Which to Improve: Repeatability or Reproducibility — A Cost-Benefit Analysis of Two Paths to Compress GR&R

When GR&R reaches 28%, the first suggestion at the review meeting is usually to "measure each piece twice more and take the average." This action is indeed effective, but only to a certain extent—it can reduce repeatability, but not reproducibility. If the wrong aspect is improved, the invested measurement time will be wasted, and the numbers on the report will still not meet the acceptable standard.

The following analysis uses a complete set of data to clarify which aspects can be saved by increasing the number of measurements, which must involve changes in clamping, judgment criteria, or measuring instruments, and how many points each path can recover.

1. Decomposition: Is the Excess Due to Equipment or People

The variance in GR&R is composed of two parts, which must be analyzed separately:

• Repeatability (EV): The scatter in measurements taken by the same person on the same piece with the same clamping within a short period. Sources include the measuring instrument itself, clamping movement, non-fixed measurement points, and environmental fluctuations.
• Reproducibility (AV): The difference in average values between different operators. Sources include clamping habits, reading and value-taking methods, judgment criteria for marginal states, and operational force.

The mathematical properties of these two aspects are entirely different, which is the foundation for all subsequent conclusions: Repeatability is a "scatter," which decreases when averaged; Reproducibility is a "bias," which does not decrease when averaged.

A table to help determine which aspect to address:

If the data for each person is stable but the means are offset, it indicates that the measurement method itself is not flawed, but rather that "people are not using the same standard"—in this case, increasing the number of repeated measurements is ineffective.

2. Increasing Measurement Times Only Reduces Half: The Limit of 1/√n

When the same evaluator measures the same piece n times and takes the average, repeatability decreases by 1/√n, while reproducibility remains unchanged:

• Repeatability: σ_E(average) = σ_E ÷ √n
• Reproducibility: σ_A remains constant
• Combined: %GRR has a lower limit determined solely by reproducibility, i.e., σ_A ÷ σ_TV

Assume a process where σ_TV = 0.10 mm, σ_E = 0.020 mm, and σ_A = 0.020 mm. The %GRR for a single measurement is 28.3%. The results of increasing the number of measurements are as follows:

Three key points to remember:

1. Increasing from 3 to 5 measurements increases measurement time by 67%, but %GRR only drops from 23.1 to 21.9—a reduction of just over one point, which still does not approach the "conditional acceptance" threshold.
2. Even with an infinite number of measurements, %GRR can only reach 20.0%—because σ_A alone accounts for 20%. This ceiling is mathematically determined, not due to a lack of effort.
3. Increasing from 2 to 3 measurements is the only clearly cost-effective option—the reduction is 1/√1.5 ≈ 0.82, or about 18%. After 3 measurements, the marginal benefit rapidly diminishes.

Therefore, "increasing the number of measurements" is only effective in one scenario: when σ_E is significantly greater than σ_A, and the gap is sufficient to bring the ceiling below 10%.

3. Example Calculation: How to Reduce 28.3% to Below 10%

Using the same example, a comparison of the benefits of three paths (σ_TV = 0.10 mm remains constant):

Interpretation: The middle three rows show that reducing reproducibility from 0.020 to 0.010 is much more effective than increasing the number of measurements from 1 to 10 (15.3% vs. 21.0%). The cost of the former often involves only standardizing measurement points, unifying value-taking methods, and adding a positioning fixture—much cheaper than replacing the measuring instrument.

The conclusion follows a sequence: First, standardize measurement points and clamping; second, unify judgment criteria and limit samples; third, increase the number of measurements; and last, consider replacing the measuring instrument. The first two steps are almost cost-free and directly impact σ_A.

4. Three Judgment Rules: When Increasing Measurement Times is Futile

Rule One: If the actual reduction is significantly less than the theoretical value, the data is not independent. In the example, increasing from 1 to 3 measurements should reduce the repeatability to 0.577 times and the combined %GRR from 28.3% to 23.1% (a reduction of 18%). If the actual reduction is only 3-5 points, it indicates that the n measurements are not independent—such as not removing and re-clamping the same piece, reading the same position each time, or the same person taking consecutive readings. Averaging correlated data does not reduce error; it just repeats the same error.

Rule Two: If the resolution is insufficient, increasing the number of measurements is meaningless. First, check the ratio of the instrument's graduation value to the process variation. If the graduation value is greater than 1/10 of the process variation, a significant portion of σ_E is quantization error due to readings only falling on the graduations. No matter how many times you measure, the results will fall into the same grid. In this case, you must first change the graduation or the measuring instrument.

Rule Three: Increasing measurement times for submission is usually not accepted by the client. The MSA report is evidence of process control, not an optimized scorecard. If the PPAP or client specifications require a specific test structure (commonly 10 pieces × 3 evaluators × 3 measurements), follow that structure. Do not arbitrarily change it to 5 measurements to make the numbers look better. During the review, if asked "How many times do you measure in daily operations?" and you cannot answer, it will be a new nonconformity.

Additionally, a commonly overlooked acceptance criterion is: ndc ≥ 5. Even if %GRR meets the standard but ndc is only 3, it indicates that the measurement system cannot distinguish between conforming and marginal pieces and is therefore unsuitable for process control.

5. Five Implementation Steps

1. Read the Variance Components: Record the contribution percentages of EV and AV separately, not just the combined %GRR.
2. Identify the Bottleneck: If EV is high, check clamping, measurement points, and resolution; if AV is high, check criteria, limit samples, training, and positioning.
3. Calculate Before Acting: Substitute the proposed measures' σ_E and σ_A into the table to see if they can bring the %GRR below 10%. If not, do not invest time; instead, modify the plan.
4. Re-measure After Improvement: Use the same batch of pieces, the same group of evaluators, and the same test structure; otherwise, the results are not comparable. The re-measurement sample should cover at least 80% of the tolerance band and have sufficient piece-to-piece variation; otherwise, %GRR will be systematically underestimated.
5. Standardize: Write the measurement points, clamping methods, value-taking rules, and measurement times into the SIP and MSA plan, and include them in the new employee training and assessment; otherwise, new employees will push σ_A back up in three months.

Record the data before and after improvement, with six fixed fields to ensure comparability:

Judgment criteria should be unified into three categories: %GRR ≤ 10% accepted; 10%～30% conditionally accepted, with an explanation of characteristic importance and measurement costs and client confirmation; > 30% not accepted, and the measurement system must be changed.

If GR&R cannot be reduced, it is often not because of too few measurements, but because the wrong aspect is being improved—first look at the AV half, then decide whether to increase the number of measurements.

Knowledge code: 6.2.1

Version: v20261024

Author: QTank QTank is dedicated to providing systematic professional knowledge, methodologies, and practical tools for quality management practitioners, helping companies continuously improve their quality capabilities.