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Graphing Calculator: Trend Lines and Regression
Data can be found in many situations. It can also be graphed and analyzed in several ways. This lesson will focus on how to analyze data that has been input into your calculator. As a review, we will also look at graphs of data as seen on the calculator screen. This lesson assumes that you already know how to put data into your calculator. If you do not know how to do this and need help click here.

Consider the following example.

Ten 25-year olds were surveyed to find out their years of formal education (first grade – college) and their annual income. The data is presented below.

Years of Education  8101212131416161819
Income (in thousands)  18192425.526.62830334045

In most cases, it is useful to look at a graph of the data. Once you have put the years of education data in L1 and the income data in L2, you want to see a graph of the data. This is done on the calculator by having the calculator graph a scatterplot.

By pressing and you get the screen which turns on statistical graphs or plots.



This screen indicates that all four of the plots are turned off. To turn on Plot 1, press and set up your screen to match the one below.



You have the option of either setting your window by hand or by having the calculator help you by pressing and then . If you use the ZoomStat option, you should see the screen below.



Many times, after data has been graphed, we want to see if there is a function (usually a line) that will explain the data. This line can be called a trend line, since it can be used to explain trends in the data. It can also be called a line of best fit. In statistics, it is referred to as the line of best fit or the regression line. All of these names simply mean that we are trying to find a line to help us describe the relationship between our data. In this case, we are trying to describe the relationship between the years of education and annual income. It makes sense that your education level is a determining factor in how much money you make.

To find this line to describe the relationship between education and income, we need to use the button and choose the CALC menu. From the CALC menu, we will use option 4:LinReg(ax+b).



When you press , you should see the LinReg(ax+b) command on your screen as shown below.



The calculator needs to know which two sets of data to use in order to find the trend line. Since our data is in L1 and L2, that’s what we will enter into the calculator. L1 and L2 are the 2ND function keys of the 1 and 2 keys. So, you need to press:

     



Not only would we like the calculator to find a trend line for us, we want to graph that line also. Remember that in order to graph anything, we must have something in Y=. To have the calculator put our line into Y1, we need the following keystrokes:

   



Now when you press the calculator will show the numbers for the trend line and put that equation into Y1 for you.



Notice that my calculator has been set to two decimal places. If your calculator did not show r and r2 there is a way to get them to appear on the screen. First let’s look at the trend line and then we will work with r and r2.

The screen you see (regardless of the r and r2 values) is telling you that the line that describes the relationship between education and income is y=2.37x-3.73. That equation is in Y1 and can be graphed along with the data points by simply pressing .



We can now use that trend line to predict the annual income of someone with 15 years of formal education. We just substitute 15 for x in our equation and get



So it is reasonable to expect someone with 15 years of formal education to make approximately 31,820. (Remember that the data was given in thousands of dollars.)

This trend line we now have is very useful for making predictions about someone’s annual income. However, we have to be careful not to go too far away from our original data. The line that we found describes the data we put into the calculator. Data that is too far away from our original numbers will not be as accurate.

Now what can we do if your calculator did not show r and r2. Use the following procedure:

Press to get the CATALOG menu. Then use the repeatedly until you see DiagnosticOn. Then press and again.



You will now have to redo the regression command, but you should now see r and r2 on your screen.

Now that we have r and r2, let’s talk about what they mean.

The r value is what is known as a correlation coefficient. It is a number to tell how well the straight line fits the data. The closer the r value is to +1 or –1, the better the line fits the data. As the r value gets closer to 0, the worse the fit gets. In our case, the r value of .96 means that our line is a very good fit to our data.

The r2 value is found just by squaring the r value. In statistics, the r2 value describes how much of the variation in the data is accounted for by the trend line. It is a very interesting topic, but also fairly complicated to understand. If you are interested in further statistical analysis of data, see your statistics teacher.
You should now be able to input data, graph it, find the trend line and graph it with the data, use the trend line to make a prediction, and find the r and r2 values for the line. These skills will be very useful to you not only in your math class, but also in your science class, or anywhere you deal with data collection and analysis.



S Taylor

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