What Are Repeatability and Reproducibility Limits (r & R) in ISO 5725-2?

If you have ever compared test results from two laboratories—or two operators in the same lab—and wondered whether the difference is real or just measurement noise, the repeatability limit (r) and reproducibility limit (R) from ISO 5725-2 give you a clear, standardized answer. They are not abstract statistics; they are practical thresholds for deciding when a difference between results is too large to be explained by random error.

What It Is

In ISO 5725-2 ("Accuracy (trueness and precision) of measurement methods and results—Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method"), repeatability and reproducibility are two measures of precision under different conditions:

• Repeatability (r) describes the precision when the same operator uses the same equipment on the same test item in the same laboratory within a short time. It captures only the smallest, unavoidable random variation.
• Reproducibility (R) describes the precision when results are obtained in different laboratories (or with different operators/equipment), capturing additional variation from lab-to-lab differences, operator skill, environmental factors, and calibration.

The repeatability limit (r) and reproducibility limit (R) are the values below which the absolute difference between two single test results is expected to lie with a probability of 95% under those respective conditions.

How It Works: The Formula and Steps

The calculation follows a standard ANOVA-based procedure in ISO 5725-2. The key formulas are:

\[
r = 2.8 \times s_r
\]

\[
R = 2.8 \times s_R
\]

Where:

• \(s_r\) = repeatability standard deviation (within-laboratory, within-operator variation)
• \(s_R\) = reproducibility standard deviation (total variation including between-laboratory effects)
• 2.8 is the coverage factor for a 95% probability, derived from \(1.96 \times \sqrt{2}\) (since we compare the difference between two independent results, each with variance \(s^2\), so the standard deviation of the difference is \(s \times \sqrt{2}\); multiplying by 1.96 gives 2.77, rounded to 2.8).

Steps to determine r and R in a collaborative study (per ISO 5725-2):

1. Organize a collaborative trial: At least 8 laboratories (ISO recommends) each test the same homogeneous material, with multiple replicate measurements per lab (typically 2–3 per level).
2. Compute within-lab variances for each laboratory.
3. Pool the within-lab variances to estimate \(s_r\) (repeatability standard deviation).
4. Estimate between-lab variance from the spread of lab means, then combine with \(s_r\) to obtain \(s_R\) (reproducibility standard deviation).
5. Apply the 2.8 multiplier to get r and R.

How to use r and R in practice:

• If two single results from the same lab differ by more than r, the results are not consistent under repeatability conditions—investigate.
• If two single results from different labs differ by more than R, the results are not consistent under reproducibility conditions—investigate.
• For the critical difference between the means of two groups of results (e.g., averages of n1 and n2 replicates), use:

\[
\text{Critical difference} = 2.8 \times \sqrt{\frac{s_r^2}{n_1} + \frac{s_R^2}{n_2}}
\]

(This is a simplified form; ISO 5725-6 provides the full treatment.)

A Worked Illustrative Example

Example data (illustrative only): Suppose a collaborative study on a chemical assay yields:

• Repeatability standard deviation: \(s_r = 0.35\) mg/L
• Reproducibility standard deviation: \(s_R = 0.62\) mg/L

Then:

\[
r = 2.8 \times 0.35 = 0.98 \text{ mg/L}
\]

\[
R = 2.8 \times 0.62 = 1.74 \text{ mg/L}
\]

Interpretation:

• If Lab A runs the same sample twice and gets 10.2 and 11.5 mg/L, the difference is 1.3 mg/L. Since 1.3 > r (0.98), the two results are not consistent under repeatability conditions—something may have changed between runs (e.g., sample degradation, operator error).
• If Lab A reports 10.2 mg/L and Lab B reports 11.8 mg/L for the same material, the difference is 1.6 mg/L. Since 1.6 < R (1.74), the results are consistent under reproducibility conditions; the difference is within normal inter-laboratory variation.

Common Pitfalls

• Confusing r and R with standard deviations: r and R are limits for the difference between two results, not the standard deviation itself. Always apply the 2.8 factor.
• Using r to judge inter-laboratory differences: r applies only under repeatability conditions (same lab, same operator, same equipment). Use R for cross-laboratory comparisons.
• Ignoring the number of replicates: The 2.8 factor applies to single results. If you compare averages of multiple runs, you must adjust using the critical difference formula above.
• Assuming 2.8 is arbitrary: It is a fixed statistical constant for 95% coverage when comparing two independent results—do not round it down to 2.0 or up to 3.0.

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To quickly compute r and R from your own \(s_r\) and \(s_R\) values, or to check whether a difference between two results exceeds the limit, use our free online tool: Repeatability/Reproducibility Limit Calculator.