Variability in Analytical Measurements

1.1 INTRODUCTION

All measurements are subject to measurement error. By error is meant the difference between the observed value and the true value of the quantity being measured. Since true values are invariably unknown, the exact magnitude of the error involved in an analytical result is also invariably unknown. It is possible, however, to estimate the likely magnitude of such errors by careful study of the properties of the analytical system. The term 'analytical system' refers to everything that impinges on a measurement: the method, the equipment, the reagents, the analyst, the laboratory environments, etc. It is fair comment that a measurement is of no value unless there is attached to it, either explicitly or implicitly, some estimate of the probable error involved. The Analytical Methods Committee of the Royal Society of Chemistry has taken a very clear position on this question in saying that 'analytical results must be accompanied by an explicit quantitative statement of uncertainty, if any definite meaning is to be attached to them or an informed interpretation made. If this requirement cannot be fulfilled, there are strong grounds for questioning whether analysis should be undertaken at all.' It gives a simple example which illustrates the necessity for measures of uncertainty: 'Suppose there is a requirement that a material must not contain more that 10 µg g-1 of a particular constituent. A manufacturer analyses a batch and obtains a result of 9 µg g-1. If the uncertainty ... in the results is 0.1 µg g-1(i.e., the true result falls within the range 8.9–9.1 µg g-1 with a high probability) then it can be accepted that the limit is not exceeded. If, in contrast, the uncertainty is 2 µg g-1 there can be no such assurance. The 'meaning' or information content of the measurement thus depends on the uncertainty associated with it."

In recent years the question of uncertainty in measurements has been of major concern to the measurement community in general, and has stimulated much activity among analytical chemists. This has been driven, in part, by the requirement by laboratory accreditation bodies for explicit statements of the quality of the analytical measurements being produced by accredited laboratories. The term 'uncertainty' has been given a specific technical meaning, viz., "a parameter associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand" and guidelines have been developed to help chemists estimate uncertainties. Uncertainty will be discussed as a technical concept in Chapter 8, but the remainder of the book may be read as a discussion of how to describe and analyze uncertainty in its usual sense of indicating a lack of certainty regarding the trueness of measured values.

Consultancy experience suggests that the importance of measurement error is often not fully appreciated. In pharmaceutical manufacturing, for instance, crucially important management decisions can be made on the basis of a single measurement or on the average of a small number of measurements, often poorly determined. Examples of such measurements would be: the purity of a raw material on the basis of which the conversion rate of a process is to be determined; the strength of an intermediate product, which will determine the dilution of the next stage of production; the potency of a final product, which will determine the amount of product to be filled into vials intended to contain a given strength of product. In the latter case lack of confidence in the quality of the measurement may lead to overfill of the vials in order to ensure conformance to statutory or customer requirements. The problem in all of these cases is not one of product variability, though this may be present also, but rather the uncertainty in a decision criterion (i.e., a test result) due to measurement variability. Since the costs of improving such measurements will generally be small compared to the costs of poor production decisions, it is clear that the economic consequences of measurement error are, as yet, not fully appreciated.

In industrial experimentation measurement error can be vitally important also. Studies are often carried out to investigate proposed process changes which it is hoped will result in small but economically important yield improvements. In such situations the ability of the experimenter to detect small yield improvements will be influenced in an important way by the number of measurements made. If the measurement error is relatively large and the number of measurements small, the effect of the process change may be missed. The determination of appropriate sample sizes for experiments is an important topic which will be discussed in Chapter 4.

This chapter begins with an example illustrating the effects of random measurement error on the results produced by a stable analytical system. The terms bias and precision are introduced to describe the nature of measurement errors; this is done in the context of an inter-laboratory study. The Normal curve is discussed as the most important statistical model for describing and quantifying analytical error. Various measures of the magnitude of analytical error are then considered: these include the standard deviation, the relative standard deviation or coefficient of variation, repeatability and reproducibility. Finally, the effect on the precision of an analytical result of averaging several measurements is discussed.

1.2 AN EXAMPLE OF MEASUREMENT VARIABILITY

Figure 1.1 is a histogram based on 119 determinations of the potency of a quality control material used in monitoring the stability of an analytical system; this system is used for measuring the potency of batches of a pharmaceutical product. The measurements were made by high performance liquid chromatography (HPLC). Here the heights of the bars represent the frequencies of values in each of the potency intervals on the horizontal axis. Very often the areas of the bars (height and area are equivalent when the bases are the same) are drawn to represent the relative frequencies, i.e., the fractions of the total number of data points that fall into each interval. In such a case the total area is one – this will be discussed again later.

The results are distributed in a roughly symmetrical bell-shaped form, with most values clustered around the centre and few values very far from the centre in either direction. This shape is often seen in laboratory data and is typical of the type of variation that results from the combined effects of a myriad of small influences. It can be shown mathematically that the cumulative effect on a system of very many small chance perturbations...