In the Classroom
A Simple Method for Illustrating Uncertainty Analysis Paul C. Yates School of Chemistry and Physics, Lennard–Jones Laboratories, Keele University, Keele, Staffordshire ST5 5BG, UK;
[email protected] The ability to correctly estimate and manipulate experimental errors, or as they are more correctly termed uncertainties, is an important skill that students need to acquire in the physical chemistry laboratory. It is reasonable to suppose that the treatment of data that students have acquired for themselves would be most effective, but when an instructor wishes to illustrate the principles involved in error analysis he or she needs a simple and fast method for generating data. Such needs have prompted others to develop exercises such as weighing pennies (1) and determining the length of an agate pestle (2). The Method Another simple way to generate data is to provide each student with a rectangle of unspecified but common dimensions drawn by the instructor. If the set of rectangles required by a class is drawn individually by hand, there should be enough variation from one to another (provided the instructor is not too careful!) to introduce slight differences into the dimensions determined by each student. Students are asked to measure the lengths of the rectangle and calculate its perimeter and area. The results of the class are then compared. If the class is small this is done by all the students present, or if larger they can be split into groups. This leads into a discussion of simple statistics culminating in a calculation and discussion of the standard deviation. An example calculation is given below, where the standard deviation is determined for both the perimeter and the area. Students are then asked to consider a scenario in which they do not have other sets of measurements. After deciding upon the precision with which each length can be determined with the ruler, they are asked to determine the uncertainty in the perimeter and area of the rectangle, using the propagation of error theorem (3). The determination of uncertainties in both perimeter and area allows a demonstration of the rules that are used to combine absolute and relative uncertainties, as shown by the example below. This method can be implemented easily for classes of any size. The students generate the data for themselves with a minimum of equipment, and various types of uncertainty analysis can be investigated. It is thus highly suitable for courses in basic mathematics or data analysis that do not contain a laboratory component. An extension to the method would be to determine the length of the diagonal of the rectangle. This could then be compared with its predicted value using the Pythagorean theorem and the measured values of the length and breadth as outlined elsewhere (4). Other features that could also be investigated include maxima, minima, and quartiles. Example Method 1. Statistical Analysis Ten readings of lengths 艎1 and 艎2 were obtained, with corresponding values of the perimeter p and area A (Table 1). 770
Beginning with the calculation of the standard deviation of the perimeter, we have the mean value p¯ = 28.28 cm. The other values needed for the calculation are shown in Table 2. From this we have Σ(p – p¯ )2 = 0.576 cm2, and so the standard deviation s is given by
s2 = Σ
p – p 2 0.576 = = 0.064 cm2; n –1 9
s = 0.252 cm
The value of p¯ expressed in terms of a 95% confidence interval (5) is then
p = 28.28 ± t .05
s = 28.28 ± 2.26 × 0.252 3 n –1
so the final value of the perimeter can be expressed as p¯ = 28.3 ± 0.2 cm Similarly, for the area, the average value A¯ is 48.31 cm2. The required values for the calculation of the standard de¯ 2 = 6.9812 cm4. viation are shown in Table 3, with Σ(A – A)
Table 1. Data for Statistical Analysis ᐉ1/cm
ᐉ2/cm
p/cm
A/cm2
8.15
5.80
27.90
47.27
8.35
5.80
28.30
48.43
8.30
5.80
28.20
48.14
8.40
5.95
28.70
49.98
8.45
5.80
28.50
49.01
8.55
5.75
28.60
49.16
8.35
5.70
28.10
47.60
8.35
5.80
28.30
48.43
8.30
5.75
28.10
47.31
8.30
5.75
28.10
47.73
Table 2. Data for SD of Perimeter (p – p¯)/cm (p – p¯)2/cm2 0.38
0.1444
0.02 0.08
Table 3. Data for SD of Area (A – A¯)/cm2 (A – A¯)2/cm4 1.04
1.0816
0.0004
0.12
0.0144
0.0064
0.17
0.0289
0.42
0.1764
1.67
2.7889
0.22
0.0484
0.70
0.4900
0.32 0.18
0.1024
0.85
0.7225
0.0324
0.71
0.5041
0.02 0.18
0.0004
0.12
0.0144
0.0324
1.00
1.0000
0.18
0.0324
0.58
0.3364
Journal of Chemical Education • Vol. 78 No. 6 June 2001 • JChemEd.chem.wisc.edu
In the Classroom
Substituting this into the formula for s 2 gives
s2 = Σ
2
A –A = 6.9812 = 0.7757 cm4 9 n –1
and so
s = 0.7757 cm4 = 0.881 cm2
The perimeter can thus be expressed as p = 27.9 ± 0.1 cm. The area is found from the equation A = 艎1 × 艎2, so the absolute error e on this quantity is given by
e = 47.27 cm2
and so finally
2
=
3.763 × 105 + 7.432 × 105 =
allowing the area at the 95% confidence level to be expressed as
A = 48.31 ± t .05
0.05 cm 2 + 0.05 cm 8.15 cm 5.80 cm
s = 48.31 ± 2.26 × 0.881 3 n –1
1.195 × 104 = 0.0106 Consequently
A¯ = 48.3 ± 0.7 cm2
Method 2. Calculation of Maximum Probable Error This example refers to the first data set in Table 1—the set of measurements that has 艎1 = 8.15 cm, 艎2 = 5.80 cm, p = 27.90 cm, A = 47.27 cm2. It is reasonable to assume that the uncertainty in each individual measurement (艎1 and 艎2) is ± 0.05 cm. Since four values with this level of uncertainty are added together to obtain the perimeter, the overall uncertainty in the perimeter is 0.05 cm 2 + 0.05 cm 2 + 0.05 cm 2 + 0.05 cm 2 = 0.10 cm
e = 0.0106 × 47.27 cm2 = 0.50 cm2 and the final expression of the area is A = 47.3 ± 0.5 cm2. Literature Cited 1. Richardson, T. H. J. Chem. Educ. 1991, 68, 310–311. 2. O’Reilly, J. E. J. Chem. Educ. 1986, 63, 894–896. 3. Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in Physical Chemistry, 6th ed.; McGraw-Hill: New York, 1996. 4. Sauls, F. C. J. Chem. Educ. 1990, 67, 958–959. 5. Wonnacott, T. H.; Wonnacott, R. J. Introductory Statistics, 5th ed.; Wiley: New York, 1990.
JChemEd.chem.wisc.edu • Vol. 78 No. 6 June 2001 • Journal of Chemical Education
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