Linear Regression
Scatter diagrams and lines of best fit y=ax+b and x=ay+b, by eye or with GDC, the Pearson correlation coefficient r, and making predictions with awareness of limitations.
Plotting approximate best fit line
Best fit lines can also be drawn approximately by eye. We start by finding the average x and y, giving the point (xˉ,yˉ). We then take a ruler and place it on this point, and adjust the slope until we find a reasonable best fit line.
Regression line y on x
Linear regression is a statistical method used to model the relationship between two variables when data is given as pairs of points (x,y). We fit a straight line (called the regression line) that minimizes the average vertical distance from the points:
The general equation of the regression line is:
where a is the slope and b is the y-intercept.
The values of a and b can be found using a calculator:
Use
Stat>Editto fill in x- and y-values into L1 and L2.Then, press
Stat, right arrow to theCALCmenu, and select4:LinReg(ax+b).
Pearson's Product-Moment Correlation Coefficient
Pearson's product-moment correlation coefficient, denoted by r, measures the strength and direction of a linear relationship between two numerical variables x and y. Its value always lies between −1 and +1:
r=+1: perfect positive linear relationship
r=−1: perfect negative linear relationship
r=0: no linear relationship
A positive value means y generally increases as x increases; a negative value means y generally decreases as x increases. The closer r is to ±1, the stronger the linear relationship.
If you clickmode, scroll to STAT DIAGNOSTICS , hover over ON, and click ENTER, then any time you perform a linear regression, the calculator will provide Pearson's coefficient in addition to the regression line.
Predicting y from x
Once we have a regression line y=ax+b, we can use it to predict y by plugging in a value of x.
Danger of extrapolation
When using a regression line to predict y from x, we need to be aware of the danger of extrapolation. This occurs when we try to predict y for a value of x far outside the range of x values in our data. For such an x, we cannot trust that the relationship is the same.
Limitations of predicting x from y
While it is possible to use a regression line y=ax+b to predict x with
this is not a reliable process. The best fit line is determined to minimize the difference between the real y’s and the predicted y’s,so the difference between real and predicted values for x may be much larger.
Regression line x on y
In the same way that we can plot a straight line minimizing the vertical distances from points (x,y), we can plot a straight line minimizing the horizontal distances. This is called an x on y regression line. We calculate an x on y regression line by switching our x and y lists while using LinReg(ax+b).
With this line we can make reliable predictions of x given y, so long as we are not extrpolating.
Nice work completing Linear Regression, here's a quick recap of what we covered:
Skills covered
Exercises checked off