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ISO approach to uncertainty evaluation

The internationally agreed approach to evaluating uncertainty is described in the ISO Guide to the expression of uncertainty in measurement (the GUM) [1]. The GUM implements a number of recommendations that were agreed during its development.

(1) The uncertainty in a measurement result generally arises from several components.

For example, the uncertainty in the volume of liquid delivered by a glass pipette will have contributions associated with:

• the calibration of the pipette;
• the random variation in filling the pipette with liquid to the calibration line and dispensing the liquid (precision of the volume delivered);
• the difference between the calibration temperature of the pipette and the temperature at which it is being used.

(2) The uncertainty components can be grouped into two categories according to how a numerical estimate of the magnitude of the uncertainty was obtained.

Type A uncertainty estimates are obtained by statistical analysis of data.

Type B uncertainty estimates are obtained by other means (e.g. from certificates, by modelling, etc).

In the case of the volume of liquid delivered by a pipette, the uncertainty associated with the pipette calibration, obtained from a calibration certificate, is a Type B uncertainty estimate. Statistical analysis of the results from several ‘fill and dispense’ experiments to evaluate the precision of the volume delivered would give a Type A estimate.

(3) Type A uncertainty estimates are characterised by estimated variances (s_i²) or standard deviations (s_i).

(4) Type B uncertainty estimates are characterised by quantities u_i² which are considered approximations of the corresponding variance (s_i²).

The quantity u_i can therefore be treated in the same way as a standard deviation s_i. In effect, all uncertainty components should be expressed in the form of a standard deviation before they are combined. Uncertainties of this type are described as standard uncertainties.

(5) Uncertainty components are combined by applying mathematical rules for the combination of variances.

The resulting uncertainty will be expressed in the form of a standard deviation and is known as a combined standard uncertainty.

(6) To obtain the required level of confidence for the quoted uncertainty, it may be necessary to multiply the combined standard uncertainty by a coverage factor (k).

The main steps in the ISO approach to evaluating uncertainty are as follows:

• Write an equation that completely describes the measurement system - include all parameters that could influence the measurement result
• Evaluate the uncertainties associated with all the parameters in the equation (Type A or Type B estimates)
• Express all the uncertainty components as standard deviations
• Combine using mathematical rules for the combination of variances
• Apply a suitable coverage factor (if required).

This approach is sometimes described as the bottom-up approach to uncertainty estimation. The measurement process is broken down into its component steps, the influence of each step on the measurement result is evaluated, and the resulting uncertainty estimates are combined to give the uncertainty in the measurement result.

The main problem with attempting to apply this approach to results obtained from chemical or biochemical methods is that it is often difficult (if not impossible) to evaluate separately all of the possible sources of uncertainty. The cause and effect diagram for the method for the determination of hydroquinone in cosmetics identifies a number of possible sources of uncertainty. It would be very time consuming to attempt to evaluate these individually. An alternative approach to evaluating uncertainties associated with chemical and biochemical test methods has therefore been developed which makes use of method performance data (e.g. data from validation studies). This approach is often referred to as the top-down approach. This approach should provide an acceptable estimate of measurement uncertainty for results produced by testing laboratories.

[1] ISO/IEC Guide 98-3:2008 Uncertainty of measurement - Part 3: Guide to the expression of uncertainty in measurement (GUM:1995)