Bivariate Data and Analysis Part D: Fitting Lines to Data (60 minutes)

Problem D4
Complete this table for the remaining 18 people. When you click “Show Answers,” the filled-in table will appear below the problem. Scroll down the page to see it.

Here are some observations about this table:

One measure of how well a particular line describes the trend in bivariate data is the total of the vertical distances. When comparing two lines, the line with the smaller total of the vertical distances is the “better” line in terms of how well it describes the linear relationship between the two variables. For the line Height = Arm Span (i.e., YL = X), this is the sum of the sixth column in the above tables combined, which is 100.

But perhaps people aren’t really “square.” Might a better prediction be that height is one centimeter shorter than arm span? Let’s see how well the line Height = Arm Span – 1 (i.e., YL = X – 1) describes the trend.

The following table shows the arm span (X), the actual observed height (Y), the predicted height based on the line 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:

Problem D5
Complete the table for the remaining 18 people. Then compute the total vertical distance for the line Height = Arm Span – 1, and compare the result to the total vertical distance for the line Height = Arm Span. Based on your calculations, which line provides the better fit? When you click “Show Answers,” the filled-in table will appear below the problem. Scroll down the page to see it.

In This Part: The SSE
Another way to see how close an individual’s data point is to a line is to square the error. This is similar to how you calculated the variance in Session 5, where you squared the distances from the mean. Like the absolute value, each squared error produces a positive number. Again, for each individual point, the smaller the squared error, the closer the actual data point is to the line. Here are the squared errors for Persons 1 through 12:

Problem D6
Complete the table to find the squared error for the remaining 12 people.

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:

Problem D7
Complete table below for the remaining 18 people. Then compute the sum of the squared errors for the line Height = Arm Span – 1, and compare the result to the sum of squared errors for the line Height = Arm Span. Based on your calculations, which line provides the better fit?

Video Segment
In this video segment, Professor Kader introduces two rules: the sum of errors and the sum of squared errors. He explains that these are used to evaluate how well any given line fits a data set and how well each line can predict the value of one variable when the value of the other variable is known.

Here is the table for the line YL = X – .7:

For this line, the sum of squared errors is 770.56, which makes it a slightly better model than the line YL = X – 1 (whose SSE was 772).

Problem D8
Here are the three lines we’ve considered, plus two new ones:

We have examined several lines that have yielded different SSEs. The lines, however, had one thing in common: they all had a slope of 1, so they were all parallel. Keep in mind that the slope of a line is often described as the ratio of rise to run. The formula for slope is: slope = (change in Y) / (change in X). Now, let’s investigate a line with a different slope to describe the trend in the data.

One such line, with slope 0.75, passes through (164, 164) and (188, 182) and near many of the other data points; its equation is YL = 0.75X + 41. Let’s compare this line to line YL = X – .7, which is the best fit we have found so far.

Note that these two lines are not parallel since they have different slopes.

Here is the scatter plot of the 24 people and the graph of the lines YL = .75X + 41 and YL = X – .7:

Here is the table to find the SSE for the line YL = .75X + 41:

The SSE for the line YL = .75X + 41 is 616.8 (as compared to 770.56). So this new line, with its different slope, turns out to be a better fit for the data set. See Note 3 below.

In This Part: Summary
In this session, we saw how the SSE can be used as criteria to determine which line best fits a set of data points. The best fit is the line with the smallest SSE. This line is referred to as the least squares line because, for a given set of data points, it is the line that minimizes the sum of the squared errors. In the following Interactive Activity, you will see how these squares can be represented graphically. The least squares line is the line that minimizes the total area of all the squares formed when the vertical distance from the data points to the line is used as the side lengths of the squares. See Note 4 below.