In the Classroom edited by
Second-Year and AP Chemistry
John Fischer Ashwaubenon High School Green Bay, WI 54303-5093
The Importance and Efficacy of Using Statistics in the High School Chemistry Laboratory
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Paul S. Matsumoto Galileo Academy of Science and Technology, San Francisco, CA 94109;
[email protected] Statistics can quantify the likelihood of differences between experimental conditions and detect a correlation between variables. Statistics is a powerful tool with applications in chemistry (1, 2), the health sciences (3), and the social sciences (4). As many students take a statistics course as take a calculus course in college (5); thus, prior exposure to statistics in high school would prepare students for a statistics course in college. In addition, statistics would be valuable to students as future citizens, since statistics provides tools to develop a better understanding of current events. For example, the limitations of polls, economic indicators, epidemiological studies, and clinical trails can be explained by statistical analyses. Despite the value of statistics, a search of articles in the Journal of Chemical Education using the keyword “statistics” produced no articles on the use of statistics in a high school chemistry course (6). While there were many papers using statistics in various college chemistry laboratory courses (e.g., 7–11), these papers described few statistical tests and experiments; as such, these articles provide a limited view of the use of statistics in a chemistry course. A preliminary survey of chemistry teachers in the San Francisco Unified School District (data not shown) showed that there was little use of statistics in chemistry courses primarily because teachers did not know how to incorporate statistics into their lessons. It is my opinion that while many high school chemistry teachers would be familiar with calculations of the mean and standard deviation, few teachers would be familiar with specific statistical tests and concepts. As such, a goal of this paper is to introduce these tools to high school chemistry teachers by describing the various statistical tools implemented in my high school chemistry classes over the past five years. In general, statistical calculations are tedious, which discourages its use in the classroom, but the availability of Web sites (12), graphing calculators, Microsoft Excel, and various commercial statistics software makes the use of statistical data analysis feasible in a high school chemistry course. In addition, since these tools are used in college, the use of statistics in high school would prepare students in the use of statistics in college. Preliminary Data Analysis In my chemistry classes, statistics serves as a data analysis tool, rather than as a course of study, thus the presentation of statistics focuses on its use and interpretation with minimal mathematical rigor. In regards to implementing sta-
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tistics, there is a need for sufficient experimental data and a convenient method to gather the data. While making repeated measurements for statistical analysis was not feasible, the required number of measurements was obtained by pooling the data among students and classes (e.g., 7, 10, 11, 13). As an introductory exercise to statistics, students rolled a die 50 times, recorded the value on the face on the die, then analyzed the data. The purpose of this activity was to introduce the concept of using a histogram (frequency versus value on the die face) and the calculation of the mean and standard deviation (of the value on the die face) to represent experimental data. In addition, this activity illustrates the concept that as the sample size increases, there is an increase in the accuracy of the data (the mean converges to the theoretical valueW). In addition to characterizing the data by the mean and standard deviation, an early step in data analysis is the removal of an outlier—a value that significantly differs from the rest of the data. A consequence of removing an outlier is a reduction of the variability in the data, thus a statistical test would be more likely to detect a difference between experimental conditions. The Q-test (1, 2, 7, 8, 11) is a statistical test to detect such outliers. In summary, preliminary data analysis consists of calculating the mean and standard deviation in a pooled class data set and the removal of any outliers using the Q-test. Students conduct such preliminary data analysis whenever there is a class data set for analysis. Graphical Analysis of Data The graphical analysis of data is a common and powerful tool in science (1, 2). An equation that describes the relationship between variables may contain various parameters whose values would be of interest. Here are three examples of the use of graphical analysis (14): 1. The Arrhenius equation, which describes the temperature dependence of the rate constant, can determine the activation energy of a reaction. 2. The Clausius–Clapeyron equation, which relates the temperature dependence of the vapor pressure, can determine the enthalpy of vaporization of a chemical. 3. The various equations describing the relationship between the concentration of a reactant in a reaction as a function of time (for zeroth-, first-, or second-order reactions with respect to a single reactant) can be used to determine the rate constants of the various reactions.
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In the Classroom Table 1. Types of Errors in a Statistical Test Conditions
Accept the Null Hypothesis
Reject the Null Hypothesis
Null Hypothesis Is Valid
Correct Decision
Type I Error
Null Hypothesis Is Invalid
Type II Error
Correct Decision
For simplicity, only linear equations (or transformation of nonlinear equations into a linear equation) were used in my chemistry course. A linear equation, y mx b, may be characterized by its slope (m) and y-intercept (b), which can be determined by least-squares regression analysis (2, 4, 8, 15) as described in the Supplemental Material.W
Regression Analyses Using Linear Equations High school students are capable of undertaking regression analysis in the chemistry laboratory. Three brief examples from my own courses follow. First, a spectrophotometric assay (using a relatively inexpensive colorimeter probe) was used to measure the concentration of Cu2 to demonstrate the Beer–Lambert law and to measure the synthesis of Cu2 in a reaction involving a copper wire in an aqueous solution of silver nitrate (to demonstrate reaction stoichiometry). The Beer–Lambert law (1, 2) states that there is a linear relationship between the absorbance of light by a Cu2-containing solution and its concentration. A standard curve or calibration curve, an important concept in analytical chemistry, was used to determine the concentration of Cu2. Another example of using a calibration curve involves the redox titration of vitamin C (16), which was modified to a microscale experiment and to use a calibration curve (Dacotah Swett; personal communication; Lowell High School, San Francisco). Students with a prior biotechnology course complained that a calibration curve was not needed to determine the concentration of a chemical in solution, since they were able to obtain the concentration of DNA using a spectrophotometer without generating a calibration curve. After this lab activity, these students realized that the spectrophotometer used to determine the concentration of DNA has a calibration curve incorporated into its operation. The value of using least-squares regression analysis to describe the calibration curve, rather than simply drawing the best-fit line and subsequent determination of the concentration of the chemical of interest—by visual inspection—is the efficiency and objective nature of the determination of the best-fit line and subsequent concentration of the chemical. A third example of using least-square regression analysis is the comparison of the density of new pennies, old pennies, and dimes (or the density of various liquids: e.g., water, sea water, and rubbing alcohol) by determining the slope— which is the density—in a graph containing the values for mass measurements on the y-axis and the corresponding volume data on the x-axis. 1650
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Regression Analysis Using Transformed Linear Equations While the preceding examples involve fitting the data to the equation y mx, the next example involves transforming a nonlinear equation to a linear equation of the form y mx b. This laboratory activity uses the ideal gas equation to demonstrate Boyle’s law, where a piece of long tubing connects a syringe to a pressure transducer (a relatively inexpensive pressure probe). An analysis of this system shows that the data can be described by the equation, Vsyringe = nR T
1 P
− Vtubing
(1)
where the slope nRT and the y-intercept Vtubing, when Vsyringe was on the y-axis and the reciprocal of the pressure was on the x-axis. The goal in this experiment was to determine the volume of the tubing and the number of moles of gas in the system. The primary motivation of transforming the nonlinear equation to a linear equation was to maintain student comfort levels by simplifying the analysis (i.e., not to use a weighted nonlinear regression analysis). Examples of Applying Statistical Tests to Lab Data The primary motivation for the use of statistical tests is to be more objective in comparing various experimental conditions, which is based on calculating the p-value (the probability of error). Statistical tests (3, 4, 15) are formulated as a choice between the null hypothesis (a statement that there is no effect, which is accepted unless there is overwhelming evidence against the null hypothesis) and the alternative hypothesis (a statement that there is an effect). The decision to accept or reject the null hypothesis is based on the p-value; the decision has two possible errors as described in Table 1. The p-value refers to the probability of a Type I error, the probability of stating that there is a difference when there is no difference. The determination of the Type II error is beyond the scope of this paper. Statistical tests are based on the Neyman–Pearson lemma (15), which is beyond the scope of this paper. By convention, when the p-value < 0.05, there is a difference between experimental conditions; conversely, when the p-value 0.05, there is no difference between experimental conditions. While there are many statistical tests, students in my classes used the two-sample t-test (1, 2, 4, 7, 12, 15), onesample t-test (1, 4, 9, 15), or one-factor analysis of variance (ANOVA; 3, 4, 10, 15) to analyze their experimental data. The description of the use of other statistical tests are beyond the scope of this paper. The two-sample t-test (also known as the unpaired ttest or the independent sample t-test) was used to compare the means of the density of old versus new (post-1982) pennies (one-cent copper-alloy coins in the U.S.), while taking into account the standard deviations of the two experimental groups and the sample size. Other examples of using the two-sample t-test include comparing the experimentally determined value of the Ka of a weak acid using two different methods, and comparing the values of chloride concentration of seawater at two sites.
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In the Classroom
The one-sample t-test (also known as the paired t-test or the correlated sample t-test) was used to statistically compare the experimentally determined value to its theoretical value. Table 2 provides a comparison of a one-sample t-test versus a two-sample t-test. One example of using the one-sample t-test compares the change in the enthalpy of dissolving solid sodium hydroxide in water to its theoretical value. Other examples would be to make these comparisons with the corresponding theoretical or accepted values: 1. Experimental determination of the heat of fusion of ice
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Table 2. The Two-Sample t-Test Compares X versus Y, – While the One-Sample t-Test Compares D to Zero Samples
Group 1
Group 2
Difference
X1 versus Y1
X1
Y1
Y1 X1
X2 versus Y2
X2
Y2
Y2 X2
X3 versus Y3
X3 and so on – X
Y3 and so on
Y3 X3 and so on – – – DYX
and so on Mean
Y
2. Actual yield of silver chloride by mixing a solution of sodium chloride and silver nitrate 3. Experimental determination of the Ka of acetic acid
Acknowledgments
One-factor ANOVA is a statistical test to compare the mean of three or more groups of data in a pair-wise manner. This was not as frequently used in my class compared to the t-test. A one-factor ANOVA test detects, but does not identify, the pair(s) of data that are different. A subsequent postANOVA multiple comparison test—for example, Tukey’s test—was used to identify the specific pair(s) of data that were different. Examples of using one-factor ANOVA in my high school chemistry class include comparing the melting point of various organic compounds, and comparing the concentration of vitamin C in various drinks.
I thank the students in my past chemistry classes for demonstrating the feasibility of using statistics in a high school chemistry course. Chris Kaegi and Amy Wong (Galileo Academy of Science and Technology) reviewed the statistics and English usage in the paper, respectively. The various Pasco probes (colorimeter, pressure, and pH) were a loan from the San Francisco Unified School District Office of Teaching and Learning; the probes were purchased through a grant from the National Science Foundation Urban Systemic Program (NSF USP).
Concluding Remarks
Literature Cited
The statistical tests and concepts discussed above have been successfully implemented by my students over the past five years in the routine analysis of experimental data. The availability of statistical tools, such as Web sites, calculators, and various computer programs have tremendously eased the use of statistics in my high school chemistry courses. These tools provide the students an opportunity to analyze their experimental data like professional scientists. Occasionally, there were situations in which the results of a statistical analysis were ambiguous, that is, different conclusions were obtained by the use of the one-tail versus two-tail p-values (see the Supplemental MaterialW for a description of these p-values). Such a situation could confront professional scientists and the situation demonstrates that the practice of science involves using one’s judgment and that statistical analysis may produce ambiguous results. In addition, the use of statistics in chemistry (or another science course) provides an excellent opportunity to illustrate the interdisciplinary nature of science to students, as well as an opportunity for collaboration between science and mathematics teachers. While the focus of this article is on the use of statistics in a high school chemistry course, a similar argument applies to the incorporation of statistics in an introductory college chemistry laboratory course.
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Supplemental Material
Theoretical background in statistics, numerical examples of statistical data analysis, and student handouts are available in this issue of JCE Online. www.JCE.DivCHED.org
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