Modern Analytical Cheymistry - Chapter 4

Chapter 4 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. 53 54 Modern Analytical Chemistry 4A Characterizing Measurements and Results Let’s begin by choosing a simple quantitative problem requiring a single measure-ment. 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 circula-tion? 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. mean The average value of a set of data (X). 4A.1 Measures of Central Tendency One way to characterize the data in Table 4.1 is to assume that the masses of indi-vidual 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 X = ån=1 Xi where Xi is the ith measurement, and nis the number of independent measurements. Table 4.1 Penny 1 2 3 4 5 6 7 Masses of Seven United States Pennies in Circulation Mass (g) 3.080 3.094 3.107 3.056 3.112 3.174 3.198 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 X = 21.821 = 3.117 g The mean is the most common estimator of central tendency. It is not consid-ered 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, Xmed, is the middle value when data are ordered from the smallest to the largest value. When the data include an odd number of measure-ments, 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. 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 (Xmed). 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. 4A.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 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 numerical difference between the largest and smallest values in a data set (w). standard deviation A statistical measure of the “average” deviation of data from the data’s mean value (s). Range The range, w, is the difference between the largest and smallest values in the data set. Range = w = Xlargest – Xsmallest 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 n (X – X)2 n 1 4.1 where Xi is one of n individual measurements, and X is the mean. Frequently, the relative standard deviation, sr, is reported. sr = s The percent relative standard deviation is obtained by multiplying sr 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 = 0.051 Chapter 4 Evaluating Analytical Data 57 The relative standard deviation and percent relative standard deviation are sr = 0.051 = 0.016 sr (%) = 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 devia-tion, or the variance. The standard deviation, rather than the variance, is usually re-ported because the units for standard deviation are the same as that for the mean value. variance The square of the standard deviation (s2). 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 = s2 = (0.051)2 = 0.0026 4B 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. 4B.1 Accuracy Accuracy is a measure of how close a measure of central tendency is to the true, or expected value, m.† Accuracy is usually expressed as either an absolute error E = X – m 4.2 or a percent relative error, Er. Er = X m m ´ 100 4.3 *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 m. Likewise, the standard deviation by experiment is given the symbol s, and its underlying true value is identified as s. ... - tailieumienphi.vn