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Unformatted text preview: 161 Supplement SUPL- 001 001 supplement Analysis of Experimental Reliability Prepared by R. F. Schneider (Revised 3/09) SUPL-001 concepts An essential part of any quantitative laboratory measurement is an understanding of the reliability of the measurement. Powerful statistical methods exist for the analysis of reliability. The exercises conducted in the introductory chemistry laboratory will utilize procedures and apparatus with well-known limits of accuracy and precision. Normally, measurements will be replicated several times in order to establish the reliability of those quantities reported as conclusions of exercises (and on which a substantial part of the earned grade will depend). The number of repetitions of a measurement will typically be relatively small (34 repetitions). Simple statistical concepts suffice to represent the reliability of measurements that are replicated small numbers of times. Supplement SUPL- 001 162 Measures of Reliability We use two quantities to characterize the reliability of a series of experimental measurements--accuracy and precision. Accuracy measures the extent to which measurements agree with a known or "true" value. Precision measures the consistency of measurements with each other, independent of their accuracy. Figure 1-1 demonstrates accuracy and precision. It shows the result of a series of attempts to hit the bull's-eye of a circular target with a bow and arrow. The objective of this exercise (i.e., its known value) is to have each arrow strike the center of the target. Concentric circles on the target mark areas which are successively further from the objective. The distance from the center of the bull's-eye represents the error in an attempt. Archer 1 shows great consistency in the part of the target that the arrows strike, but they are all distant from the center of the target. The data are precise, but not accurate. Archer 2 shows inconsistency in the location of the arrows on the target, but there are about as many of them on the right as on the left and on the top of the target as on its bottom. This archer demonstrates some measure of accuracy, but with low precision. Archer 3 demonstrates both accuracy and precision. How does accuracy play a role in experiments whose results are qualitative? If on the basis of a series of experimental results, a sample is asserted to be potassium iodide, the assertion is accurate (i.e., true) if the material was, in fact, potassium iodide. If the material was another substance, the assertion is inaccurate, which in this case means false. If the data from which the identity was concluded does not permit distinguishing potassium iodide from sodium iodide, asserting that the substance may be one OR the other would be an accurate (i.e., true) statement if the substance was actually one or the other. d ay H Archer 1 Precise Archer 2 Accurate Archer 3 Precise and Accurate Figure 1-1. en -M cNe il , LLC Analysis of Experimental Reliability In quantitative laboratory exercises, as with the archers, the precision of a result tells us something about how reproducibly the procedure was conducted. Are there limits on the precision attainable in laboratory exercises? Yes. Each device and each procedure that is used in an exercise has an intrinsic (built-in) precision. E.g., the top loading balance can be read to the nearest 0.01 g, but the balance manufacturer guarantees only that if we weigh the same object several times independently, (i.e., weigh the object, remove it from the balance, replace it on the balance pan and weigh it again) the indicated weights will not differ by more than 0.03 g. No matter how experienced the individual using the balance may be, the measured weight is uncertain because the balance mechanism introduces small uncertainties. The intrinsic precision of the devices used in an exercise place limits on the overall precision of the exercise. 163 Quantitative Measures of Reliability Suppose an exercise calls for the determination of the density of a sample of matter by measuring its volume and mass. The results of two measurements of the same volume result in 31.70 mL and 32.50 mL. What is the "best" value to use for the volume? If the experimenter has no reason to believe that one of the measurements is "better" than the other, the best value to report is the average (sometimes called arithmetic mean, or simply, mean) of the two results, Vavg of the values. I.e., V1 V2 n Vn Vavg , or, in the case of two measurements, Vavg = V1 +V2 / 2 Note that the average value has the same units as the individual values (i.e., if each volume is measured in mL, the mean will also be in mL). How shall we measure the precision of the two volume measurements? For small numbers of repetitions, a reasonable measure of precision is the average deviation. To calculate the average deviation, calculate the difference between each individual measurement and the average (the difference is called a deviation), take the absolute value of the deviations* (i.e., disregard the sign of the deviation) and average the deviations. For the volume measurements given earlier, the calculation yields: Avg Dev =[ |(31.70 32.10)| |(32.50 32.10)| ]/2 = [ |0.40| |0.40| ]/2 = 0.40 mL V = 32.10 0.40 mL The numerical value of the average deviation reveals a lot about the precision of the volume measurement. However, it is also useful to consider how large the deviation * If we do not take the absolute value of the deviations, the average deviation will always be zero (0). The Supplement SUPL- 001 164 is compared to the quantity being measured. This is accomplished by calculating the percent deviation (percent means parts per hundred) of the measurement, namely, % Deviation = 100 0.40 mL/32.10 mL = 1.3 % Note that the units of the measurement appear in both the numerator and the denominator, so that the resulting percent deviation has no physical units. The second part of a density determination consists of measuring the mass of the object. In this case, three measurements of the same object as above are made resulting in measurements of 41.61 g, 41.72 g and 41.65. The average mass is then Massavg = (41.61 and the average deviation is Average Deviation = (0.05 The mass should be reported as 41.66 0.06 0.01)/3 = 0.04 g. 41.72 41.65)/3 = 41.66 g 0.04 g. The percent deviation of the mass is Percent Deviation = 100 0.04 g/41.66 g = 0.1%. What is a quantitative measure of accuracy? The experimental result above suggests 31.70 mL and 32.10 32.50 mL. Suppose the true volume of the sample is 31.80 mL. The deviation of the average tion of the experimental result from the true value is 100 When exercises result in numerical values, grades will generally depend on the percent deviation of the reported result as a measure of the precision with which the exercise has been conducted. We will often use the percent deviation of the mean of reported results from the true value as a measure of accuracy. Computing with Uncertainties While computing with uncertainties can be treated rigorously, we use a simple approach in this illustration. We examine the extremes that result from using the worst combinations of the mass and volume. The largest computed density will be that using the largest mass and the smallest volume, i.e., dmax = 41.70/31.70 = 1.315 g/mL The smallest density will result from using the smallest mass and the largest volume, i.e., dmin = 41.62/32.50 = 1.281 g/mL Analysis of Experimental Reliability The mean of these two values is 1.298. The average deviation of these two values of the density is 0.017 g/mL from the mean. The computed density should then be expressed as 1.298 0.017 g/mL The percent deviation of the density is computed to be 100 0.017/1.298 165 general rule, namely, that the percent deviation in a quantity computed as the quotient of two quantities will be the sum of the percent deviations of the two individual quantities. The same rule applies to quantities computed as the product of two quantities. What about the deviations of computed quantities involving the sum or difference between two uncertain numbers? Consider the measurement of net volumes delivered mL. Suppose that, due to the intrinsic precision of the buret, each of these readings has an uncertainty of 0.02 mL. The net volume is Vnet = (35.87 0.02) (0.74 0.02) = 35.13 mL What uncertainty should we ascribe to the net volume? Again, we look at the worst volume (35.87 These values each differ from the earlier result (35.13 mL) by 0.04 mL producing an average deviation of 0.04. The net volume should, therefore, be reported as Vnet = 35.13 0.04 mL Note that the average deviation of the result is the sum of the average deviations of the two quantities. What about the percent deviation? The percent deviation of the tial volume but the uncertainties are the same, its percent deviation is much smaller. that the percent deviation of the result is much closer to the percent deviation of the reflects a general rule, namely, the average deviation of the sum or difference of two numbers is the sum of the average deviations of each of them. Types of Errors Two types of errors contribute to experimental uncertainty--random errors and systematic errors. Random errors are those over which the experimenter has little or no control. They may result from intrinsic precision of the measuring devices (e.g., the same object weighed on the same analytical balance can give weights that differ by 0.0002 g simply due to the balance mechanism). They may also arise from Supplement SUPL- 001 166 the lack of ability to control an experimental condition (e.g., a sample of a material that is analyzed may have small variations in composition from one sample to the next because of the way it was produced or constructed). Using averages of several samples will tend to cancel out random errors. This may cause an increase in the numerical value of the expected precision of a measurement, but will generally not affect the accuracy of the result. The word homogeneous (and the corresponding property, homogeneity) describes the extent to which a material has uniform properties no matter which sample is used. When you are given an unknown material to analyze or reagents to use in an exercise, considerable effort is expended to insure that the samples are homogeneous to minimize such errors. When you dissolve a known weight of a substance to make a solution of accurately known concentration, it is important that the solution be mixed until it is homogeneous. If it is not homogeneous, different samples of equal volumes of Substances which are not of uniform composition are called heterogeneous. Systematic errors are errors that are introduced by the experimenter or the experimental procedure. Suppose a small amount is lost in the process of transferring a weighed amount of a solid material from a container to a flask. The transferred material is used to make a homogeneous solution of known concentration. The concentration of the solution will be uniformLy incorrect because of the lost sample. The subsequent procedure may be carried out with high precision, but the result is guaranteed to be inaccurate. A common problem in the introductory chemistry laboratory occurs in the use of stock solutions of a precisely known concentration. If the student uses a wet beaker to transfer the solution, the water in the beaker causes the concentration of the solution used to be less than the indicated concentration. The exercise may result in very precise results which are inaccurate because of the decreased concentration. Results of high precision but low accuracy, are an indication of the occurrence of a systematic error. Decimal Digits, Significant Figures and Experimental Measurements ing standards for exercises and quizzes invariably include a substantial number of points that deal with the proper implementation of this concept. We encourage the use of electronic calculators for facilitating the computations associated with experimental numbers. Even inexpensive calculators are designed to deal with and display numbers having 10 or more digits. Everyday laboratory measurements rarely permit many of these digits to represent reality. number of decimal Analysis of Experimental Reliability If it is a computed quantity, the number of significant digits reflects the number of digits in the numbers from which it is computed. not simply related to the intrinsic precision of a device. The buret permits a volume to be read to - 167 volume is between 10.00 and 50.00 (the maximum capacity of the buret), the numBuret readings must always be reported to two decimal places, regardless of the volume reading. Similarly, the display of an analytical balance displays weights to the nearest 0.0001 g. weight of an object. Nevertheless, weights determined on an analytical balance must always be reported to four decimal places, regardless of the weight. ferent from those of the quantities from which it is computed. Consider a container which together with its contents weighs 35.2749 g. A sample of the contents is trans- now used in the calculation of a concentration by dividing it by a volume of 20.32 mL. In general, the number of significant figures in a multiplication or division of quantities will be that of the quantity with the least number of significant figures. We will pay great attention to the use of significant figures in the grading of laboratory reports**. They reflect an understanding of the measurement limitations of the devices used in the exercises. You are urged to master this concept early in the laboratory course. Most introductory chemistry texts treat this topic and provide exercises to permit you to evaluate your understanding of their use. Logarithms, Exponentials and Scientific Notation rithms and conversions between them, deserves some special discussion. We are ace.g., Avogadro's number is almost always written in the form 6.022 1023 mol1 to avoid having to write the large number of 0's which serve only to specify the location this course, their proper use represents the student's understanding of the precision of the measuring devices used in the exercises. Supplement SUPL- 001 168 its. If we write it to almost the full precision to which it is known, namely 6.022141 1023 - quantity. It may, however, not be a convenient representation when we are asked to add a series of numbers like 3.442 101 5.645 102 1.739 102 7.21. This is particularly evident when these numbers are added on a typical calculator, the result - doing the computing. Logarithms (and the exponentials to which they are related) pose a similar problem. pH ). If a solution has Consider the relation between a pH and its associated [H ]? This is 4.15 (5 0.85) 5 0.85 5 10 . The best recognized by writing [H following, where we increase and decrease the final digit in the pH by 1 unit (i.e., to 4.16 and 4.14) 104.16 = 105 100.84 = 6.92 105 and 104.14 = 105 100.86 = 7.24 105 than one unit in the second decimal place of [H ] calculated from the pH. We can formulate a rule of thumb that: in the process of converting a logarithm to the numWhat happens if we take the logarithm of a number? [H ] = 3.35 103 log (3.35 103 ) = 3 log 3.35 = 3 0.525 = 2.475 ] gives us: log (3.36 103) = 3 0.526 = 2.474 and log (3.34 103) = 3 0.524 = 2.476 Analysis of Experimental Reliability Summary Every experimental measurement involves uncertainty. Uncertainty can be due to a measuring device or due to the way it is used. Uncertainty is measured by accuracy and precision. Accuracy measures the deviation from the true value. For qualitative determinations, accuracy often means "true" or "false." For quantitative determinations, accuracy measures the deviation of a result from its "true" or accepted value. Precision measures the extent to which measurements that can be duplicated agree with one another. For small numbers of repetitions, the average deviation is an acceptable measure of precision. When we calculate derived results from quantities with known precision, the precision of the calculated values depends on which arithmetic operations are involved. For addition and subtraction, the average deviation of the result is the sum of the average deviations of the quantities. For multiplication and division, the percent deviation of the result is the sum of the percent deviations of the quantities. The number of decimal digits in a measured quantity is normally determined by the computed quantity depends arithmetic operations are involved. 169 Supplement SUPL- 001 170 Analysis of Experimental Reliability [Pre-laboratory Questions] Name Date Course/Section 171 1. A student reports the following data in several determinations of the density of a homogeneous solid plastic sphere. RUN 1 Mass of Sphere (g) Volume of Sphere (mL) 8.30 6.3 RUN 2 8.34 6.4 RUN 3 8.25 6.3 RUN 4 8.32 6.6 a. How should the student report the mass of the sphere? What is the percent deviation? b. How should the student report the volume of the sphere? What is the percent deviation? c. How should the student report the density of the sphere? What is the percent deviation? d. If the intrinsic error of the mass determination is 0.03 g and the intrinsic error of the volume determination is 0.1 mL, what is the expected intrinsic error in the density? e. If the true density of the sphere is 1.26, what are the deviation and percent deviation of the accuracy of the density determination compared with the true density? (over) Supplement SUPL- 001 172 f. Comment on how random or systematic errors might have been involved in this determination. g. The grading standard for this exercise was as follows. What grade should be assigned? Accuracy 50 points 40 points 30 points 25 points Precision 50 points 40 points 30points 25 points 2. What are the average, average deviation and percent deviation of the last 6 digits of your student ID? QUANTITY 0.003441 1.003441 3441.00 6.023060 3.441 103 1.2345090 3.441 103 10.4566 2 200 NUMBER OF SIGNIFICANT FIGURES ...
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This note was uploaded on 10/24/2009 for the course CHE 134 taught by Professor Staff during the Spring '08 term at SUNY Stony Brook.

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