Estimating the Average Slope

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The slope is usually the parameter of primary importance in a simple linear regression. If the straight line model gives a poor fit to the data, one can consider the average slope of the non-linear response. In this paper, we show that if the response is quadratic, then the average slope can be obtained by simply using the slope from a straight line fit. In fact, if the slope of the best fitting line to a smooth non-linear function equals the average slope of the function over an arbitrary interval, then the function must be quadratic. This paper illustrates the case where intentionally fitting a wrong model (in this case, a straight line) gives the correct result (the average slope). The example which motivated this study is used to illustrate the results.