Determinative errors

Determinate errors may be divided into four categories sampling errors, method errors, measurement errors, and personal errors. 

A determinate error whose value is the same for all samples. 

Identifying Determinate Errors Determinate errors can be difficult to detect. Without knowing the true value for an analysis, the usual situation in any analysis with meaning, there is no accepted value with which the experimental result can be compared. Nevertheless, a few strategies can be used to discover the presence of a determinate error. 

Effect of Constant Positive Determinate Error on Analysis of Sample Containing 50% Analyte (%w/w)... 

A proportional determinate error, in which the error s magnitude depends on the amount of sample, is more difficult to detect since the result of an analysis is independent of the amount of sample. Table 4.6 outlines an example showing the effect of a positive proportional error of 1.0% on the analysis of a sample that is 50.0% w/w in analyte. In terms of equations 4.4 and 4.5, the reagent blank, Sreag, is an example of a constant determinate error, and the sensitivity, k, may be affected by proportional errors. 

Effect of a constant determinate error on the reported concentration of analyte. 

Analytical chemists make a distinction between error and uncertainty Error is the difference between a single measurement or result and its true value. In other words, error is a measure of bias. As discussed earlier, error can be divided into determinate and indeterminate sources. Although we can correct for determinate error, the indeterminate portion of the error remains. Statistical significance testing, which is discussed later in this chapter, provides a way to determine whether a bias resulting from determinate error might be present. 

Uncertainty expresses the range of possible values that a measurement or result might reasonably be expected to have. Note that this definition of uncertainty is not the same as that for precision. The precision of an analysis, whether reported as a range or a standard deviation, is calculated from experimental data and provides an estimation of indeterminate error affecting measurements. Uncertainty accounts for all errors, both determinate and indeterminate, that might affect our result. Although we always try to correct determinate errors, the correction itself is subject to random effects or indeterminate errors. 

A second example is also informative. When samples are obtained from a normally distributed population, their values must be random. If results for several samples show a regular pattern or trend, then the samples cannot be normally distributed. This may reflect the fact that the underlying population is not normally distributed, or it may indicate the presence of a time-dependent determinate error. For example, if we randomly select 20 pennies and find that the mass of each penny exceeds that of the preceding penny, we might suspect that the balance on which the pennies are being weighed is drifting out of calibration. 

In the previous section we noted that the result of an analysis is best expressed as a confidence interval. For example, a 95% confidence interval for the mean of five results gives the range in which we expect to find the mean for 95% of all samples of equal size, drawn from the same population. Alternatively, and in the absence of determinate errors, the 95% confidence interval indicates the range of values in which we expect to find the population s true mean. 

If evidence for a determinate error is found, as in Example 4.16, its source should be identified and corrected before analyzing additional samples. Failing to reject the null hypothesis, however, does not imply that the method is accurate, but only indicates that there is insufficient evidence to prove the method inaccurate at the stated confidence level. 

The data we collect are characterized by their central tendency (where the values are clustered), and their spread (the variation of individual values around the central value). Central tendency is reported by stating the mean or median. The range, standard deviation, or variance may be used to report the data s spread. Data also are characterized by their errors, which include determinate errors... 

Signals are measured using equipment or instruments that must be properly calibrated if Sjneas is to be free of determinate errors. Calibration is accomplished against a standard, adjusting S eas until it agrees with the standard s known signal. Several common examples of calibration are discussed here. 

Example showing how an improper use of a single-point standardization can lead to a determinate error in the reported concentration of analyte. 

The method of standard additions can be used to check the validity of an external standardization when matrix matching is not feasible. To do this, a normal calibration curve of Sjtand versus Cs is constructed, and the value of k is determined from its slope. A standard additions calibration curve is then constructed using equation 5.6, plotting the data as shown in Figure 5.7(b). The slope of this standard additions calibration curve gives an independent determination of k. If the two values of k are identical, then any difference between the sample s matrix and that of the external standards can be ignored. When the values of k are different, a proportional determinate error is introduced if the normal calibration curve is used. 

In a single-point standardization, we assume that the reagent blank (the first row in Table 5.1) corrects for all constant sources of determinate error. If this is not the case, then the value of k determined by a singlepoint standardization will have a determinate error. 

Effect of a Constant Determinate Error on the Value of k Calculated Using a Single-Point Standardization... 

Table 5.2 demonstrates how an uncorrected constant error affects our determination of k. The first three columns show the concentration of analyte, the true measured signal (no constant error) and the true value of k for five standards. As expected, the value of k is the same for each standard. In the fourth column a constant determinate error of +0.50 has been added to the measured signals. The corresponding values of k are shown in the last column. Note that a different value of k is obtained for each standard and that all values are greater than the true value. As we noted in Section 5B.2, this is a significant limitation to any single-point standardization. 

That all four methods give a different result for the concentration of analyte underscores the importance of choosing a proper blank but does not tell us which of the methods is correct. In fact, the variation within each method for the reported concentration of analyte indicates that none of these four methods has adequately corrected for the blank. Since the three samples were drawn from the same source, they must have the same true concentration of analyte. Since all four methods predict concentrations of analyte that are dependent on the size of the sample, we can conclude that none of these blank corrections has accounted for an underlying constant source of determinate error. 

To obtain accurate results we must eliminate determinate errors affecting the measured signal, S ieas the method s sensitivity, k, and any signal due to the reagents, Sjeag-... 

A reagent blank corrects the measured signal for signals due to reagents other than the sample that are used in an analysis. The most common reagent blank is prepared by omitting the sample. When a simple reagent blank does not compensate for all constant sources of determinate error, other types of blanks, such as the total Youden blank, can be used. 

Is the failure to correct for buoyancy a constant or proportional source of determinate error ... 

Selecting a sample introduces a source of determinate error that cannot be corrected during the analysis. If a sample does not accurately represent the population from which it is drawn, then an analysis that is otherwise carefully conducted will yield inaccurate results. Sampling errors are introduced whenever we extrapolate from a sample to its target population. To minimize sampling errors we must collect the right sample. 

Statement that a periodic signal must be sampled at least twice each period to avoid a determinate error in measuring its frequency. 

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