In the Laboratory
Statistical Comparison of Data in the Analytical Laboratory
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Michael J. Samide Department of Chemistry, Butler University, Indianapolis, IN 46208;
[email protected] Statistical analysis is often taught in the analytical laboratory within the context of a broader problem. For example, evaluation of sampling techniques (1) employs methods of statistical analysis as does the comparison of different analytical methods to determine the most appropriate procedure for analysis of an unknown (2–4). Finally, students can statistically compare an experimentally determined result with the known value to determine accuracy and validate a method (5, 6). These methods provide the students with valuable experience in evaluation of their data and evaluation of the method of analysis. However, one problem with the contextbased approach to statistical analysis lies in the complexity of the procedures being performed. If difficult procedures are required, students with little laboratory experience might unknowingly introduce error into their measurements. Time and cost issues can also discourage the implementation of method comparisons as a means to perform statistical analysis. Examples of more simplistic procedures designed to promote the use of statistical analysis can be found in the literature. May (7) reported on the comparison of micropipets to classical volumetric pipets. Rohrbach and Pickering (8) had their students compare the time it took to take a weight measurement on an electronic balance and a mechanical balance. Hartman (9) described an in-class demonstration of sampling techniques using candy. The simplicity of these procedures allows the student to focus on analysis of the data rather than implementation of the procedure. This article contains a description of a one-week experiment that is designed to provide students in an analytical chemistry laboratory course with an experience in statistical treatment of data. Through a comparison of 11 different volume-measurement techniques, students learn how to process raw data, check for statistical outliers, and compare data to known values as well as to other data sets. In this procedure, students must obtain six replicate measurements for each technique and then analyze their data by calculating common statistical parameters through the use of a computer spreadsheet program. A set of questions provides specific guidelines for the students to follow during the analysis of their data. Furthermore, students with little or no experience in the laboratory are able to gain valuable hands-on skills through the use of different types of analytical glassware. Experimental
Advanced Preparation Before the start of the experiment, 11 different stations were created, each with a different measuring device. (Table 1 lists the stations used for this experiment, along with the expected nominal volume to be delivered.) In addition, a large volume of water was set aside so that it could reach thermal equilibrium with the temperature of the room. Finally, each station was explained to the students and the proper use of each piece of glassware was demonstrated. www.JCE.DivCHED.org
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Student Involvement The laboratory exercise continued with the students employing each technique to deliver the specified volume of water into a clean, dry, preweighed 100-mL beaker. The weight of the beaker plus water was measured on an analytical balance to the precision of the balance. Using the density of water at a measured temperature, students converted the mass values to volumes of water delivered. Once all 11 stations were visited, and the data were collected, the students processed their data on a computer according to specific guidelines. Data Analysis and Results In all cases, the error introduced through subtraction of masses is considered to be negligible in comparison to the error introduced through student use of the measuring glassware. Therefore, we assume no error was introduced from the balances, but only from student use of the volume-measuring glassware. For each data set, a Q-test was employed to determine whether a measurement could be considered a statistical outlier. If so, the point in question was not used in subsequent calculations. Once the data were deemed acceptable, a mean, standard deviation, standard deviation of the mean, and confidence intervals at the 95% and 99% levels were calculated. Then, by working through a series of questions, students were guided through three different case scenarios involving the mathematical evaluation of their data. In each case, a t-value or an F-value was calculated (at the 95% confidence level) and compared with the appropriate values from the literature (10).
Table 1. Various Volume-Measuring Devices and the Corresponding Nominal Volume To Be Delivered by Each Techniquea
Nominal Volume/mL
Station Identification A
10-mL Graduated cylinder
10.00
100-mL Graduated cylinder
10.00
B
50-mL Beaker
10.00
C
50-mL Buretb
10.00
D
50-mL Buretb
10.00
E
10-mL Mohr pipet
10.00
F
10-mL Volumetric pipet
10.00
G
10-mL Volumetric flask
10.00
H
50-mL Repipet
10.00
I
50-mL Buretb
05.00
J
50-mL Buretb
01.00
K
a
Class A glassware used where appropriate.
b
Burets are identical to one another.
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In the Laboratory Table 2. Average Volume, Standard Deviation, and Standard Deviation of the Mean for Two Students’ Data Station ID
Student 1 x
Sx
Sx
Comparison of Student 1 to Student 2
Student 2 Accuracya
x
Sx
Sx
Accuracya
Precisionb (F-test)
Accuracyb (t-test)
A
9 .9 c
0.3
0.1
S (0.568)
9.65
0.1
0.04
D (9.798)
D (9.0)
S (1.94)
B
8.2
0.5
0.2
D (8.531)
7.72
0.1
0.03
D (80.80)
D (25)
D (2.31)
C
9.4
0.6
0.2
S (2.394)
7.3
0.8
0.3
D (8.427)
S (0.56)
D (5.14)
D
10.00
0.09
0.04
S (0.000)
9.98
0.1
0.04
S (0.500)
S (0.81)
S (0.364)
E
10.01
0.09
0.04
S (0.230)
9.98
0.2
0.07
S (0.321)
S (0.20)
S (0.335)
F
10.01
0.05
0.02
S (0.665)
10.02
0.03
0.01
S (1.188)
S (2.8)
S (0.420)
G
9.98
0.02
0.01
S (1.939)
10.004
0.009
0.004
S (1.050)
S (4.9)
D (2.68)
H
9.80
0.03
0.01
D (17.34)
0.02
0.01
D (15.87)
S (2.3)
D (2.72)
Id
10.15
0.06
0.02
D (6.433)
0.3
0.1
D (5.993)
S (0.040)
D (2.60)
9.84 10.6
a
This refers to a comparison of the mean volume with the expected target volume of 10 mL. For each trial, a t-value was calculated (shown in parentheses) and compared to the literature at the 95% confidence level (ttable = 2.571 for 5 degrees of freedom). S = statistically identical values and D = statistically different values. b These columns refer to a comparison of the two students’ use of a particular measuring device. For each device, an F-value and t-value were calculated (both shown in parentheses) and these were compared to the literature at the 95% confidence level (Ftable = 5.05 and ttable = 2.228 for 10 degrees of freedom). S = statistically identical values and D = statistically different values. c All values have units of mL. dStations J and K are not included because the delivery volume is not 10 mL and no student–student comparisons were made with these data.
Case 1. Comparison of Two Students’ Results with a “Known” Value The mean, the standard deviation, and the standard deviation of the mean for two students’ individual trials for stations A through I are displayed in Table 2. The “Accuracy” column in Table 2 indicates whether the reported mean volumes were statistically different or statistically identical to the target volume of 10 mL. As can be seen, only the calibrated (class A) volumetric glassware consistently delivered accurate volumes of water (tcalc < 2.571), though in some cases careful technique allowed for good precision with beakers and graduated cylinders. Case 2. Comparison of Two Students’ Technique with a Single Device In this set of comparisons, students examined whether a specific measuring device gives identical results even if two different students perform the measurement. First, an F-test (Ftable = 5.05) was applied to determine whether the precision of each data set was statistically similar. Then, an appropriate t-value was calculated and compared with the literature (ttable = 2.228). This method of comparison was applied to each specific measuring device separately and the final two columns of Table 2 list the results of these comparisons. The students seemed to use each device with identical precision, but accuracy varied for most glassware. Case 3. Comparison of One Student’s Use of Multiple Measuring Devices To compare the different measuring devices with one another, an F-test (Ftable = 5.05) was performed so that the precision obtained through the use of each measuring device could be compared. Then the calculated t-value was compared
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to the literature value for 10 degrees of freedom (ttable = 2.228). The results in Table 3 as well as those discussed below are calculated using one student’s data, but the same general trend can be seen for most students who perform this experiment. Techniques A and B delivered statistically different volumes of water (tcalc = 21.01), whereas techniques D and E dispensed statistically identical volumes (tcalc = 0.709). When comparing the two pipets, it is seen that they both delivered statistically identical volumes of water (tcalc = 1.365), but there was a significant difference in the precision (F = 6.25). This was most likely due to the fact that one must start and stop the liquid level with the Mohr pipet, compared to simply starting the liquid for the volumetric pipet. While the pipets and the burets delivered the same volume of water, there was
Table 3. Comparison of the Precision and Accuracy of One Student’s Use of Different Measuring Techniques Comparison
Precision (F-test)a
Accuracy (t-test)a
A vs B
S (2.78)
D (21.01)
D vs E
S (1.00)
S (0.709)
F vs G
D (6.25)
S (1.365)
D vs F
S (3.24)
S (0.238)
D vs G
D (20.3)
S (0.531)
a These columns refer to a comparison of two different measuring devices. For each comparison, an F-value and t-value were calculated (both shown in parentheses) and these were compared to the literature a t t h e 9 5 % c o n f i d e n c e l e v e l ( Ft a b l e = 5 . 0 5 a n d t t a b l e = 2 . 2 2 8 f o r 1 0 degrees of freedom). S = statistically identical values and D = statistically different values.
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In the Laboratory
Summary
4
Relative Error (%)
3
2
1
0 0
2
4
6
8
10
Delivered Volume / mL Figure 1. A plot of relative error versus volume showing the expected trend of increased relative error for small volume measurements.
This laboratory exercise proved useful in several specific areas. First, students gained familiarity with analytical glassware that they had not seen or used before. Because wetchemical methods involve proper use of analytical glassware, the skills learned during this experiment will benefit the students throughout the semester. Furthermore, most students were able to generate data that allowed them to identify trends in accuracy and precision. In many instances, this experiment opened students eyes so that they will no longer blindly trust that a volume-measuring device will repeatedly deliver the desired volume with any sort of accuracy. This fact was driven home by one student who commented in his concluding statements, “The only dilutions I would use a beaker for is in making Kool-Aid.” Finally, students gained experience with a spreadsheet program. First, students had to create their own spreadsheets by programming the appropriate formulas. Once completed, they then processed their data with that spreadsheet. For most students, this was their first experience using computer software to analyze data and calculate values. W
a large difference in the precision of the buret versus the precision of the volumetric pipet. Technique C, the 50-mL beaker, is not a calibrated piece of glassware. As a result of the large standard deviation, this technique should not be used to deliver exactly 10.00 mL of liquid. Technique H is expected to deliver less volume than expected because this flask is calibrated to contain 10.00 mL rather than to deliver 10.00 mL.
Supplemental Material
Instructions for the students, notes for the instructor, and a materials and equipment list are available in this issue of JCE Online. Literature Cited 1. 2. 3. 4.
Examination of Relative Error
5.
Figure 1 shows a plot of relative error versus volume from one student’s data for techniques D, J, and K. Techniques D, J, and K are the delivery of 10.00, 5.00, and 1.00 mL of liquid using a 50-mL buret. This graph reveals that as volume decreases, relative error in the measured volume increases. Here we assume no error was introduced from the balances, but only from the volume-measuring glassware.
6. 7. 8.
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9. 10.
Harvey, D. J. Chem. Educ. 2002, 79, 360–363. Park, B. J. Chem. Educ. 1958, 35, 516–517. Beilby, A. L. J. Chem. Educ. 1972, 49, 679–681. Harrison, A. M.; Peterman, K. E. J. Chem. Educ. 1989, 66, 772–773. Edmiston, P. L.; Williams, T. R. J. Chem. Educ. 2000, 77, 377– 379. Sheeran, D. J. Chem. Educ. 1998, 75, 453–456. May, L. J. Chem. Educ. 1980, 57, 483. Rohrbach, D. F.; Pickering, M. J. Chem. Educ. 1982, 59, 418. Hartman, J. R. J. Chem. Educ. 2000, 77, 1017–1018. Harris, D. C. Quantitative Chemical Analysis, 6th ed.; W. H. Freeman and Company: New York, 2003.
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