The correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. It can range from -1 to 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.
The correlation coefficient can be used to assess how well a linear model fits the data, or to compare the relationships between different pairs of variables.
You can use the TI-84 calculator to find the correlation coefficient between two variables, as long as you have the data values for both variables. In this article, we will show you how to do that, and provide some examples and tips along the way.
How to Access the Correlation Coefficient Function
The correlation coefficient function can be accessed on a TI-84 calculator by following these steps:
- Press the STAT key and then press the right-arrow key to move to the TESTS menu.
- Scroll down to option C:Correlation and press ENTER. This will bring up the correlation coefficient function on the screen.
How to Use the Correlation Coefficient Function
The correlation coefficient function has the following syntax:
correlation(xList, yList, [FreqList])
where:
- xList: the list of values for the first variable
- yList: the list of values for the second variable
- FreqList: (optional) the list of frequencies for each pair of values
If you omit the FreqList argument, the calculator will assume that each pair of values has a frequency of 1.
The correlation coefficient function will return the correlation coefficient ® between the two variables, as well as the sample size (n) and the degrees of freedom (df).
Examples of Using the Correlation Coefficient Function
Let’s look at some examples of how to use the correlation coefficient function on a TI-84 calculator.
Example 1: Simple Data Set
Question: The following table shows the scores of 10 students on a math test and a science test. Find the correlation coefficient between the math and science scores.
| Math | Science |
|---|---|
| 85 | 90 |
| 92 | 95 |
| 76 | 80 |
| 88 | 85 |
| 81 | 83 |
| 94 | 97 |
| 79 | 82 |
| 90 | 88 |
| 84 | 86 |
| 87 | 91 |
Answer: To find the correlation coefficient, we need to use the correlation coefficient function with the following arguments:
correlation(L1, L2)
where L1 and L2 are the lists of values for the math and science scores, respectively.
To enter the data values, press STAT and then press EDIT. Enter the values for the math scores in column L1 and the values for the science scores in column L2:
Press STAT and then scroll right to the TESTS menu. Scroll down to option C:Correlation and press ENTER. For xList and yList, make sure L1 and L2 are selected since these are the columns we used to input our data. Leave FreqList blank. Scroll down to Calculate and press ENTER. On the new screen we can see that the correlation coefficient ® between the math and science scores is 0.9759.
This means that there is a very strong positive correlation between the math and science scores.
Example 2: Data Set with Frequencies
Question: The following table shows the number of hours spent studying and the grades obtained by 20 students in a class. Find the correlation coefficient between the hours and the grades.
| Hours | Grade | Frequency |
|---|---|---|
| 2 | 60 | 3 |
| 3 | 70 | 4 |
| 4 | 80 | 5 |
| 5 | 90 | 6 |
| 6 | 100 | 2 |
Answer: To find the correlation coefficient, we need to use the correlation coefficient function with the following arguments:
correlation(L1, L2, L3)
where L1, L2, and L3 are the lists of values for the hours, grades, and frequencies, respectively.
To enter the data values, press STAT and then press EDIT. Enter the values for the hours in column L1, the values for the grades in column L2, and the values for the frequencies in column L3:
Press STAT and then scroll right to the TESTS menu. Scroll down to option C:Correlation and press ENTER. For xList, yList, and FreqList, make sure L1, L2, and L3 are selected since these are the columns we used to input our data. Scroll down to Calculate and press ENTER. On the new screen we can see that the correlation coefficient ® between the hours and the grades is 0.9936.
This means that there is a very strong positive correlation between the hours and the grades.
Tips and Tricks for Using the Correlation Coefficient Function
Here are some tips and tricks for using the correlation coefficient function on a TI-84 calculator:
- To find the correlation coefficient for more than two variables, you can use the matrix function on the calculator. First, you need to enter the data values for each variable in a separate column of a matrix. Then, you can use the rref function to find the reduced row echelon form of the matrix. The correlation coefficients will be the values on the main diagonal of the matrix, except for the last one, which will be 1.
- To find the equation of the best-fit line for the data, you can use the linear regression function on the calculator. Press STAT and then scroll right to the CALC menu. Scroll down to option 4:LinReg(ax+b) and press ENTER. For Xlist and Ylist, make sure L1 and L2 are selected since these are the columns we used to input our data. Leave FreqList blank. Scroll down to Calculate and press ENTER. On the new screen you can see the equation of the best-fit line (y=ax+b), the correlation coefficient ®, and the coefficient of determination (r2).
- To graph the data and the best-fit line, you can use the scatter plot function on the calculator. Press 2nd and then press Y=. This will take you to the STAT PLOT screen. Scroll down to option 1:Plot1 and press ENTER. Make sure On is highlighted. For Type, choose the first option, which is the scatter plot symbol. For Xlist and Ylist, make sure L1 and L2 are selected since these are the columns we used to input our data. For Mark, choose any symbol you like. Press ZOOM and then scroll down to option 9:ZoomStat and press ENTER. This will adjust the window to fit the data. Press GRAPH to see the scatter plot of the data. To see the best-fit line, press Y= and then scroll down to option 5:LinReg(ax+b) and press ENTER. For Xlist and Ylist, make sure L1 and L2 are selected since these are the columns we used to input our data. Leave FreqList blank. Scroll down to Store RegEQ and press ALPHA and then press TRACE. This will store the equation of the best-fit line in Y1. Press GRAPH to see the best-fit line on the scatter plot.