In the Laboratory
Quantitative Comparison of Three Standardization Methods Using a One-Way ANOVA for Multiple Mean Comparisons W Russell D. Barrows Department of Chemistry, Metropolitan State College, Denver, CO 80217;
[email protected] The use of an external standard calibration curve (1–3), standard addition (4, 5), and internal standard (6, 7) methods for quantitatively determining the concentration of analytes is important to the study of analytical chemistry, especially in the area of instrumental analysis. However, as important as these three methods are, few analytical or instrumental laboratory programs have the time for students to gain adequate laboratory experience with each of the three methods. The following laboratory experiment allows for students to gain experience with all three quantitative methods in a single laboratory period using the gas chromatograph (GC), and moves their exposure to these methods from traditionally scripted laboratory exercises to a more inquiry-based, comparative-learning experience (8). This is achieved by requiring students to determine the concentration of three long-chain hydrocarbons, in a single unknown sample, by each of the three methods of standardizations using only 10 GC vials and an experimental procedure of their own design. To determine whether or not the three standardization methods are statistically different in determining the concentration of the three paraffin analytes, a one-way ANOVA is performed. Knowledge of rudimentary statistical methods and direct comparison of experimental data is vital for undergraduate chemistry students. This Journal supports this end through a number of statistical experiments (9, 10). Students must be able to determine for themselves the precision or accuracy of their final numbers or results (11, 12). Unless they determine the “goodness” of their numbers, the answers they generate will not help students assimilate the analytical techniques presented, nor aid their self-evaluation of what they have learned. This is one of the main points to inquiry-based learning (13). In particular, comparison of means between two or more sets of data are commonly used in analytical chemistry to find out whether the means are the same or different at a predetermined confidence level (14). Over the past 20 years, a number of articles have shown the benefits of requiring students to design and test their own analytical procedures as opposed to merely performing an analysis where individual steps are provided, and their grade is based solely on how close their determined results approach an accepted value, as told to them by the instructor (8, 15– 17). One of the current forefronts in the pedagogy of analytical chemistry is to show students how to systematically solve problems (18, 19). Toward this effort, a number of innovative problem-based laboratory projects have been reported that emphasize the explicit use of the scientific method and invite students to generate and test their own hypotheses (15, 18, 20); however, a common feature of many of the problem-based laboratories is their length (1). Independent projects involving the analysis of “real-world” samples severely crowd an instrumental laboratory program that needs to expose students to the many facets of the field (e.g., chromatography, spectroscopy, atomic absorption, electrochemistry, X-ray analysis, www.JCE.DivCHED.org
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and statistical analysis) (21, 22). Therefore, a need exists for shorter laboratory experiments emphasizing the scientific inquiry process so that students may be successful in later problem-based projects (1, 23). To these ends, the goals of the experiment are for students (i) to gain meaningful experience with the three commonly used methods of standardization in a single laboratory period, (ii) to use an experimental procedure of their own design, and (iii) to perform a statistical analysis on their results to determine for themselves whether they have been successful in their experimental design and execution. The following laboratory exercise is recommended for instrumental analysis students. Experimental To give students exposure to the three quantitative methods in a non-scripted laboratory exercise, the experiment asks students to determine the concentration by GC of three longchain hydrocarbons—octadecane (C18), eicosane (C20), and docosane (C22)—contained in a single unknown sample. As the students are given only ten GC vials to perform the entire experiment, they are forced to compare and contrast the experimental requirements of the three standardization methods to devise their own analytical procedure to determine the concentrations of the three analytes. The lab exercise can be performed in a single lab period with the students working in pairs. Matrix effects are not a concern in this experiment because the solvent used in all samples is heptane, and the analytes and the internal standard are all long chain hydrocarbons. Therefore, from the homogeneity of the samples, it is extremely unlikely that the analytes would give variable responses in the GC analysis of the ten allowed vials. However, in real-world samples with serious matrix effects, the one-way ANOVA method would reveal discrepancies between the three standard methods since the internal standards and standard addition approaches are both designed in part to compensate for matrix variability.1
Pre-Lab Completion of the lab in one laboratory period depends on the experimental procedure that students develop before coming to the laboratory class. The more successful designs often use the following configuration. Four of the ten vials create the external standard calibration curves and determine the detector response factors needed for the internal standard method. Heptadecane, C17, is used as the internal standard. The next four vials are used to prepare standard addition curves for the three analytes, leaving two vials for the unknown mixed with the internal standard, C17.2 The laboratory directions inform students that their unknown sample will consist of 100 to 300 ppm of octadecane
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In the Laboratory
(C18), eicosane (C20), and docosane (C22) in heptane. Stock solutions of approximately 7,000 ppm of C18, C20, and C22, in heptane are provided, along with an approximately 7,000 ppm of heptadecane, C17, in heptane to be used as the internal standard.
GC Analysis Once the students have prepared their ten GC vials, they are placed in the tray of a Hewlett Packard (HP) 6890 auto sampler connected to a HP 6890 GC fitted with a flame ionization detector (FID), controlled by HP ChemStation. Helium, the carrier gas, is set at a constant flow of 4.0 mLs in a HP-5 (cross-linked 5% PH Me siloxane) 30 m × 0.32 mm (0.25 µm film) column. The oven program was initialized at 60 ⬚C for 1 min then programmed to a temperature of 200 ⬚C at a rate of 20 ⬚Cmin with a final time of 5 min. Splitless injection of 2 µL is made with a split delay of 0.5 min (purge flow 20:1). Results are stored in a HP ChemStation file and retrieved the following day.
Table 1. Data Matrix for One-Way ANOVA Standard Curve (ppm)
Internal Standard (ppm)
Standard Addition (ppm)
C18
179.3
176.4
135.3
C20
169.4
162.7
142.8
C22
176.1
172.2
151.1
Table 2. ANOVA Summary Groups
Count
Sum
Average
Variance
Standard Curve
3
524.8
174.933
25.523
Internal Standard
3
511.3
170.433
49.263
Standard Addition
3
429.2
143.067
62.463
Table 3. ANOVA Results
Hazards There are no significant hazards associated with this laboratory beyond those normally associated with an alkane solvents and long-chain paraffins. Preparation of heptane solutions should be performed in a well-ventilated hood to avoid inhalation of heptane vapors, which could lead to throat irritation and may cause headache, dizziness, and drowsiness. Exposure limits of 10,000 ppm may cause intoxication and narcosis. Sample prep area must be free of all flames and ignition sources: heptane flash point, ᎑4 ⬚C, auto ignition temperature, 204 ⬚C. Students should use appropriate caution when handling any chemical; safety glasses are mandatory and safety gloves are recommended. Analysis of the Results To determine whether or not the three standardization methods are statistically different in determining the concentration of the three paraffin analytes, a one-way ANOVA is performed using the Microsoft Excel’s statistical macro ANOVA: Single Factor (located in the Data Analysis ToolPak of Excel) (24). Students realize that by receiving only one unknown sample that the three quantitative methods should give similar values for the three analytes, at a 95% confidence level. This is the null hypothesis (H0: ⫺ X1 = ⫺ X2 = ⫺ X3) that the students work toward. If students find that their results do not support the null hypothesis then they use the Tukey multiple-comparison method for obtaining confidence intervals for the differences between means to show where their experimental design or execution failed (25).1,3 However, visual inspection of the data, or graphic representation of the data, is often sufficient to identify the problem area. As an example of the statistical analysis students perform on their results, consider the following data in Table 1. Using Microsoft Excel’s statistical macro, ANOVA: Single Factor (α = 0.05) gives the following results in Tables 2 and 3 (24). The results in Table 3 show that the null hypothesis should be rejected because the calculated P value of 0.002 is less than the set α value of 0.05, and the F value (19.50) is greater than the F(crit) value (5.14). With the rejection of the null hypoth840
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Source of Variation
SS
df
MS
F
P Value
F(crit)
Between Groups 1784.7 2 892.33 19.505 0.0024 5.143 Within Groups
0274.5 6 045.75
Total
2059.2 8
NOTE: The null hypothesis is rejected: F(19.50) > F(crit)(5.143), and P value (0.002) < α = 0.05.
esis, a student knows that an error was made either in their analytical procedure or in the preparation of their GC vials. To find the source of error, students perform a Tukey multiple-comparison method for obtaining confidence intervals for the differences between the means (25). The first step is to obtain the critical value for the comparison of two of the three means. This value, qα, can be found in a studentized range distribution, or simply the q-curve distribution (α = 0.05, q0.05) using the two parameters κ = k, where k is the number of factors, and ν = n − k, where n is the total number of observations (26). Using the values from Table 1, κ = k = 3 and ν = n − k = 9 − 3 = 6. In a q-curve distribution table (α = 0.05), qα = q0.05 = 4.34. Next, the endpoints of the confidence interval for –x 1 − –x 2 (mean of standard curve – mean of internal standard) are obtained using
(x i
)
− x j ±
S =
qα
MS =
2
S
1
ni
+ 1
nj
(1)
45.75
(Table 3, MS within groups) The endpoints of the confidence interval for –x 1 − –x 2, with qα = 4.34 (α = 0.05), from eq 1, are
(174.93
− 170.4 43) ±
4.34 45.75 1 3 + 1 3 2
= 4.50 ± 16.95
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In the Laboratory
or ᎑12.45 to 21.45. Proceeding in the same way, the remaining confidence intervals for the three quantitative methods are shown in Table 4. Two means are said to be equal (x–1 − –x 2 = 0) if the confidence interval for their difference contains zero and not equal if the difference does not contain zero: (x–1 − –x 3 > 0) and (x–2 − –x > 0) (27). Therefore, the student knows that an error was 3 either made in their analytical procedure for standard addition or preparation of the GC sample for this method. Summary An approach is presented that allows students to gain laboratory experience with three important standardization methods—external standard calibration curves, internal standards, and standard additions—in a single laboratory period. The described laboratory exercise asks students to combine the three methods in a single analytical procedure of their own design to determine the concentration of three long-chain hydrocarbons in a single unknown sample. What forces students to compare and contrast the experimental requirements of the three quantitative methods is that they are given only ten GC vials to perform the entire GC analysis. Using a oneway ANOVA statistical analysis, students determine for themselves whether or not the three standardization methods are statistically different (at a 95% confidence level) in measuring the concentration of the three analytes, C18, C20, and C22. W
Supplemental Materials
The following materials are available in this issue of JCE Online: • The instructional handout given to students, which gives a simple explanation on how to perform a one-way ANOVA analysis using Microsoft Excel and use of the Tukey multiple-comparison method • A one-way ANOVA analysis with a student’s data that does not reject the null hypothesis • A mathematical description of a one-way ANOVA analysis using the lab data shown in Table 1 • Appendix A gives individual steps from standard and unknown sample preparation to the final one-way ANOVA analysis, including steps for calculating the concentrations of the three analytes by the three standardization methods.
Notes 1. See Sample Matrix Effects in the Supplemental Material.W 2. See Appendix A in the Supplemental Material.W 3. Calculation of the parameters in a one-way ANOVA, the Tukey multiple-comparison method, and their meaning are covered in lecture before the experiment is performed.
Literature Cited 1. Harvey, David T. J. Chem. Educ. 2002, 79, 613–615. 2. Galbán, Javier. J. Chem. Educ. 2004, 81, 1053–1057. 3. Taylor, V. A. An Introduction to Error Analysis:The Study of Uncertainties in Physical Measurements, 2nd ed.; University Science Books: Mill Valley, CA, 1977; Chapter 8.
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Table 4. Simultaneous 95% Confidence Intervals for the Difference between the Three Quantitative Methods Standard Curve Internal Standard
᎑12.45 to 21.45
Standard Addition
14.92 to 48.81
Internal Standard 10.42 to 44.32
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