Modern Analytical Cheymistry - Chapter 4 doc

CChhaapptteerr 4 53 Evaluating Analytical Data Aproblem dictates the requirements we place on our measurements and results. Regulatory agencies, for example, place stringent requirements on the reliability of measurements and results reported to them. This is the rationale for creating a protocol for regulatory problems. Screening the products of an organic synthesis, on the other hand, places fewer demands on the reliability of measurements, allowing chemists to customize their procedures. When designing and evaluating an analytical method, we usually make three separate considerations of experimental error. 1 First, before beginning an analysis, errors associated with each measurement are evaluated to ensure that their cumulative effect will not limit the utility of the analysis. Errors known or believed to affect the result can then be minimized. Second, during the analysis the measurement process is monitored, ensuring that it remains under control. Finally, at the end of the analysis the quality of the measurements and the result are evaluated and compared with the original design criteria. This chapter is an introduction to the sources and evaluation of errors in analytical measurements, the effect of measurement error on the result of an analysis, and the statistical analysis of data. 1400-CH04 9/8/99 3:53 PM Page 53 54 Modern Analytical Chemistry mean The average value of a set of data ( – X). 4 A Characterizing Measurements and Results Let’s begin by choosing a simple quantitative problem requiring a single measurement. The question to be answered is—What is the mass of a penny? If you think about how we might answer this question experimentally, you will realize that this problem is too broad. Are we interested in the mass of United State pennies or Cana- dian pennies, or is the difference in country of importance? Since the composition of a penny probably differs from country to country, let’s limit our problem to pennies minted in the United States. There are other considerations. Pennies are minted at several locations in the United States (this is the meaning of the letter, or absence of a letter, below the date stamped on the lower right corner of the face of the coin). Since there is no reason to expect a difference between where the penny was minted, we will choose to ignore this consideration. Is there a reason to expect a difference between a newly minted penny not yet in circulation, and a penny that has been in circulation? The answer to this is not obvious. Let’s simplify the problem by narrow- ing the question to—What is the mass of an average United States penny in circulation? This is a problem that we might expect to be able to answer experimentally. A good way to begin the analysis is to acquire some preliminary data. Table 4.1 shows experimentally measured masses for seven pennies from my change jar at home. Looking at these data, it is immediately apparent that our question has no simple answer. That is, we cannot use the mass of a single penny to draw a specific conclusion about the mass of any other penny (although we might conclude that all pennies weigh at least 3 g). We can, however, characterize these data by providing a measure of the spread of the individual measurements around a central value. 4 A.1 Measures of Central Tendency One way to characterize the data in Table 4.1 is to assume that the masses of individual pennies are scattered around a central value that provides the best estimate of a penny’s true mass. Two common ways to report this estimate of central tendency are the mean and the median. Mean The mean, – X, is the numerical average obtained by dividing the sum of the individual measurements by the number of measurements where X i is the i th measurement, and n is the number of independent measurements. X X n i i n = = ∑ 1 Table 4 .1 Masses of Seven United States Pennies in Circulation Penny Mass (g) 1 3.080 2 3.094 3 3.107 4 3.056 5 3.112 6 3.174 7 3.198 1400-CH04 9/8/99 3:53 PM Page 54 Chapter 4 Evaluating Analytical Data 55 EXAMPLE 4 .1 What is the mean for the data in Table 4.1? SOLUTION To calculate the mean, we add the results for all measurements 3.080 + 3.094 + 3.107 + 3.056 + 3.112 + 3.174 + 3.198 = 21.821 and divide by the number of measurements The mean is the most common estimator of central tendency. It is not considered a robust estimator, however, because extreme measurements, those much larger or smaller than the remainder of the data, strongly influence the mean’s value. 2 For example, mistakenly recording the mass of the fourth penny as 31.07 g instead of 3.107 g, changes the mean from 3.117 g to 7.112 g! Median The median, X med , is the middle value when data are ordered from the smallest to the largest value. When the data include an odd number of measurements, the median is the middle value. For an even number of measurements, the median is the average of the n/2 and the (n/2) + 1 measurements, where n is the number of measurements. EXAMPLE 4 .2 What is the median for the data in Table 4.1? SOLUTION To determine the median, we order the data from the smallest to the largest value 3.056 3.080 3.094 3.107 3.112 3.174 3.198 Since there is a total of seven measurements, the median is the fourth value in the ordered data set; thus, the median is 3.107. As shown by Examples 4.1 and 4.2, the mean and median provide similar esti- mates of central tendency when all data are similar in magnitude. The median, however, provides a more robust estimate of central tendency since it is less sensi- tive to measurements with extreme values. For example, introducing the transcrip- tion error discussed earlier for the mean only changes the median’s value from 3.107 g to 3.112 g. 4 A.2 Measures of Spread If the mean or median provides an estimate of a penny’s true mass, then the spread of the individual measurements must provide an estimate of the variability in the masses of individual pennies. Although spread is often defined relative to a specific measure of central tendency, its magnitude is independent of the central value. Changing all X == 21 821 7 3 117 . . g median That value for a set of ordered data, for which half of the data is larger in value and half is smaller in value ( – X med ). 1400-CH04 9/8/99 3:53 PM Page 55 56 Modern Analytical Chemistry measurements in the same direction, by adding or subtracting a constant value, changes the mean or median, but will not change the magnitude of the spread. Three common measures of spread are range, standard deviation, and variance. Range The range, w, is the difference between the largest and smallest values in the data set. Range = w = X largest – X smallest The range provides information about the total variability in the data set, but does not provide any information about the distribution of individual measurements. The range for the data in Table 4.1 is the difference between 3.198 g and 3.056 g; thus w = 3.198 g – 3.056 g = 0.142 g Standard Deviation The absolute standard deviation, s, describes the spread of individual measurements about the mean and is given as 4.1 where X i is one of n individual measurements, and – X is the mean. Frequently, the relative standard deviation, s r , is reported. The percent relative standard deviation is obtained by multiplying s r by 100%. EXAMPLE 4 . 3 What are the standard deviation, the relative standard deviation, and the percent relative standard deviation for the data in Table 4.1? SOLUTION To calculate the standard deviation, we obtain the difference between the mean value (3.117; see Example 4.1) and each measurement, square the resulting differences, and add them to determine the sum of the squares (the numerator of equation 4.1) (3.080 – 3.117) 2 = (–0.037) 2 = 0.00137 (3.094 – 3.117) 2 = (–0.023) 2 = 0.00053 (3.107 – 3.117) 2 = (–0.010) 2 = 0.00010 (3.056 – 3.117) 2 = (–0.061) 2 = 0.00372 (3.112 – 3.117) 2 = (–0.005) 2 = 0.00003 (3.174 – 3.117) 2 = (+0.057) 2 = 0.00325 (3.198 – 3.117) 2 = (+0.081) 2 = 0.00656 0.01556 The standard deviation is calculated by dividing the sum of the squares by n – 1, where n is the number of measurements, and taking the square root. s . = − = 0 01556 71 0 051. s s X r = s XX n i i n = − = ∑ (–) 1 2 1 standard deviation A statistical measure of the “average” deviation of data from the data’s mean value (s). range The numerical difference between the largest and smallest values in a data set (w). 1400-CH04 9/8/99 3:53 PM Page 56 Chapter 4 Evaluating Analytical Data 57 s r == 0 051 3 117 0 016 . . . The relative standard deviation and percent relative standard deviation are s r (%) = 0.016 × 100% = 1.6% It is much easier to determine the standard deviation using a scientific calculator with built-in statistical functions.* Variance Another common measure of spread is the square of the standard deviation, or the variance. The standard deviation, rather than the variance, is usually reported because the units for standard deviation are the same as that for the mean value. EXAMPLE 4 . 4 What is the variance for the data in Table 4.1? SOLUTION The variance is just the square of the absolute standard deviation. Using the standard deviation found in Example 4.3 gives the variance as Variance = s 2 = (0.051) 2 = 0.0026 4 B Characterizing Experimental Errors Realizing that our data for the mass of a penny can be characterized by a measure of central tendency and a measure of spread suggests two questions. First, does our measure of central tendency agree with the true, or expected value? Second, why are our data scattered around the central value? Errors associated with central tendency reflect the accuracy of the analysis, but the precision of the analysis is determined by those errors associated with the spread. 4 B.1 Accuracy 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, E r . 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 (s 2 ). 1400-CH04 9/8/99 3:53 PM Page 57 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 characterized by a systematic deviation from the true value; that is, all the individual measurements are either too large or too small. A positive determinate error results 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 negative determinate errors may affect the result of an analysis, with their cumulative effect 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 errors when 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 associated 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 S meas = kn A+ S reag (total analysis method) 4.4 S meas = kC A+ S reag (concentration method) 4.5 method errors exist when the sensitivity, k, and the signal due to the reagent blank, S reag , are incorrectly determined. For example, methods in which S meas is the mass of a precipitate containing the analyte (gravimetric method) assume that the sensitivity is defined by a pure precipitate of known stoichiometry. When this assumption fails, a determinate error will exist. Method errors involving sensitivity are minimized by standardizing the method, whereas method errors due to interferents present in reagents are minimized by using a proper reagent blank. Both are discussed 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 Chemistry sampling 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. 1400-CH04 9/8/99 3:53 PM Page 58 Chapter 4 Evaluating Analytical Data 59 Table 4 .2 Measurement Errors for Selected Glassware a Measurement Errors for Volume 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 a Specifications 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 Pipets Volume Measurement Error Pipet Range (mL or µL) a (±%) 10–100 µL b 10 1.0 50 0.6 100 0.6 200–1000 µL c 200 1.5 1000 0.8 1–10 mL d 1 0.6 5 0.4 10 0.3 a Units for volume same as for pipet range. b Data for Eppendorf Digital Pipet 4710. c Data for Oxford Benchmate. d Data for Eppendorf Maxipetter 4720 with Maxitip P. Table 4 . 3 Measurement Errors for Selected Balances Capacity Measurement Balance (g) Error Precisa 160M 160 ±1 mg A & D ER 120M 120 ±0.1 mg Metler H54 160 ±0.01 mg 1400-CH04 9/8/99 3:53 PM Page 59 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 volumetric 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 samples. The presence of a constant determinate error can be detected by running several 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 Chemistry Table 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. 1400-CH04 9/8/99 3:53 PM Page 60 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 analyte 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 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, S reag , 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 sample 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 example, 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 results of the two methods can be compared. Once identified, the source of a determinate error can be corrected. The best prevention against errors affecting 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 reported 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 equipment 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. 1400-CH04 9/8/99 3:53 PM Page 61 4 B.2 Precision Precision is a measure of the spread of data about a central value and may be expressed as the range, the standard deviation, or the variance. Precision is commonly divided into two categories: repeatability and reproducibility. Repeatability is the precision obtained when all measurements are made by the same analyst during a single period of laboratory work, using the same solutions and equipment. Repro- ducibility, on the other hand, is the precision obtained under any other set of con- ditions, including that between analysts, or between laboratory sessions for a single analyst. Since reproducibility includes additional sources of variability, the reproducibility of an analysis can be no better than its repeatability. Errors affecting the distribution of measurements around a central value are called indeterminate and are characterized by a random variation in both magnitude and direction. Indeterminate errors need not affect the accuracy of an analysis. Since indeterminate errors are randomly scattered around a central value, positive and negative errors tend to cancel, provided that enough measurements are made. In such situations the mean or median is largely unaffected by the precision of the analysis. Sources of Indeterminate Error Indeterminate errors can be traced to several sources, including the collection of samples, the manipulation of samples during the analysis, and the making of measurements. When collecting a sample, for instance, only a small portion of the available material is taken, increasing the likelihood that small-scale inhomogeneities in the sample will affect the repeatability of the analysis. Individual pennies, for example, are expected to show variation from several sources, including the manufacturing process, and the loss of small amounts of metal or the addition of dirt during circulation. These variations are sources of indeterminate error associated with the sampling process. 62 Modern Analytical Chemistry SRM Type Unit of issue 2694a Simulated rainwater Set of 4: 2 of 50 mL at each of 2 levels Constituent element parameter 2694a-I 2694a-II pH, 25°C 4.30 3.60 Electrolytic Conductivity (S/cm, 25°C) 25.4 129.3 Acidity, meq/L 0.0544 0.283 Fluoride, mg/L 0.057 0.108 Chloride, mg/L (0.23)* (0.94)* Nitrate, mg/L (0.53)* 7.19 Sulfate, mg/L 2.69 10.6 Sodium, mg/L 0.208 0.423 Potassium, mg/L 0.056 0.108 Ammonium, mg/L (0.12)* (1.06)* Calcium, mg/L 0.0126 0.0364 Magnesium, mg/L 0.0242 0.0484 * Values in parentheses are not certified and are given for information only. Figure 4.2 Analysis sheet for Simulated Rainwater (SRM 2694a). Adapted from NIST Special Publication 260: Standard Reference Materials Catalog 1995–96, p. 64; U.S. Department of Commerce, Technology Administration, National Institute of Standards and Technology. Simulated Rainwater (liquid form) This SRM was developed to aid in the analysis of acidic rainwater by providing a stable, homogeneous material at two levels of acidity. repeatability The precision for an analysis in which the only source of variability is the analysis of replicate samples. reproducibility The precision when comparing results for several samples, for several analysts or several methods. indeterminate error Any random error that causes some measurements or results to be too high while others are too low. 1400-CH04 9/8/99 3:53 PM Page 62 [...]... (g) 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 3.073 3.0 84 3. 148 3. 047 3.121 3.116 3.005 3.115 3.103 3.086 3.103 3. 049 2.998 3.063 3.055 3.181 3.108 3.1 14 3.121 3.105 3.078 3. 147 3.1 04 3. 146 3.095 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 3.101 3. 049 3.082 3. 142 3.082 3.066 3.128 3.112 3.085 3.086 3.0 84 3.1 04 3.107 3.093 3.126 3.138 3.131... 3.010–3.028 3.029–3. 047 3. 048 –3.066 3.067–3.085 3.086–3.1 04 3.105–3.123 3.1 24 3. 142 3. 143 –3.161 3.162–3.180 3.181–3.199 2 0 4 19 15 23 19 12 13 1 2 140 0-CH 04 9/8/99 3: 54 PM Page 79 Chapter 4 Evaluating Analytical Data 79 25 20 Frequency 15 10 5 0 2.991 to 3.009 3.010 to 3.028 3.029 to 3. 047 3. 048 3.067 3.086 3.105 3.1 24 to to to to to 3.066 3.085 3.1 04 3.123 3. 142 Weight of pennies (g) 3. 143 to 3.161 3.162... degrees 140 0-CH 04 9/8/99 3: 54 PM Page 81 Chapter 4 Evaluating Analytical Data of freedom The value of t from Table 4. 14, is 2 .45 Substituting into equation 4. 13 gives µ = X ± ts = 3.117 ± (2 .45 )(0.051) n = 3.117 ± 0. 047 g 7 Thus, there is a 95% probability that the population’s mean is between 3.070 and 3.1 64 g Table 4. 14 Values of t for the 95% Confidence Interval Degrees of Freedom t 1 2 3 4 5 6 7... population’s mean, therefore, is µ = X ± zσ n 4. 11 140 0-CH 04 9/8/99 3: 54 PM Page 77 Chapter 4 Evaluating Analytical Data EXAMPLE 4. 14 What is the 95% confidence interval for the analgesic tablets described in Example 4. 13, if an analysis of five tablets yields a mean of 245 mg of aspirin? SOLUTION In this case the confidence interval is given as µ = 245 ± (1.96)(7) = 245 mg ± 6 mg 5 Thus, there is a 95% probability... showing the effect of random error 140 0-CH 04 9/8/99 3: 54 PM Page 74 74 Modern Analytical Chemistry (a) f (x ) (b) (c) Figure 4. 6 Normal distributions for (a) µ = 0 and σ2 = 25; (b) µ = 0 and σ2 = 100; and (c) µ = 0 and σ2 = 40 0 –50 40 –30 –20 –10 0 10 Value of x 20 30 40 50 Examples of normal distributions with µ = 0 and σ2 = 25, 100 or 40 0, are shown in Figure 4. 6 Several features of these normal... Single United States Penny in Circulation Replicate Number Mass (g) 1 2 3 4 5 6 7 8 9 10 3.025 3.0 24 3.028 3.027 3.028 3.023 3.022 3.021 3.026 3.0 24 Background noise in a meter obtained by measuring signal over time in the absence of analyte 140 0-CH 04 9/8/99 3: 54 PM Page 64 64 Modern Analytical Chemistry 4B.3 Error and Uncertainty Analytical chemists make a distinction between error and uncertainty.3... is 140 0-CH 04 9/8/99 3: 54 PM Page 75 f (mg aspirin) Chapter 4 Evaluating Analytical Data 0.08 0.07 0.06 0.05 0. 04 0.03 0.02 0.01 0 75 Figure 4. 7 210 220 240 230 z low = 250 260 Aspirin (mg) 270 280 290 Normal distribution for population of aspirin tablets with µ = 250 mg aspirin and σ2 = 25 The shaded area shows the percentage of tablets containing between 243 and 262 mg of aspirin 243 − 250 = −1 .4 5... how individual measurements and results are distributed around a central value 140 0-CH 04 9/8/99 3: 54 PM Page 71 Chapter 4 Evaluating Analytical Data Table 4. 10 Results for a Second Determination of the Mass of a United States Penny in Circulation Penny Mass (g) 1 2 3 4 5 3.052 3. 141 3.083 3.083 3. 048 – X s 3.081 0.037 4D.1 Populations and Samples In the previous section we introduced the terms “population”... calculate *N! is read as N-factorial and is the product N × (N – 1) × (N – 2) × × 1 For example, 4! is 4 × 3 × 2 × 1, or 24 Your calculator probably has a key for calculating factorials 140 0-CH 04 9/8/99 3: 54 PM Page 73 Chapter 4 Evaluating Analytical Data 73 the probability, we substitute appropriate values into the binomial equation P(0, 27) = 27! × (0.0111)0 × (1 − 0.0111)27 − 0 = 0. 740 0!(27 − 0)! There... 18 C ± 1 C 4C .4 Uncertainty for Mixed Operations Many chemical calculations involve a combination of adding and subtracting, and multiply and dividing As shown in the following example, the propagation of uncertainty is easily calculated by treating each operation separately using equations 4. 6 and 4. 7 as needed 140 0-CH 04 9/8/99 3: 54 PM Page 67 Chapter 4 Evaluating Analytical Data EXAMPLE 4. 7 For a . measurements. X X n i i n = = ∑ 1 Table 4 .1 Masses of Seven United States Pennies in Circulation Penny Mass (g) 1 3.080 2 3.0 94 3 3.107 4 3.056 5 3.112 6 3.1 74 7 3.198 140 0-CH 04 9/8/99 3:53 PM Page 54 Chapter 4 Evaluating Analytical. 0672 22 . s R s A s B s C RABC =       +       +       222 0 0085 19 9 84 100 0 043 . . .%×= 140 0-CH 04 9/8/99 3: 54 PM Page 66 Chapter 4 Evaluating Analytical Data 67 EXAMPLE 4 . 7 For a concentration technique the relationship. or measurement. 140 0-CH 04 9/8/99 3: 54 PM Page 64 Chapter 4 Evaluating Analytical Data 65 It is easy to see that combining uncertainties in this way overestimates the total uncertainty. Adding