In the Laboratory
Using Pooled Data and Data Visualization To Introduce Statistical Concepts in the General Chemistry Laboratory
Stoichiometric Ratio:
As our department worked to strengthen the laboratory component of the first-semester general chemistry course, we decided to link several experiments more closely with conceptual statistics as a unifying theme. Here I describe revising a standard stoichiometry experiment by adapting graphical techniques (1–5) associated with exploratory data analysis. It is the emphasis on graphical techniques that differentiates our approach from other papers about statistics in the introductory laboratory (6–8) that have appeared in the Journal. The basic stoichiometry experiment is preparation of magnesium oxide by direct reaction of the elements (magnesium is burned in air) and subsequent determination of its empirical formula. We have found the inclination of students to report an empirical formula consistent with an expected result rather than with their data to be a persistent difficulty. For example, many students would associate an O/Mg ratio (amounts are per unit amount compound) of 0.845 with MgO rather than Mg4O3 or Mg5O4. The instructor controls the list of reasonable formulas in our graphical approach and can include discussion of how the list was determined at his or her discretion.
amount O amount Mg
Robert J. Olsen Division of Natural and Mathematical Sciences, The Richard Stockton College of New Jersey, Pomona, NJ 08240-0195;
[email protected] MgO1.000 (MgO)
0.9
0.8
MgO0.800 (Mg5O4) MgO0.750 (Mg4O3)
0.7
Figure 1. Example number line for O/Mg = 0.845 (amounts indicated are per unit amount compound). The reference lines labeled to the right correspond to empirical formulas selected by the instructor as reasonable. Line segments emanating from the data point to the left are drawn assuming empirical formulas MgxOy with 1 ≤ x,y ≤ 5 are allowed; the nearest empirical formula is Mg5O4. The data point to the right illustrates the outcome if allowed empirical formulas are restricted to 1 ≤ x,y ≤ 4, in which case the nearest empirical formula is Mg4O3.
Method
1.3
MgO1.250 (Mg4O5)
amount O amount Mg
1.2
Stoichiometric Ratio:
Each student calculates the O/Mg ratio from his or her data and then plots a point at the calculated O/Mg ratio on the provided number line (see Figure 1). Next, each student draws line segments outward from this point in both directions until a labeled reference line is reached, thus locating the empirical formula in best agreement with the data. A student whose individual result leads to an empirical formula other than MgO may be reluctant to report the formula that is consistent with the data. This reluctance can be overcome by moving beyond individual results, which we do by pooling the class’s data and continuing the analysis with this larger data set. Many recent papers (9–15) in this Journal have featured data pooling in the introductory laboratory. To analyze the pooled data, we use the same number line as before, although now the data for an entire class is plotted as a 1-D scatterplot (see Figure 2). Both the central tendency and the dispersion of the data are evident in a scatterplot. Students repeat the steps they used to analyze their individual data—applying these steps to the mean of the class data— and go on to attach confidence to this result by identifying the number of allowed empirical formulas that fall in the interval containing the middle 80% of the points. The only empirical formula in this range in the scatterplot to the left is MgO, so students in this section can say that they have found the empirical formula of magnesium oxide to be MgO with a reasonable degree of confidence. By comparison, two additional empirical formulas (Mg4O3 and Mg5O4) fall within this range in the scatterplot to the right, so students in this section must conclude
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1.0
1.1
1.0
MgO1.000 (MgO)
0.9
0.8
MgO0.800 (Mg5O4) MgO0.750 (Mg4O3)
0.7
MgO0.667 (Mg3O2) 0.6
MgO0.600 (Mg5O3)
Figure 2. Example scatterplots of pooled data. Points plotted as filled circles make up the middle 80% of the students’ data measuring O/Mg ratios (ratio amounts are per unit amount compound). Points plotted as open circles are the smallest and largest 10% of the students’ data measuring O/Mg ratios. The mean is plotted as a thick, dashed, horizontal line.
Journal of Chemical Education • Vol. 85 No. 4 April 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Laboratory
that they have not unambiguously determined the empirical formula of magnesium oxide. These two data sets highlight that imprecise data necessarily leads to imprecise conclusions, a lesson that we feel is more valuable than confirming that magnesium oxide is MgO. The instructor can exercise latitude regarding the choice of the percentiles that determine the subset of points being analyzed. Restricting the analysis to points between the 10th and 90th percentiles balances retention of a reasonably large fraction of the students’ data against elimination of extreme outliers, whose presence can unduly increase the number of identified empirical formulas. Summary We have found that 1-D scatterplots are an effective way to emphasize the central tendency and dispersion of a data set, two notions basic to understanding its statistical properties. More experienced students are engaged by this new angle rather than feeling that they are repeating their high school work. At the same time, the new aspects of the data analysis do not increase the workload of the experiment significantly for the remaining students because a visual, intuitive approach to statistics puts them on a comparable footing with their more experienced peers. Approaching data analysis and interpretation through data visualization converts the experiment from a conventional confirmatory format (i.e., verifying that the formula of magnesium oxide is MgO) to a style akin to that of a discovery experiment. Appropriate customization of the basic 1-D scatterplot allows a similar graphical component to be added readily to the data analysis and interpretation associated with other experiments typical of the general chemistry laboratory. We include graphical data analysis in an experiment comparing the volume of water delivered by a beaker, a graduated cylinder, and a pipette; in an experiment comparing the acid-neutralizing capacity of two brands of antacid; and in an experiment in which the gas constant R is determined. We plan to include it an experiment exploring Hess’s law and an experiment in which the equilibrium constant of a reaction is measured. Hazards Magnesium burns with an extremely bright flame that can cause permanent eye damage if it is viewed directly. Porcelain crucibles should be handled with care throughout the experiment because they remain hot enough to cause serious burns long after they are no longer glowing red.
Acknowledgments I would like to thank Tom Olsen, Brian Rogerson, and Simeen Sattar for helpful discussions and critical readings of the manuscript; Justine Ciraolo, Doreen Goldberg, Ed Paul, and Brian Rogerson for adopting a pooled-data approach to this experiment in their laboratory sections; and the students in the many sections of Chemistry 2115 who have done this experiment as it has evolved to its present form. Literature Cited 1. Cleveland, W. S. The Elements of Graphing Data, revised ed.; Hobart Press: Summit, NJ, 1994. 2. Cleveland, W. S. Visualizing Data; Hobart Press: Summit, NJ, 1993. 3. Tufte, E. R. The Visual Display of Quantitative Information, 2nd ed.; Graphics Press: Cheshire, CT, 2001. 4. Tufte, E. R. Envisioning Information; Graphics Press: Cheshire, CT, 1990. 5. Tufte, E. R. Visual Explanations: Images and Quantities, Evidence and Narrative; Graphics Press: Cheshire, CT, 1997. 6. Spencer, R. D. J. Chem. Educ. 1984, 61, 555–563. 7. Marino, F. J. Chem. Educ. 1988, 65, 445–446. 8. Salzsieder, J. C. J. Chem. Educ. 1995, 72, 623. 9. Ricci, R. W.; Ditzler, M. A. J. Chem. Educ. 1991, 68, 228–231. 10. Ricci, R. W.; Ditzler, M. A.; Jarret, R.; McMaster, P.; Herrick, R. S. J. Chem. Educ. 1994, 71, 404–405. 11. Ditzler, M. A.; Ricci, R. W. J. Chem. Educ. 1994, 71, 685–688. 12. Herrick, R. S.; Nestor, L. P.; Benedetto, D. A. J. Chem. Educ. 1999, 76, 1411–1413. 13. Sadoski, R. C.; Shipp, D.; Durham, B. J. Chem. Educ. 2001, 78, 665–666. 14. Moss, D. B.; Cornely, K. J. Chem. Educ. 2001, 78, 1260–1262. 15. Sanger, M. J.; Geer, K. J. Chem. Educ. 2002, 79, 994–996.
Supporting JCE Online Material http://www.jce.divched.org/Journal/Issues/2008/Apr/abs544.html Abstract and keywords Full text (PDF) with links to cited JCE articles Supplement Student handouts: experiment write-up and data spreadsheet
Instructor notes, including: further discussion of the experiment; a summary of the results obtained by 17 laboratory sections during six semesters (Fall 2002–Spring 2006); suggestions about how analysis of the pooled data can be extended; a description of the spreadsheet used to pool the data; and information about assessment of the experiment during the 2006–2007 academic year
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 4 April 2008 • Journal of Chemical Education
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