Accuracy is a measure of how close a measure of central tendency is to the true, or expected value, µ.†Accuracy is usually expressed as either an absolute error
E= –
X–µ 4.2
or a percent relative error, Er.
4.3
E X
r = −µ ×
µ 100
*Many scientific calculators include two keys for calculating the standard deviation, only one of which corresponds to equation 4.3. Your calculator’s manual will help you determine the appropriate key to use.
†The standard convention for representing experimental parameters is to use a Roman letter for a value calculated from experimental data, and a Greek letter for the corresponding true value. For example, the experimentally determined mean is X, and its underlying true value is µ. Likewise, the standard deviation by experiment is given the symbol s,– and its underlying true value is identified as σ.
variance
The square of the standard deviation (s2).
Although the mean is used as the measure of central tendency in equations 4.2 and 4.3, the median could also be used.
Errors affecting the accuracy of an analysis are called determinate and are char- acterized by a systematic deviation from the true value; that is, all the individual measurements are either too large or too small. A positive determinate errorresults in a central value that is larger than the true value, and a negative determinate error leads to a central value that is smaller than the true value. Both positive and nega- tive determinate errors may affect the result of an analysis, with their cumulative ef- fect leading to a net positive or negative determinate error. It is possible, although not likely, that positive and negative determinate errors may be equal, resulting in a central value with no net determinate error.
Determinate errors may be divided into four categories: sampling errors, method errors, measurement errors, and personal errors.
Sampling Errors We introduce determinate sampling errorswhen our sampling strategy fails to provide a representative sample. This is especially important when sampling heterogeneous materials. For example, determining the environmental quality of a lake by sampling a single location near a point source of pollution, such as an outlet for industrial effluent, gives misleading results. In determining the mass of a U.S. penny, the strategy for selecting pennies must ensure that pennies from other countries are not inadvertently included in the sample. Determinate errors as- sociated with selecting a sample can be minimized with a proper sampling strategy, a topic that is considered in more detail in Chapter 7.
Method Errors Determinate method errors are introduced when assumptions about the relationship between the signal and the analyte are invalid. In terms of the general relationships between the measured signal and the amount of analyte
Smeas=knA +Sreag (total analysis method) 4.4
Smeas=kCA +Sreag (concentration method) 4.5
method errors exist when the sensitivity, k,and the signal due to the reagent blank, Sreag, are incorrectly determined. For example, methods in which Smeasis the mass of a precipitate containing the analyte (gravimetric method) assume that the sensitiv- ity is defined by a pure precipitate of known stoichiometry. When this assumption fails, a determinate error will exist. Method errors involving sensitivity are mini- mized by standardizing the method, whereas method errors due to interferents present in reagents are minimized by using a proper reagent blank. Both are dis- cussed in more detail in Chapter 5. Method errors due to interferents in the sample cannot be minimized by a reagent blank. Instead, such interferents must be sepa- rated from the analyte or their concentrations determined independently.
Measurement Errors Analytical instruments and equipment, such as glassware and balances, are usually supplied by the manufacturer with a statement of the item’s maximum measurement error,or tolerance.For example, a 25-mL volumetric flask might have a maximum error of ±0.03 mL, meaning that the actual volume contained by the flask lies within the range of 24.97–25.03 mL. Although expressed as a range, the error is determinate; thus, the flask’s true volume is a fixed value within the stated range. A summary of typical measurement errors for a variety of analytical equipment is given in Tables 4.2–4.4.
58
Modern Analytical Chemistrysampling error
An error introduced during the process of collecting a sample for analysis.
heterogeneous
Not uniform in composition.
method error
An error due to limitations in the analytical method used to analyze a sample.
determinate error
Any systematic error that causes a measurement or result to always be too high or too small; can be traced to an identifiable source.
measurement error
An error due to limitations in the equipment and instruments used to make measurements.
tolerance
The maximum determinate measurement error for equipment or instrument as reported by the manufacturer.
Chapter 4 Evaluating Analytical Data
59
Table 4.2
Measurement Errors for Selected Glasswarea Measurement Errors forVolume Class A Glassware Class B Glassware
Glassware (mL) (±mL) (±mL)
Transfer Pipets 1 0.006 0.012
2 0.006 0.012
5 0.01 0.02
10 0.02 0.04
20 0.03 0.06
25 0.03 0.06
50 0.05 0.10
Volumetric Flasks 5 0.02 0.04
10 0.02 0.04
25 0.03 0.06
50 0.05 0.10
100 0.08 0.16
250 0.12 0.24
500 0.20 0.40
1000 0.30 0.60
2000 0.50 1.0
Burets 10 0.02 0.04
25 0.03 0.06
50 0.05 0.10
aSpecifications for class A and class B glassware are taken from American Society for Testing and Materials E288, E542 and E694 standards.
Table 4.4
Measurement Errors for Selected Digital PipetsVolume Measurement Error
Pipet Range (mL or µL)a (±%)
10–100µLb 10 1.0
50 0.6
100 0.6
200–1000µLc 200 1.5
1000 0.8
1–10 mLd 1 0.6
5 0.4
10 0.3
aUnits for volume same as for pipet range.
bData for Eppendorf Digital Pipet 4710.
cData for Oxford Benchmate.
dData for Eppendorf Maxipetter 4720 with Maxitip P.
Table 4.3
Measurement Errors for Selected BalancesCapacity Measurement
Balance (g) Error
Precisa 160M 160 ±1 mg
A & D ER 120M 120 ±0.1 mg
Metler H54 160 ±0.01 mg
Volumetric glassware is categorized by class. Class A glassware is manufactured to comply with tolerances specified by agencies such as the National Institute of Standards and Technology. Tolerance levels for class A glassware are small enough that such glassware normally can be used without calibration. The tolerance levels for class B glassware are usually twice those for class A glassware. Other types of vol- umetric glassware, such as beakers and graduated cylinders, are unsuitable for accu- rately measuring volumes.
Determinate measurement errors can be minimized by calibration. A pipet can be calibrated, for example, by determining the mass of water that it delivers and using the density of water to calculate the actual volume delivered by the pipet. Al- though glassware and instrumentation can be calibrated, it is never safe to assume that the calibration will remain unchanged during an analysis. Many instruments, in particular, drift out of calibration over time. This complication can be minimized by frequent recalibration.
Personal Errors Finally, analytical work is always subject to a variety of personal errors, which can include the ability to see a change in the color of an indicator used to signal the end point of a titration; biases, such as consistently overestimat- ing or underestimating the value on an instrument’s readout scale; failing to cali- brate glassware and instrumentation; and misinterpreting procedural directions.
Personal errors can be minimized with proper care.
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.
Some determinate errors can be detected experimentally by analyzing several samples of different size. The magnitude of a constant determinate error is the same for all samples and, therefore, is more significant when analyzing smaller sam- ples. The presence of a constant determinate error can be detected by running sev- eral analyses using different amounts of sample, and looking for a systematic change in the property being measured. For example, consider a quantitative analysis in which we separate the analyte from its matrix and determine the analyte’s mass.
Let’s assume that the sample is 50.0% w/w analyte; thus, if we analyze a 0.100-g sample, the analyte’s true mass is 0.050 g. The first two columns of Table 4.5 give the true mass of analyte for several additional samples. If the analysis has a positive constant determinate error of 0.010 g, then the experimentally determined mass for
60
Modern Analytical ChemistryTable 4.5
Effect of Constant Positive Determinate Error on Analysis of Sample Containing 50% Analyte (%w/w)Mass Sample True Mass of Analyte Constant Error Mass of Analyte Determined Percent Analyte Reported
(g) (g) (g) (g) (%w/w)
0.100 0.050 0.010 0.060 60.0
0.200 0.100 0.010 0.110 55.0
0.400 0.200 0.010 0.210 52.5
0.800 0.400 0.010 0.410 51.2
1.000 0.500 0.010 0.510 51.0
constant determinate error
A determinate error whose value is the same for all samples.
personal error
An error due to biases introduced by the analyst.
any sample will always be 0.010 g, larger than its true mass (column four of Table 4.5). The analyte’s reported weight percent, which is shown in the last column of Table 4.5, becomes larger when we analyze smaller samples. A graph of % w/w ana- lyte versus amount of sample shows a distinct upward trend for small amounts of sample (Figure 4.1). A smaller concentration of analyte is obtained when analyzing smaller samples in the presence of a constant negative determinate error.
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 in- dependent of the amount of sample. Table 4.6 outlines an example showing the ef- fect 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.
Potential determinate errors also can be identified by analyzing a standard sam- ple containing a known amount of analyte in a matrix similar to that of the samples being analyzed. Standard samples are available from a variety of sources, such as the National Institute of Standards and Technology (where they are called standard reference materials) or the American Society for Testing and Materials. For exam- ple, Figure 4.2 shows an analysis sheet for a typical reference material. Alternatively, the sample can be analyzed by an independent
method known to give accurate results, and the re- sults of the two methods can be compared. Once identified, the source of a determinate error can be corrected. The best prevention against errors affect- ing accuracy, however, is a well-designed procedure that identifies likely sources of determinate errors, coupled with careful laboratory work.
The data in Table 4.1 were obtained using a calibrated balance, certified by the manufacturer to have a tolerance of less than ±0.002 g. Suppose the Treasury Department reports that the mass of a 1998 U.S. penny is approximately 2.5 g. Since the mass of every penny in Table 4.1 exceeds the re- ported mass by an amount significantly greater than the balance’s tolerance, we can safely conclude that the error in this analysis is not due to equip- ment error. The actual source of the error is re- vealed later in this chapter.
Chapter 4 Evaluating Analytical Data
61
Amount of sample
% w/w analyte
Negative constant error Positive constant error
True % w/w analyte
Figure 4.1
Effect of a constant determinate error on the reported concentration of analyte.
Table 4.6
Effect of Proportional Positive Determinate Error on Analysis of Sample Containing 50% Analyte (%w/w)Mass Sample True Mass of Analyte Proportional Error Mass of Analyte Determined Percent Analyte Reported
(g) (g) (%) (g) (%w/w)
0.200 0.100 1.00 0.101 50.5
0.400 0.200 1.00 0.202 50.5
0.600 0.300 1.00 0.303 50.5
0.800 0.400 1.00 0.404 50.5
1.000 0.500 1.00 0.505 50.5
proportional determinate error A determinate error whose value depends on the amount of sample analyzed.
standard reference material A material available from the National Institute of Standards and Technology certified to contain known
concentrations of analytes.