Readily available programs, such as Excel, allow one to determine the line of best fit for a set of data with the click of a button. While this "black-box" approach removes the drudgery of calculation, it does little to facilitate conceptual understanding. The linear least-squares expressions, which are used to determine the line of best fit, may be obtained and understood using analogies from elementary mechanics. Two different mechanical analogies are used to obtain the linear least-squares expressions. The first analogy demonstrates that if the difference between an experimental value and the line of best fit is considered as a force, and if the product of this experimental value and the corresponding difference is considered as a moment of this force, the linear least-squares expressions may be obtained by setting the net resultant force, and the net momentum, equal to zero. The second analogy demonstrates that if the line of best fit is represented by a thin rod, the experimental values are represented by fixed pegs on a board, and the differences between the experimental values and the line of best fit are represented by springs which connect the pegs to the rod, the linear least-squares expressions may be obtained by determining which expressions correspond to the minimum potential energy in the springs. Since only elementary algebra is necessary in each case, this approach facilitates conceptual understanding and enables the introduction of these concepts at the high-school and introductory college level.

Conceptual understanding of the linear least-squares expressions by means of mechanical analogs

255th National Meeting (2018)

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