Eric Trompier explains why a low standard deviation or relative standard deviation does not necessarily correspond to a high level of accuracy nor the result obtained correspond to the true value of your sample.
Good laboratory practice requires not just the justification of traceability but also the expression of the quality of results obtained. Standard deviation (SD) or relative standard deviation (RSD) are the concepts most frequently used. But a low SD or RSD does not necessarily correspond to a high level of accuracy nor the result obtained correspond to the true value of your sample.
When applied to Karl Fischer analysis, the ideal solution is a volumetric titration system which automatically expresses measurement quality with the corresponding accuracy.
The science of measurement, ametrology' aims to express the true value of a measurement. The mean value associated with its expanded uncertainty is calculated and traced to national or international standards (traceability).
Differentiation
It is important to differentiate between repeatability and accuracy. To illustrate this, let us consider that shooting at a target is like performing a measurement: the bull's eye is the true value of your sample or of a known reference standard.
Each shot corresponds to one result. The four targets show the possibilities: shots on or off target, close to or far from the bull's eye, widely distributed, etc. Take the case of repeatable analyses that give a good SD with closely grouped shots. How far are these shots from the bull's eye? It is impossible to say based only on the SD!
It is necessary to take into account measurement errors and introduce another parameter to express measurement quality and accuracy: expanded uncertainty (Fig. 1).
Errors and uncertainty
If a set of repeated measurements is performed and the corresponding histogram plotted, we may have a normal or Gaussian distribution. The top of the curve corresponds to the arithmetic mean value of the measurements.
If we represent the ainaccessible' true value and select another measurement value (eg n+1), the difference between these results corresponds to the error (systematic and random).
The systematic error corresponds to the bias, the deviation that can be known on some parameters of the measurement, eg instrument or sensor calibration.
To reduce systematic errors, known corrections must be applied based on a calibration certificate. Random errors are due to changing parameters during a series of measurements such as instrument resolution or environmental conditions.
They are reduced by increasing measurements and taking the average value because the sign of each error can be positive or negative and the average is close to zero.
Plotted on this Gaussian curve, the confidence level can be expressed as being given by one, two or three times the standard deviation of the corresponding distribution.
A corresponding coverage factor k equal to 1, 2 or 3 can be set, meaning there is a 68.3 per cent, 95.4 per cent and 99.7 per cent chance of finding results in the specified interval.
Uncertainty
Remember that the error is the difference between the result and the true value whereas the uncertainty characterises the dispersion around the measurement result values of the measurand.
For example, the result of a measurement after correction can be close to the unknown value of the measurand despite a large uncertainty
(see Fig. 2).
Uncertainty calculations must be considered in the context of the guide to the expression of the uncertainty in measurement (GUM: ENV13005) which defines requirements for analysing and computing the uncertainty of a measurement. The mathematical model of the measurement process needs to be defined first. A result is always a function of one or more input parameters. The model gives the relationship between these inputs and the final result.
With the model function, the uncertainty calculation is based on the determination of the first order terms of its Taylor series.
Sensitivity coefficient
Computing the partial derivative of each input gives the sensitivity coefficient of the corresponding input. Each sensitivity coefficient is combined with the standard uncertainty au' of each parameter, including SD as', in a quadratic sum to express the combined standard uncertainty auc'.
It is then possible to compute the expanded uncertainty aU'as it is the combined standard uncertainty auc' multiplied by the k factor (the coverage factor), usually equal to 2, that is a confidence level close to 95 per cent.
In other words, there is a 95 per cent chance of having the true value included in the expression of the interval associated with your result, for example, 5.023 ± 0.015 mg/l, k=2 (Fig. 3).
Taking the effort out of metrology
Following GUM requirements and calculating the expanded uncertainty for results is both time-consuming and complicated on a day-to-day basis.
A major breakthrough in the expression of measurement quality is obtained with the TIM550 Volumetric Karl Fischer Titrator from Radiometer Analytical (Fig. 4).
This system, dedicated to water content determination, automatically expresses the quality of measurements and corresponding accuracy by associating the expanded uncertainty with each result.
Minimising errors
Calibration data are embedded and automatically taken into account in systematic error calculation, increasing result accuracy. Repeating measurements is simple and a binary choice helps operators minimise random errors.
ENQUIRY No 46
Eric Trompier is a product specialist with Radiometer Analytical SA, Villeurbanne Cedex, France. www.radiometer-analytical.com
