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Another measure of how well a particular line describes the relationship in bivariate data is the total of the squared errors. When comparing two lines, the line with the smaller total of the squared errors is the "better" line in terms of how well it describes the linear relationship between the two variables. For the line Height = Arm Span, this is the sum of the sixth column in the above table, which is 784.
This quantity, the sum of squared errors (SSE), is what statisticians prefer to use when comparing different lines for potential fit. If you could consider all possible lines, then the one with the smallest SSE is called the least squares line; it may also be referred to as the line of best fit.
Before we determine the SSE for the line Height = Arm Span - 1 (i.e., YL = X - 1), let's take a look at Person 1 and the line YL = X - 1:
Person 1's squared error can be represented on the graph as a square with a side whose length is |Y - YL|:
The following is the scatter plot for the data and a graph of the line YL = X - 1.
Note once again that a point above the line is indicated by a positive error; a point below the line is indicated by a negative error; and a point is on the line when the error is 0.
The following table shows the arm span (X), the observed height (Y), the predicted height based on the line Height = Arm Span - 1 (i.e., YL = X - 1), the error, and the vertical distance between the person's observed height (Y) and predicted height (YL) for Persons 1 through 6 in our study:
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