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Determinate Errors: These Are Systematic

Two main categories of errors can influence the accuracy or precision of a measured quantity. Determinate, or systematic, errors are nonrandom and arise when something is intrinsically wrong in the measurement.

Determinate errors can be determined, as the name suggests, and can presumably be either avoided or corrected. They may be constant, as in the case of an uncalibrated pipet that is used for all volume deliveries. Or they may be variable but of such a kind that they can be accounted for and corrected, such as a buret whose volume readings are in error by different amounts at different volumes.

The error may be proportional to sample size or may vary in a more complex manner. More often than not, the variation is unidirectional, as in the case of solubility loss of a precipitate owing to its solubility (negative error). It can, however, be random in sign, that is, a positive or negative error. One such example is the change in solution volume and concentration that occurs with changes in temperature. This can be corrected for by measuring the temperature of the solution. Such measurable determinate errors are classified as systematic errors.

Some common determinate errors are:

1. Instrumental errors. These involve faulty equipment, for example uncalibrated glassware.
2. Operative errors. These involve personal errors and can be reduced through the analyst’s experience and care in the physical manipulations involved. Operative errors can be minimized by using a checklist of operations. Operations where these errors may occur include transfer of solutions, effervescence and “bumping” during sample dissolution, incomplete drying of samples, and so on. These are difficult to correct for. Other personal errors include mathematical errors in calculations and prejudice in estimating measurements.
3. Errors of the method. These are the most serious errors of an analysis. Most of the above errors can be minimized or corrected for, but errors that are inherent in the method cannot be altered unless the conditions of the determination are modified. Certain sources of methodical errors include coprecipitation of impurities, limited solubility of a precipitate, side reactions, incomplete reactions, and impurities present in reagents. Occasionally, correction can be fairly straightforward, for instance, by running a reagent blank. A blank determination is an analysis performed on the added reagents only. It is standard practice to run such blanks and to subtract their results from those obtained for the sample. However, a good blank analysis by itself cannot guarantee accurate measurements. If the method, for instance, responds to an analyte present in the sample other than the intended analyte, the method must be changed. Therefore, when errors become intolerable, another approach to the analysis must be adopted. Sometimes, however, we are compelled to accept a given method in the absence of a better one.

Determinate errors may be additive or multiplicative, depending on the nature of the error or how it enters into the calculation. To detect systematic errors in an analysis, it is common practice to add a known amount of standard to a sample (a “spike”) and assess how well it is recovered. Note, however, that a good spike recovery is unable, on its own, to correct for a response coming from an unintended analyte (that is, an interference). Analyzing reference samples helps protect against method errors or instrumental errors.

Indeterminate Errors: They Are Random

This second category of errors comprises the indeterminate errors, frequently referred to as accidental or random errors, which reflect the experimental uncertainty present in every measurement. Such errors show up as small differences between successive measurements taken by the same analyst under essentially identical conditions, and they are impossible to predict or estimate. These accidental errors will exhibit a random distribution; consequently, mathematical laws of probability can be used to reach some conclusion about the most probable result of a series of measurements.

A full treatment of mathematical probability lies beyond this text’s scope, but we can state that indeterminate errors ought to follow a normal distribution, or Gaussian curve. The symbol σ stands for the standard deviation of an infinite population of measurements, and this measure of precision defines the spread of the normal population distribution. It is clear that there should be few very large errors and that there should be an equal number of positive and negative errors.

Indeterminate errors actually arise from the limited ability of the analyst to control or make corrections for external conditions, or from the inability to recognize the appearance of factors that will lead to errors. Some random errors arise from the intrinsic nature of things. For example, consider that a sample of the radionuclide ¹²⁹I is taken. The isotope is long lived, and within a short time there will be no perceptible change in its number. But if a sufficient amount is taken, then based on the half-life, you can expect a decay to occur every 60 s. In reality, this may not occur every 60 s, but can fluctuate, with an average of 60 s. Sometimes, by changing conditions, some unknown error will disappear. Of course, it will be impossible to eliminate all possible random errors in an experiment, and the analyst must be content to minimize them to a tolerable or insignificant level.