Section 5: The Line of Best Fit

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 3.5 The Line of Best Fit 313 Version: Fall 2007 3.5 The Line of Best Fit When gathering data in the real world, a plot of the data often reveals a “linear trend,” but the data don’t fall precisely on a single line. In this case, we seek to find a linear model that approximates the data. Let’s begin by looking at an extended example. Aditya and Tami are lab partners in Dr. Mills’ physics class. They are hanging masses from a spring and measuring the resulting stretch in the spring. See Table 1 for their data. m (mass in grams) 10 20 30 40 50 x (stretch in cm) 6.8 10.2 13.9 21.2 24.2 Table 1. Aditya and Tami’s data set. The goal is to find a model that describes the data, in both the form of a graph and of an equation. The first step is to plot the data. Recall some of the guidelines provided in the first section of the current chapter. Guidelines. When plotting real data, we follow these guidelines. 1. You don’t want small graphs. It’s best to scale your graph so that it fills a full sheet of graph paper. This will make it much easier to read and interpret the graph. 2. You may have different scales on each axis, but once chosen, you must remain consistent. 3. You want to choose a scale which facilitates our first objective, but which also makes the data easy to plot. Aditya and Tami are free to choose the masses which they hang on the spring. Hence, the mass m is the independent variable. Consequently, we will scale the hori- zontal axis to accommodate the mass. The distance the spring stretches depends upon the amount of mass that is hanging from the spring, so the distance stretched x is the dependent variable. We will scale the vertical axis to accommodate the distance stretched. On the horizontal axis, we need to fit the masses 10, 20, 30, 40, and 50 grams. To avoid a smallish graph, we will let every 5 boxes represent 10 grams. On the vertical axis, we need to fit distances ranging from 6.8 centimeters up to and including 24.2 centimeters. Making each box represent 1 cm gives a nice sized graph and will allow for easy plotting of our data points, which we’ve done in Figure 1 (a). Note the linear trend displayed by the data in Figure 1 (a). It’s not possible to draw a single line that will pass through every one of the data points, so a linear model will not exactly “fit” the data. However, the data are “approximately linear,” so let’s try to draw a line that “nearly fits” the data. It is not our goal here to try to draw a line that passes through as many data points as possible. If we do, then we are essentially saying that the points through which
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314 Chapter 3 Linear Functions Version: Fall 2007 the line does not pass have no influence on the overall model, nor do they have any influence on any predictions we might make with our model. This is not a reasonable assumption.
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