Limits in Analytical Accuracy

You might recall from our previous column (1) how Horwitz throws down the gauntlet to analytical scientists, stating that a general equation can be formulated for the representation of analytical precision based upon analyte concentration (2). He states this as follows:

CV(%) = 2^{(1–0.5logC)}

where C is the mass fraction as concentration expressed in powers of 10 (for example, 0.1% analyte is equal to C = 10^–3).

A paper published by Hall and Selinger (3) points out an empirical formula relating the concentration (c) to the coefficient of variation (CV), also known as the precision (σ). They derive the origin of the "trumpet curve" using a binomial distribution explanation. Their final derived relationship becomes

They further simplify the Horwitz trumpet relationship in two forms as follows:

CV(%) = 0.006c^–0.5

and

σ = 0.006c^0.5

They then derive their own binomial model relationships using Horwitz's data with variable apparent sample size.

CV(%) = 0.02c^–0.15

and

σ = 0.02c^0.85

Both sets of relationships depict relative error as inversely proportional to analyte concentration.

In yet a more detailed incursion into this subject, Rocke and Lorenzato (4) describe two disparate conditions in analytical error: concentrations near zero and macro level concentrations, say greater than 0.5% for argument's sake. They propose that analytical error comprises two types, additive and multiplicative. So their derived model for this condition is

x = μe^η + ε

where x is the measured concentration; μ is the true analyte concentration; and η is a normally distributed analytical error with mean 0 and standard deviation σ_η. It should be noted that η represents the multiplicative or proportional error with concentration and ´ represents the additive error demonstrated at small concentrations.

Using this approach, the critical level at which the CV is a specific value can be found by solving for x using the following relationship:

(CVx)² = (σ_ηx)² + (σ_ε)²

where x is the measured analyte concentration as the practical quantitation level (PQL used by the U.S. Environmental Protection Agency [EPA]). This relationship is simplified to