(صورة توضيحية فقط)
The repeatability limit, [latex]r[/latex], is a critical value derived from the repeatability standard deviation ([latex]s_r[/latex]). It represents the maximum expected absolute difference between two single test results, obtained under repeatability conditions, with a 95% probability. It is commonly calculated as [latex]r = 2.8 \times s_r[/latex]. If the difference exceeds [latex]r[/latex], the results are considered suspect.
The repeatability limit provides a practical tool for judging the acceptability of two test results. Its statistical foundation lies in the properties of the normal distribution. The difference between two measurements drawn from the same normal distribution with standard deviation [latex]s_r[/latex] is also normally distributed with a mean of zero and a standard deviation of [latex]\sqrt{s_r^2 + s_r^2} = \sqrt{2}s_r[/latex]. To encompass 95% of these differences, we use a coverage factor. For a normal distribution, this factor is approximately 1.96. Therefore, the 95% limit is [latex]1.96 \times \sqrt{2} \times s_r \approx 2.77s_r[/latex], which is often rounded to [latex]2.8s_r[/latex] for simplicity in standards like ISO 5725.
A more precise calculation uses the Student’s t-distribution, especially when [latex]s_r[/latex] is estimated from a small number of measurements. The formula becomes [latex]r = t_{(1-\alpha/2, \nu)} \times \sqrt{2} \times s_r[/latex], where [latex]t_{(1-\alpha/2, \nu)}[/latex] is the critical value from the t-distribution for a confidence level of [latex]1-\alpha[/latex] (e.g., 95%) and [latex]\nu[/latex] degrees of freedom used to estimate [latex]s_r[/latex]. In practice, if a lab runs two tests on the same sample and the difference is greater than [latex]r[/latex], it’s a signal to investigate potential issues like procedural errors, sample contamination, or instrument malfunction.
Repeatability Limit (stats)
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