bar{X}_C = 1.24998"$
• Part Averages Range: $R_p = \bar{X}{P,\max} - \bar{X}{P,\min} = 1.25600" - 1.24350" = 0.01250"$

Constants from AIAG Tables:

• For $r = 3$ trials: $K_1 = 0.5908$
• For $k = 3$ operators: $K_2 = 0.5231$
• For $n = 10$ parts: $K_3 = 0.3146$

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Step 1: Calculate Repeatability (Equipment Variation, $EV$)

$EV = \bar{\bar{R}} \times K_1 = 0.00085" \times 0.5908 = \mathbf{0.0005022"}$

Step 2: Calculate Reproducibility (Appraiser Variation, $AV$)

First, find the maximum difference between operator averages: $\bar{X}_{\text{diff}} = \bar{X}_A - \bar{X}_C = 1.25040" - 1.24998" = 0.00042"$ Now compute $AV$: $AV = \sqrt{(\bar{X}_{\text{diff}} \times K_2)^2 - \frac{EV^2}{n \times r}}$ $(\bar{X}_{\text{diff}} \times K_2)^2 = (0.00042" \times 0.5231)^2 = (0.0002197)^2 \approx 4.8269 \times 10^{-8}$ $\frac{EV^2}{n \times r} = \frac{(0.0005022)^2}{10 \times 3} = \frac{2.5220 \times 10^{-7}}{30} \approx 8.4068 \times 10^{-9}$ $AV = \sqrt{4.8269 \times 10^{-8} - 8.4068 \times 10^{-9}} = \sqrt{3.9862 \times 10^{-8}} \approx \mathbf{0.0001997"}$

Step 3: Calculate Total Gage R&R ($GRR$)

$GRR = \sqrt{EV^2 + AV^2} = \sqrt{(0.0005022)^2 + (0.0001997)^2}$ $GRR = \sqrt{2.5220 \times 10^{-7} + 3.9880 \times 10^{-8}} = \sqrt{2.9208 \times 10^{-7}} \approx \mathbf{0.0005404"}$

Step 4: Calculate Part Variation ($PV$)

$PV = R_p \times K_3 = 0.01250" \times 0.3146 = \mathbf{0.0039325"}$

Step 5: Calculate Total Variation ($TV$)

$TV = \sqrt{GRR^2 + PV^2} = \sqrt{(0.0005404)^2 + (0.0039325)^2}$ $TV = \sqrt{2.9208 \times 10^{-7} + 1.5465 \times 10^{-5}} = \sqrt{1.5757 \times 10^{-5}} \approx \mathbf{0.0039695"}$

Step 6: Evaluate $%GRR$ Acceptance Criteria

1. $%GRR$ Relative to Total Variation (Process): $\%GRR_{TV} = \frac{GRR}{TV} \times 100\% = \frac{0.0005404"}{0.0039695"} \times 100\% = \mathbf{13.62\%}$
2. $%GRR$ Relative to Specification Tolerance: $\%GRR_{\text{Tol}} = \frac{6 \times GRR}{\text{Tolerance}} \times 100\% = \frac{6 \times 0.0005404"}{0.0200"} \times 100\% = \frac{0.0032424"}{0.0200"} \times 100\% = \mathbf{16.21\%}$

• Evaluation: Both $%GRR_{TV}$ ($13.6%$) and $%GRR_{\text{Tol}}$ ($16.2%$) fall cleanly between $10%$ and $30%$. The measurement system is Conditionally Acceptable.

Step 7: Calculate Number of Distinct Categories ($ndc$)

$ndc = 1.41 \times \left(\frac{PV}{GRR}\right) = 1.41 \times \left(\frac{0.0039325"}{0.0005404"}\right) = 1.41 \times 7.277 = 10.26$

• Truncating to the nearest integer: $ndc = 10$.
• Discrimination Evaluation: Because $ndc = 10 \ge 5$, the gage possesses excellent discrimination, capable of dividing part variation into 10 distinct categories. It is fully qualified for Statistical Process Control (SPC) charting.

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Technician Inspection Scenarios & Common Exam Traps

Real-World Shop Scenario: Investigating High %GRR

A quality technician conducts a Gage R&R on an optical shaft measurement gage and finds $%GRR = 38%$ (unacceptable). The technician's manager immediately demands that all operators be sent to an off-site training course to improve their measurement technique. The technician reviews the data sheet and notes:

• $EV = 0.0018"$
• $AV = 0.0003"$
• Root Cause Diagnosis: The manager's decision is fundamentally flawed. $EV$ accounts for over $97%$ of the total measurement variance ($EV^2 / GRR^2 = (0.0018)^2 / [(0.0018)^2 + (0.0003)^2] = 0.973$). Appraiser variation ($AV$) is tiny. Training operators will not improve $EV$ by even one percent! The problem is mechanical equipment variation: worn spindle bearings, loose optical stage guide rails, or camera vibration. The technician must rebuild the mechanical fixture.

Common Exam Traps for CQT Candidates

• Exam Trap 1: Truncating $ndc$: Always round down (truncate) $ndc$ to the integer. If $ndc = 4.95$, it is 4, which fails the requirement of $ndc \ge 5$. Never round up to 5!
• Exam Trap 2: Variance Summation vs. Standard Deviations: Remember that variances add directly ($\sigma^2_{GRR} = \sigma^2_{EV} + \sigma^2_{AV}$), but standard deviations do not add linearly ($GRR \ne EV + AV$). You must take the square root of the sum of squares: $GRR = \sqrt{EV^2 + AV^2}$.
• Exam Trap 3: Handpicked "Golden" Parts in Study Design: If an exam question asks: "What happens if an engineer hand-selects 10 parts that are all machined near nominal?" Handpicking identical parts slashes Part Variation ($PV$). Because $%GRR = \frac{GRR}{\sqrt{GRR^2 + PV^2}}$, a tiny $PV$ forces the denominator to shrink, causing $%GRR$ to skyrocket falsely.
• Exam Trap 4: ANOVA Interaction Interpretation: If an ANOVA Gage R&R indicates a significant $p$-value for the Operator $\times$ Part interaction ($p < 0.05$), the root cause is that appraisers apply different techniques to different part sizes (e.g., operator deflection on thin parts, or inconsistent sight angles on tall parts).