4.7 Measurement System Analysis (MSA)

4.7 Measurement System Analysis (MSA)

Measurement System Analysis (MSA) is an experimental and mathematical procedure used to quantify the variation contributed by a measurement process to the overall observed process variation. If a measurement system is noisy or biased, good product may be scrapped (Producer's Risk) or bad product shipped (Consumer's Risk).

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Components of Total Observed Variation

The total observed variation ($\sigma^2_{Total}$) in any production dataset is the sum of actual part-to-part process variance ($\sigma^2_{PV}$) and measurement system variance ($\sigma^2_{GRR}$):

$$\sigma^2_{Total} = \sigma^2_{PV} + \sigma^2_{GRR} = \sigma^2_{PV} + \sigma^2_{EV} + \sigma^2_{AV}$$

• Repeatability (Equipment Variation, EV): The variance observed when one appraiser uses the same instrument to measure the same part repeatedly under identical conditions.
• Reproducibility (Appraiser Variation, AV): The variance in the average of measurements made by different appraisers using the same instrument measuring the same parts.
• Gage Repeatability & Reproducibility ($GRR$): The combined standard deviation of Repeatability and Reproducibility: $$GRR = \sqrt{EV^2 + AV^2}$$

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Variable Gage R&R: ANOVA vs. X-bar & R Method

Two principal methodologies are used to conduct variable Gage R&R studies (typically employing $a = 3$ operators, $n = 10$ representative process parts, and $r = 3$ trials per part):

Formulas for Range Method (6-Sigma / 5.15-Sigma Spread)

Under AIAG 4th Edition ($6\sigma$ spread multiplier):

1. Equipment Variation (Repeatability): $$EV = 6 \cdot \sigma_{EV} = 6 \cdot \left(\frac{\bar{\bar{R}}}{d_2^*}\right) = K_1 \cdot \bar{\bar{R}}$$
2. Appraiser Variation (Reproducibility): $$AV = \sqrt{(K_2 \cdot \bar{X}_{diff})^2 - \frac{EV^2}{n \cdot r}}$$ (If the quantity under the radical is negative, $AV$ is set to 0)
3. Part-to-Part Variation: $$PV = K_3 \cdot R_p$$
4. Total Variation: $$TV = \sqrt{GRR^2 + PV^2}$$

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AIAG Acceptance Criteria & %GRR

The performance of a measurement system is evaluated using Percent Gage R&R (%GRR), calculated either relative to Total Variation ($TV$) or Specification Tolerance ($Tolerance = USL - LSL$):

$$\%GRR_{TV} = \left( \frac{\sigma_{GRR}}{\sigma_{Total}} \right) \times 100\% \quad \text{or} \quad \%GRR_{Tol} = \left( \frac{6 \cdot \sigma_{GRR}}{USL - LSL} \right) \times 100\%$$

AIAG Guidelines for %GRR

Number of Distinct Categories ($ndc$)

The Number of Distinct Categories ($ndc$) represents the number of non-overlapping confidence intervals of process variation that the gage can resolve:

$$ndc = \lfloor 1.41 \cdot \left( \frac{\sigma_{PV}}{\sigma_{GRR}} \right) \rfloor$$

• Acceptance Rule: $ndc \ge 5$ is required for process monitoring and control. If $ndc < 5$, the measurement system lacks sufficient resolution to discriminate part differences.

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Attribute Agreement Analysis (Kappa Statistic)

When inspection yields attribute data (e.g., Pass/Fail visual surface defects), an Attribute Agreement Analysis (AAA) is conducted using 2 to 3 inspectors evaluating 30 to 50 parts multiple times (with a known master standard).

Cohen's Kappa Statistic ($\kappa$)

Cohen's Kappa measures inter-rater agreement above and beyond agreement expected by chance alone:

$$\kappa = \frac{P_o - P_e}{1 - P_e}$$

• $P_o$: Observed proportion of concordant agreements between appraisers.
• $P_e$: Expected proportion of agreement occurring by random chance.

Interpretation of Kappa ($\kappa$)

Attribute Effectiveness Benchmark Thresholds

• Within-Appraiser Agreement (Repeatability): $\ge 90\%$ required (preferably $\ge 95\%$).
• Between-Appraiser Agreement (Reproducibility): $\ge 90\%$.
• Appraiser vs. Standard Agreement (Accuracy): $\ge 90\%$.

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Worked Numerical Example: Variable Gage R&R

Problem: A Gage R&R study conducted on a digital micrometer yields the following standard deviations from ANOVA calculation:

• Part-to-Part Standard Deviation: $\sigma_{PV} = 0.0120\text{ mm}$
• Repeatability Standard Deviation: $\sigma_{EV} = 0.0018\text{ mm}$
• Reproducibility Standard Deviation: $\sigma_{AV} = 0.0014\text{ mm}$

Evaluate:

1. Total Gage R&R Standard Deviation ($\sigma_{GRR}$).
2. Total Variation Standard Deviation ($\sigma_{TV}$).
3. Percent Gage R&R ($\%GRR_{TV}$).
4. Number of Distinct Categories ($ndc$).

Step 1: Calculate $\sigma_{GRR}$

$$\sigma_{GRR} = \sqrt{\sigma_{EV}^2 + \sigma_{AV}^2} = \sqrt{(0.0018)^2 + (0.0014)^2} = \sqrt{0.00000324 + 0.00000196} = \sqrt{0.00000520} = 0.00228\text{ mm}$$

Step 2: Calculate Total Variation Standard Deviation ($\sigma_{TV}$)

$$\sigma_{TV} = \sqrt{\sigma_{PV}^2 + \sigma_{GRR}^2} = \sqrt{(0.0120)^2 + (0.00228)^2} = \sqrt{0.000144 + 0.00000520} = \sqrt{0.0001492} = 0.012215\text{ mm}$$

Step 3: Calculate $\%GRR_{TV}$

$$\%GRR_{TV} = \left( \frac{\sigma_{GRR}}{\sigma_{TV}} \right) \times 100\% = \left( \frac{0.00228}{0.012215} \right) \times 100\% = 18.67\%$$

• Evaluation: Since $10\% < 18.67\% < 30\%$, the measurement system is conditionally acceptable (marginal).

Step 4: Calculate Number of Distinct Categories ($ndc$)

$$ndc = \left\lfloor 1.41 \cdot \left( \frac{\sigma_{PV}}{\sigma_{GRR}} \right) \right\rfloor = \left\lfloor 1.41 \cdot \left( \frac{0.0120}{0.00228} \right) \right\rfloor = \lfloor 1.41 \cdot 5.263 \rfloor = \lfloor 7.42 \rfloor = 7$$

• Evaluation: Since $ndc = 7 \ge 5$, the system has adequate discrimination resolution.