|
Because statistical concepts such as least-squares fitting are often experienced as abstract, their visualization can be a helpful didactic tool, provided that this is done correctly. Unfortunately, the example in Figure 1 on page 1884 of the December 2006 issue of JCE (1) fails the latter test. The example shows six data points, including zero absorbance A at zero concentration C. However, the graph of those data, labeled "Beer's Law Plot", shows only five, missing the point at the origin. When one analyzes the six data points according to the Beer–Lambert law, that is, with A = kC where k (in the example spreadsheet called the "slope") is the product of molar absorptivity and optical path length, one finds that k = 4.084 with a standard deviation in k of 0.059, and a standard deviation of 0.016 for the overall fit of the function to these data. But the actual example given does two things: it apparently excludes the first data point from the analysis (because otherwise one does not recover the numerical answers shown), and it adds a constant "intercept" term to the analysis. This leads to a slope of 3.99 ±0.13, and an intercept of 0.014 ±0.016, where the values are given with ± one standard deviation. Clearly, when the intercept has an absolute value that is smaller than its standard deviation, it has no statistical significance. Moreover, the Beer–Lambert law contains no such intercept. Both the data set (by excluding its first data point) and the model used to analyze that set (A = a + kC instead of A = kC) are incorrect. This even shows in the overall fit, which now has a standard deviation of 0.019, larger than the 0.016 found with fitting all six data points properly to the Beer–Lambert law. Incidentally, the data are described as "data and results from Beer–Lambert experiments", and the reader must therefore assume that they are experimental rather than "made up" data. When looked at closely, they exhibit a clearly nonlinear trend, which is approximately described by a quadratic expression: A = kC + k'C2. Indeed, when the data are fitted to such a model (which also has two adjustable parameters, slope and curvature rather than slope and intercept) one finds k = 4.53 ±0.12 and k' = 2.67 ±0.72, with a standard deviation of the overall fit of only 0.0086. Therefore, even on a purely empirical basis, the two-parameter model used in the example is nonoptimal for the data shown. Instead, the latter analysis strongly suggests that more attention may have to be paid to using correct instrumental settings in measuring the absorbance before these are subjected to any statistical analysis. Finally, the heading and text of the accompanying editorial comments (2) seem to imply that least-squares analysis on a spreadsheet is synonymous with trendline. This is unfortunate, because such an implication would be incorrect. Excel (3) has two scientifically useful least-squares routines, the function LinEst and the macro Regression. It also has Trendline, a tool for financial predictions. While all three rely on the same algorithm, Trendline is the only one that does not display uncertainty estimates for its coefficients, and is therefore unsuitable for scientific applications, in which one not only needs to have numerical answers, but also estimates of their individual uncertainties. Literature Cited- Kim, M. S.; Burkart, M.; Kim, M.-H. J. Chem. Educ. 2006, 83, 1884.
- Coleman, W. F.; Fedosky, E. W. J. Chem. Educ. 2006, 83, 1884.
- Microsoft Office Excel Home Page (accessed Dec 2007).
|
Always in the