Finding the Average Rates of Change of a Set of Data using Differences

 x 1 2 5 7 8 9 10 y 1 3 9 13 15 17 19

When the x-differences are all constant, then the y-differences (or first differences) are useful in telling if a set of data is nearly linear or not. In this case, the x-differences of the values in the table are not constant. Thus we will use average rates of change to determine if the data is nearly linear or not. Recall that the average rate of change is the differences in the y's over the difference of the x's or .

To find the average rates of change, press STAT, select 1:EDIT by pressing ENTER. Enter the x values in L1, and the y values in L2 as shown below.

Next, with the flashing cursor in the L3 column, press the UP ARROW once so that L3 is highlighted. Press 2nd LIST[STAT] and RIGHT ARROW to OPS menu.

From the OPS menu select 7: DLIST by pressing a 7. DLIST now appears at the bottom of the screen beside L3=. Press 2nd L2[2] to type L2, and then press the ) key followed by the key. That is the numerator of the average rate of change fraction and the fraction line as seen in the second frame below.

Once again press 2nd LIST[STAT] and RIGHT ARROW to OPS menu. From the OPS menu select 7: DLIST by pressing a 7.

Press 2nd L1[1] to type L1, and then press the ) key. That completes the denominator of the average rate of change formula. Press ENTER and the consecutive average rates of change will appear in the L3 column.

Examining the average rates of change in L3 column, we see that they are exactly equal to each either. This means that that data is exactly linear.