. Elementary biophysics: selected topics. Biophysics. 20 THE MATHEMATICAL TREATMENT OF DATA TREATMENT OF EXPERIMENTAL DATA 1. Data-fitting using the least-squares principle The least-squares principle was used to obtain an index of the varia- tion in data. It has other uses, one of which is the subject of this sec- tion. The problem of finding a theoretical expression to fit data utilizes this principle to choose the parameters of the curve so as to minimize the variation around the curve. We illustrate the approach by finding the expressions for the parameters of a straight line which is to b
. Elementary biophysics: selected topics. Biophysics. 20 THE MATHEMATICAL TREATMENT OF DATA TREATMENT OF EXPERIMENTAL DATA 1. Data-fitting using the least-squares principle The least-squares principle was used to obtain an index of the varia- tion in data. It has other uses, one of which is the subject of this sec- tion. The problem of finding a theoretical expression to fit data utilizes this principle to choose the parameters of the curve so as to minimize the variation around the curve. We illustrate the approach by finding the expressions for the parameters of a straight line which is to be fitted to the data. The equation of a straight line is y = a + bx. Consider the difference between the theoretically and the experimentally determined y: 2/th — */exP = a + bx — yQXp. These differences are sketched in Fig. 8. Our principle of least squares tells us to square this difference, add all the squared differences for all the points, and then to choose a and b so as to minimize the sum. For those with mathematical facility, the deri- vation of a and 6 will be presented. For other readers, the results are given below. !lth = (i+bx. Fig. 8. A plot of some experimental data relating y and x and a theoretical line whose slope and intercept are to be chosen to minimize the total deviation of the theoretical values of y from the experimental values of y. The short vertical lines are the deviations themselves: ?/th — ? Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Epstein, Herman T. (Herman Theodore), 1920-. Reading, Mass. , Addison-Wesley Pub. Co
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