In the Laboratory
The Penny Experiment Revisited: An Illustration W of Significant Figures, Accuracy, Precision, and Data Analysis Joseph Bularzik Department of Chemistry and Physics, Purdue University Calumet, Hammond, IN 46323-2094;
[email protected] The penny experiment has frequently been used to calculate density (1–5). These experiments compared the penny minted before and after 1982, showing two objects similar in material and shape but having different densities. The discontinuous density can be used as a discovery laboratory process for students. Some investigations describe the origin of this difference, such as comparing the melting points and visual observations of pennies cut in half. Other investigations focused on data collection and error analysis (6–9). At Purdue University Calumet the penny experiment is used as a general chemistry laboratory to illustrate the difference between accuracy and precision. The students measure the volumes of the pennies both by a volumetric and a linear measurement method. Because of the limited number of measurements each student makes, the values may appear random and incorrect. Comparing their data to the whole class’s average density and standard deviation alleviates this misunderstanding. The collected data illustrate the difference between precision and accuracy and also shows how the number of significant figures in the measurement technique affects these measurements. Experiment Students measure the density of pennies by four different procedures. An example procedure is included in the Supplemental Material.W Materials needed are pre-1982 and post-1982 pennies, a top-loading balance, 50 mL graduated cylinders that are calibrated with 1 mL intervals, and vernier calipers with the lowest calibration of 1 mm. Using the 10% measurement rule—a person can estimate to about 10% of the lowest graduation—the uncertainty of measurement for the calipers is ±0.1 mm and the uncertainty of measurement for the graduated cylinder is ±0.1 mL. The mass of the pennies is measured on a top-loading balance that has an uncertainty of ±0.0001 g. The density is calculated from the measured mass and volume of the pennies, δ = mass兾volume = g兾mL = g兾cm3. The volume is determined by one of the following four methods: 1. Measure the height and diameter of a stack of ten pennies using a vernier calipers. The stack height ranges from 14 to 15 mm and the diameter is about 19 mm. 2. Place the stack of ten pennies into a 50 mL graduated cylinder. The volume of the stack is about 3.5 ml. 3. Measure the thickness and width of a single penny using a vernier calipers. 4. Place a single penny into the graduated cylinder and measure the displacement.
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Steps 3 and 4 are repeated for several individual pennies. All four methods are repeated for pre-1982 and post-1982 pennies. Because of the change of the composition, copper to zinc ratios, pennies minted before 1982 have a density of 8.8 g兾mL and after 1982 are have a density of 7.2 g兾mL (4). Hazards There are no significant hazards associated with this laboratory experiment. Results and Discussion The calculation for the volume from methods 1 and 3 is V = (area)(height) = (π)(r)2(h), where r = d兾2. In method 1 the measured values for calculating the volume are the diameter (19.x mm ± 0.1 mm) and height (14.x mm ± 0.1 mm), each measured with three significant figures. The radius of a penny is 9.x mm ± 0.1 mm with two significant figures. Using the radius (r) to calculate the volume, V = (π)(r)2(h), the calculated volume has two significant figures. In method 2 the volume is the measured value (3.x mL ± 0.1 mL), with two significant figures. The calculated density has the same number of significant figures as the volume because the mass measurement has more significant figures. The precision from the two methods is different, as demonstrated by actual student measurements. The uncertainty for the radius is 0.1 mm out of 9.5 mm, where the uncertainty for the graduated cylinder measurement is 0.1 mL out of 3.5 mL, a factor of about 3 times more. In reality this difference is probably even greater because a student can actually read the vernier calipers to within ±0.1 mm but the graduated cylinder only to ±0.2 mL or more. Therefore, the class average measurements from method 1 should have a lower standard deviation than for method 2. The average densities from method 2 should be more accurate. The caliper measurement method measures the high features of the pennies and calculates a volume that is too large. Using either of these two methods, the students should be able to distinguish between the two types of pennies. The class average density and standard (or average) deviation demonstrate these differences. The data for 66 stacks of pennies from several laboratory sections are shown in Table 1. More data are supplied in the Supplemental Material.W The density standard deviations from the caliper measurements are less than the standard deviations of the measurements using the graduated cylinder. As expected, the graduated cylinder provides more accurate measurements. The data for six laboratory sections are shown in Figure 1. Each section shows the expected behavior.
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In the Laboratory
Table 1. Results for Density Measurements Density of Stacks of Pennies (g/mL) a Penny
by Dimensions Ave
by Displacement
Std Dev
Ave
Std Dev
Pre-1982
7.2 (8.8)
b
0.4
8.2 (8.8)
1.4
Post-1982
6.4 (7.2)
0.3
7.3 (7.2)
1.0
a
Density measurements of 66 stacks of 10 pennies. bActual density shown in parentheses.
Figure 2. The calculated value of density if the measurement of the height in cm or the volume in mL is measured different from the actual value.
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Figure 1. The average density and standard deviation of stacks of pre-1982 pennies for several laboratory sections. The actual density is shown by the dashed line.
Figure 3. The average calculated density measurements of pre-1982 pennies for 60 students.
Using method 3 the measurement of the thickness of an individual penny has only two significant figures because the thickness of a penny is about 1.4 mm ±0.1mm. Having an uncertainty of about 10% for the measured thickness of the penny, the measured average densities for the two types of pennies may be within one standard deviation of each other. The typical value for the volume of a penny is 0.35 mL. Using method 4 students measure within ±0.1 mL; therefore, no individual measurement can calculate the correct density and the measurement has only one significant figure. Assuming a mass of 2.5 g for the newer pennies, volumes of 0.3 mL and 0.4 mL gives densities of 8 g兾mL and 6 g兾mL, respectively. Measuring half at 0.4 mL and half at 0.3 mL, the calculated density, using one significant figure, is 7 g兾mL, the correct value. Using a range of measurements of ±0.2 mL, the density will range from 12 g兾mL to 5 g兾mL. With such a large range, this method becomes useless to differentiate the density of the different pennies. Figure 2 shows how the calculated densities of the pennies change as the measurements of the volumes vary from 0.35 mL. For the dimension measurements using the calipers, if the height is measured slightly larger than 1.4 mm,
the calculated density will be slightly lower than the correct value. Conversely if the height is measured slightly smaller, a slightly larger density will be calculated. Having random height measurements of ±0.3 mm, the average density value will still be correct. But this is not true for the displacement volume measurements. A volume slightly larger than 0.35 mL results in a slightly lower density. A slightly smaller volume results in a much higher density. Having a random volume measurement of ±0.3 mL, the average density value will be much greater than the correct value. The most accurate and most precise methods can be determined from the average and standard deviation of the class data. A density value for a single penny that is much different from the accepted value could be from the large uncertainty in the measurement techniques. Reading one unit value different from the real value in the volume measurement will calculate a very different density. With an uncertainty of ±0.1 cm, reading the volume of 0.2 mL gives a much different density than 0.3 mL. Plotting the individual density values provides a visual aid to help the students understand accuracy and precision. Figure 3 shows the average density calculations for 60 students. The better precision gained by using the dimension
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measurements is illustrated. Figure 4 shows the different density values measured by the displacement method compared to the mass values of the individual pennies. The quantized measurements of the density depend on the quantized measurement of the volumes of ±0.1 mL. The variation of the volume has a much greater effect on the variation of the density than the variation of the mass. Some students will carefully record the measurements while others will write down the same measurement value after they measure one or two. Performing an analysis of variance, ANOVA, can illustrate the differences or similarities of the students’ measurements (8, 10, 11). An example is given in the Supplemental Material.W Conclusion A simple measurement laboratory procedure is used to teach students about accuracy, precision, and measurement bias. This procedure illustrates two different methods of measurements, where one method is more accurate and the other method is more precise. Also it shows that random errors of measurement will not always give an average close to the actual value. An important lesson is how the uncertainty of the measured values will depend upon the measuring tools and techniques. W
Supplemental Material
Instructions for the students including postlab questions and ANOVA analysis are available in this issue of JCE Online. Literature Cited 1. Miller, J. M. J. Chem. Educ. 1983, 60, 142.
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Figure 4. Density values for pennies measured by the displacement method compared to the mass.
2. Ricci, Robert W.; Ditzler, Mauri A. J. Chem. Educ. 1991, 68, 228–231. 3. Sardella, Dennis J. J. Chem. Educ. 1992, 69, 933. 4. Mauldin, Robert F. J. Chem. Educ. 1997, 74, 952–955. 5. Stolzberg, Richard J. J. Chem. Educ. 1998, 75, 1453–1455. 6. Nelson, Lloyd S. J. Chem. Educ. 1956, 33, 126–131. 7. Cunningham, Cindy C.; Brown, G. R.; St Pierre, L. E. J. Chem. Educ. 1981, 58, 509–511. 8. O’Reilly, James E. J. Chem. Educ. 1986, 63, 894–896. 9. Richardson, T. H. J. Chem. Educ. 1991, 68, 310–311. 10. Skoog, D. A.; West, D. M.; Holler, J. F.; Crouch, S. R. Fundamentals of Analytical Chemistry, 8th ed.; Thompson: Belmont, CA, 2004; pp 160–169. 11. Harvey, D. Modern Analytical Chemistry; McGraw Hill: Boston, 2000; pp 693–696.
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