10 Analytical Instrumentation
UNDERSTANDING AND LEARNING TO CALCULATE REPEATABILITY AND REPRODUCIBILITY USING ASTM D6300
Introduction In the world of petroleum and lubricant testing, few qualities are as important as precision. Precision refl ects the reliability of a procedure to produce consistent results under set conditions, and two key elements of precision are repeatability (r) and reproducibility (R). Repeatability refers to the variability of multiple test results by an individual analyst using a single apparatus to test replicate specimens from a single sample. Reproducibility measures the variability of test results among diff erent analyses, each using a diff erent apparatus, when testing specimens from sub-samples of a sample that have been shipped to multiple laboratories.
Both types of variability are crucial for understanding the confi dence limits for a test method and minimizing the risk of incorrectly assessing a property to either be out of specifi cation when it is actually in specifi cation (known as a Type 1 error) or, conversely, assessing the property to be in specifi cation when it is actually not (known as a Type II error). These values then inform precision, which can establish the expected resolution between two values such that researchers performing condition monitoring tests can minimize the risk of either type of error. With an established range, researchers can more confi dently test the results of an experiment and gauge whether the process itself is operating within specifi ed parameters.
To numerically determine these parameters, ASTM International developed the ASTM D6300 standard, which provides guidelines for determining precision and bias for test methods involving petroleum products, liquid fuels, and lubricants1
. ASTM D6300 has become an indispensable tool
for gauging variability in test results. The formulas for calculating repeatability and reproducibility are of great importance to researchers in this fi eld. This paper covers the calculation methods for both values, as outlined in the ASTM D6300 standard, explains the rationale behind key steps and provides a framework for manual calculation of repeatability (r) and reproducibility (R).
The Signifi cance of Repeatability and Reproducibility
The importance of repeatability and reproducibility should be reinforced signifi cantly. While the two values do not govern the outcome of a procedure, they act as measures to examine the variability of a testing method. Repeatability and reproducibility allow analyses to establish tolerance thresholds by indicating when two replicate test results are and are not “within normal expectation,” with a standard expectation cutoff representing 95% of the reference distribution for the “absolute difference” variable (2). This is necessary for researchers performing condition monitoring tests, as an observable measure of replicability is always welcome for applications machines can automate, as mechanisms performing outside of acceptable error ranges are typically in need of maintenance and inspection.
Without the ability to gauge the irregularity of the output, researchers would be venturing blindly into a maze of Type I and Type II errors, unable to gauge whether the machine is operating inside or outside specifi ed ranges and thus unqualifi ed to determine whether an apparatus is working or malfunctioning. This would make system maintenance a nightmare, as without a basis to gauge the expected range of products, it would be much more diffi cult to determine if a mechanism requires maintenance just by observing its output. This drastically increases maintenance time and cost in factories, since each machine would then require manual inspection for errors at regular intervals to avoid the risk of catastrophic failure. With the application of repeatability and reproducibility, researchers can simply observe the precision of a baseline machine and compare it to the output of the machines under observation to gauge whether they are within acceptable margins.
Preparing Samples for Calculations:
Before properly beginning calculations, we must prepare a valid and informative data set for the experiment. According to standard practice, one would prepare enough material to supply at least 10% more than necessary for each laboratory involved in the experiment2
. The samples This serves as a measure of spread across multiple labs using the average values as a baseline, . Each unit will be
labelled with a letter for the material and a sequential number. For instance, for 10 laboratories and 2 test results of a material B, the test samples would be labeled B1 through B223
themselves should include specimens in which the measured property is just above the detection limit, within the detection range, and just below the maximum measurable value without dilution.
While other precision metrics are tested and calculated within the experiment, this paper will focus solely on the formulas related to the calculation of repeatability and reproducibility.
Just like Repeatability Standard Defi nition, this is the standard deviation of the test results obtained under reproducibility conditions.
and we can use it to further extrapolate precision results across labs. With SA calculated, we then determine the Reproducibility Standard Deviation (SR) using the following formula:
Repeatability standard deviation is the standard deviation of the given test under repeatability conditions. However, this value does not account for 95% confi dence. This will be rectifi ed in the next
step.This value will be further refi ned in the next step.
With Sr calculated, we then multiply the answer by 2.8 to calculate repeatability (r). This is to account for 95% confi dence, as 1.96 standard deviations multiplied by √2 is roughly equivalent to 2.8.we now must account for the “normal expectation” cutoff, which falls within the 95th percentile, or 95% of all data. To do so, we multiply the repeatability standard deviation by 2.8, which brings the value within those bounds and thus provides our value for repeatability.
With this procedure, you can now calculate repeatability for each of the samples in the data set by simply inputting the data from that specifi c sample.
Formula for Reproducibility (R):
Reproducibility is found after repeatability, as Repeatability Standard Deviation is required for calculations. First, we calculate the mean of a sample in each lab within the set, as well as the total average of all the sample data across all labs (A). With the means, we calculate the Standard Deviation of the Averages (SA) using the following formula:
Formula for Repeatability (r):
Of the various measures of accuracy, the fundamental precision statistics are Repeatability Standard Deviation (Sr) and Reproducibility Standard Deviation (SR). These two values are central to calculating repeatability and reproducibility of the subject, respectively. Repeatability concerns the results of multiple analyses by a single operator on specimens from a single sample. The expected data set will contain one or more samples undergoing multiple trials of the same procedure across multiple labs. As the calculations for reproducibility require repeatability, repeatability will be calculated fi rst.
First, we must calculate the mean of test results from a single sample as tested by a single analyst. The formulas for computing the mean and standard deviation (S) of replicate tests are:
This gives us the average value of the sample within the relevant data set. With the mean, we can calculate the standard deviation of that sample with the next formula:
The standard deviation indicates how dispersed the data is relative to the mean. This serves as a general indicator of how precise the data is before any further calculations. We will then calculate the standard deviation for the same sample in each of the remaining labs in the data set using the same formulas above. With that set of standard deviations, we will calculate the repeatability standard deviation (Sr) pooled over all labs using the following formula:
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