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A Good Enough Fit?

head shot of andrewThe residual values will have a sum of approximately 0.  When analyzing residuals, you want residuals to be as small as possible.  The smaller the residuals, the closer the line (or curve) is to the actual values. 

It is important to also examine the residual plot. You can find and analyze residuals with your graphing calculator, graphing applet or the Shodor Applet opens in new window.

Andrew plots the residuals for the linear model and sees the following:

This graph is a residual plot displaying the residuals for the Boys 2 A Cross Country state championship when a linear regression equation is applied to the data.  The horizontal axis represents the years since 2007 and extends from negative 1 to 7.  The vertical axis represents the residual values and extends from negative 0.6 to 1.  The graph displays the following ordered pairs:  (0, negative 0.47), (1, negative 0.014), (2, 0.788), (3, 0.102), (4, negative 0.274), and (5, negative 0.16).

 

On the residual plot, the points should appear to be randomly dispersed around a horizontal axis.  If a pattern emerges from the plot, this is an indication that the model selected may not be a best fit for the data.

In the residual plot for winning times, the data is fairly small and fairly random.  This indicates that a line may be the best association for this data set. 

Here are some samples of residuals plots that are not random:

Five residual plots are displayed, each of which extend on the horizontal axis from 0 to 12 and on the vertical axis from negative 1 to 1.  Graph 1 includes ordered pairs (1, 0.05), (2.9, 0.2), (3, 0.2), (5, 0.4), (6, negative 0.5), (7, 0.7), (8.9, 0.9), (9.1, negative 0.9), (10.9, 1), (11.2, negative 1).  Graph 2 includes ordered pairs (1, negative 1), (3, 1), (5, negative 0.8), (6, 0.6), (7, negative 0.5), (8.9, 0.4), (9, negative 0.2), (11, 0.2), (11, 0.05).  Graph 3 includes ordered pairs (1, negative 0.8), (3, negative 0.5), (5, 0.3), (6, 0.5), (7, 0.5), (9, 0.3), (11, negative 0.1), (12, negative 0.75).  Graph 4 includes ordered pairs (1, negative 0.5), (3, negative 0.3), (5, negative 0.1), (6, 0.1), (7, 0.3), (9, 0.5), (11, 0.75), (12, 0.85).  Graph 5 includes ordered pairs (1, 0.8), (3, 0.7), (5, 0.5), (6, 0.4), (7, 0.1), (9, negative 0.1), (11, negative 0.3), (12, negative 0.5).

Real-Life Scenarios



> Text version for Real-Life Scenario opens in new window

Andrew must also examine the correlation coefficient before making a final decision about the strength of the model.

 

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