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Linear Regression and Correlation: The Regression Equation is a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean. Contributions from Roberta Bloom include instructions for finding and graphing the regression equation and scatterplot using the LinRegTTest on the TI-83,83+,84+ calculators.

Data rarely fit a straight line exactly. Usually, you must be satisfied with rough predictions. Typically, you have a set of data whose scatter plot appears to "fit" a straight line. This is called a Line of Best Fit or Least Squares Line .

Optional collaborative classroom activity

If you know a person's pinky (smallest) finger length, do you think you could predict that person's height? Collect data from your class (pinky finger length, in inches). Theindependent variable, x , is pinky finger length and the dependent variable, y , is height.

For each set of data, plot the points on graph paper. Make your graph big enough and use a ruler . Then "by eye" draw a line that appears to "fit" the data. For your line, pick two convenient points and use them to find the slope of the line. Find the y-intercept ofthe line by extending your lines so they cross the y-axis. Using the slopes and the y-intercepts, write your equation of "best fit". Do you think everyone will have the sameequation? Why or why not?

Using your equation, what is the predicted height for a pinky length of 2.5 inches?

A random sample of 11 statistics students produced the following data where x is the third exam score, out of 80, and y is the final exam score, out of 200. Can you predict the final exam score of a random student if you know the third exam score?

x (third exam score) y (final exam score)
65 175
67 133
71 185
71 163
66 126
75 198
67 153
70 163
71 159
69 151
69 159
Table showing the scores on the final exam based on scores from the third exam.
Scatterplot of exam scores with the third exam score on the x-axis and the final exam score on the y-axis.
Scatter plot showing the scores on the final exam based on scores from the third exam.

The third exam score, x , is the independent variable and the final exam score, y , is the dependent variable. We will plot a regression line that best "fits" the data. If each of youwere to fit a line "by eye", you would draw different lines. We can use what is called a least-squares regression line to obtain the best fit line.

Consider the following diagram. Each point of data is of the the form ( x , y ) and each point of the line of best fit using least-squares linear regression has the form ( x , y ^ ) .

The y ^ is read "y hat" and is the estimated value of y . It is the value of y obtained using the regression line. It is not generally equal to y from data.

Scatterplot of the exam scores with a line of best fit tying in the relationship between the third exam and final exam scores. A specific point on the line, specific data point, and the distance between these two points are used in order to show an example of how to compute the sum of squared errors in order to find the points on the line of best fit.

The term y 0 - y ^ 0 = ε 0 is called the "error" or residual . It is not an error in the sense of a mistake. The absolute value of a residual measures the vertical distance between the actual value of y and the estimated value of y . In other words, it measures the vertical distance between the actual data point and the predicted point on the line.

If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y . If the observed data point lies below the line, the residual is negative, and the line overestimates that actual data value for y .

In the diagram above, y 0 - y ^ 0 = ε 0 is the residual for the point shown. Here the point lies above the line and the residual is positive.

Questions & Answers

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Source:  OpenStax, Collaborative statistics (custom online version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11476/1.5
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