When Barbara, Carmen, and Denice looked at the following plot, they thought the points lay more or less in a line. They each drew a different line, using different ideas about what line would fit the data best.
You might have seen this in the problem set, “Tennis, anyone?”
Top Earnings, WTA (Women’s Tennis) 2001

When statisticians and other mathematicians want to fit a line (or another curve) to data, they often use numerical methods and strict guidelines about what the curve of best fit would be.

Carmen found a line with equation y = -0.16x + 2.44. Denice found a line with equation y = -0.19x + 2.58. Both lines miss each point. The difference

For example, Carmen’s line predicts the ninth ranked player’s earnings to be $1 million: -0.16(9) + 2.44 = 1. The actual earnings was $0.946385. The residual is 0.946385 - 1, which is -0.053615. Denice’s line predicts the earnings to be $0.87, so the residual is 0.946385 - 0.87, which is 0.033615.

1. Residuals are calculated: actual value - predicted value. What does the sign on the residual tell you?
2. What does the absolute value of the residual represent?
3. Calculate the absolute values of the residuals and complete the following table. For the last row, find the sums of the last two columns.
   

   Rank	Actual	Predicted Earnings		|Residuals|
   	Earnings	Carmen	Denice	Carmen	Denice
   1	2.522610
   2	2.238624
   3	2.086263
   4	1.832242
   5	1.465116
   6	1.314659
   7	1.155716
   8	0.995704
   9	0.946385
   10	0.867702
   			Sum:

4. Which line, Carmen’s or Denice’s, has a lower sum of the absolute value of the residuals? Why would that line be better to use for predicting players’ earnings than the other? Using the absolute value of the residuals gives you a way to decide if one line is better than another. However, it’s difficult to use absolute values to generalize the method so that you can find the best line to use for any data set.

   One line of “best” fit is the one that makes the squares of the residuals as small as possible. A formula was found for this line, using techniques to make the sum of the squares of the residuals as small as possible. The resulting line is called the least squares regression line. (“Regression” simply means that some numerical way is used to decide how well the line fits the data. This is “least squares regression” because the way to decide is to make squares of the residuals the least possible.) The calculations are not really complicated, but even with just a few data points, it can get messy. Fortunately, graphing calculators and statistical software can use the formula to find the least squares regression line easily.
   Suppose you have a data set {(x₁, y₁), ..., (x_n, y_n)}. The standard deviation in the x variable is S_x and in the y variable is S_y. Let x and y be the means of the x and y values, respectively. Then the least squares regression line is y = a + bx, where

   and
   
5. Use a graphing calculator or other software to find the least squares regression line. (This is sometimes just called linear regression.)
6. Use the line you just found to complete the following table, similar to the one in problem 3.
   You can copy Denice’s predicted earnings column from problem 3, but you’ll need to calculate values for the other columns.
   

   Rank	Actual	Predicted Earnings		Residuals2
   	Earnings	Denice	Regression	Denice	Regression
   1	2.522610
   2	2.238624
   3	2.086263
   4	1.832242
   5	1.465116
   6	1.314659
   7	1.155716
   8	0.995704
   9	0.946385
   10	0.867702
   			Sum:

1. If the residual is negative, the predicted value is too high. (That is, the point is below the line.) If the residual is positive, the predicted value is too low. (The point is above the line.)
2. The (vertical) distance from the point to the line. Or, the distance from the predicted point to the actual point.
3. The table is as follows:
   Teacher’s Note: You might have students complete both tables in pairs, or using spreadsheets or the data list capabilities of a graphing calculator.
   

   Rank	Actual	Predicted Earnings		|Residuals|
   	Earnings	Carmen	Denice	Carmen	Denice
   1	2.522610	2.28	2.39	0.242610	0.132610
   2	2.238624	2.12	2.2	0.118624	0.038624
   3	2.086263	1.96	2.01	0.126263	0.076263
   4	1.832242	1.8	1.82	0.032242	0.012242
   5	1.465116	1.64	1.63	0.174884	0.164884
   6	1.314659	1.48	1.44	0.165341	0.125341
   7	1.155716	1.32	1.25	0.164284	0.094284
   8	0.995704	1.16	1.06	0.164296	0.064296
   9	0.946385	1	0.87	0.053615	0.076385
   10	0.867702	0.84	0.68	0.027702	0.187702
   			Sum:	1.269861	0.972631

4. Denice’s line has a lower sum. That means the total distance between the predicted and actual points is smaller than for Carmen’s line--so overall, Denice’s line comes closer to the points than Carmen’s.
5. The line (using only the first five digits for each parameter) is y = -0.19135x + 2.59492.
6. The table is as follows:
   To calculate the squares of the residuals, you subtract each predicted value from the corresponding actual (observed) value, then square the result.
   

   Rank	Actual	Predicted Earnings		Residuals2
   	Earnings	Denice	Regression	Denice	Regression
   1	2.522610	2.39	2.4036	0.01759	0.01417
   2	2.238624	2.2	2.2122	0.00149	0.00070
   3	2.086263	2.01	2.0209	0.00582	0.00428
   4	1.832242	1.82	1.8295	0.00015	0.00001
   5	1.465116	1.63	1.6382	0.02719	0.02995
   6	1.314659	1.44	1.4468	0.01571	0.01747
   7	1.155716	1.25	1.2555	0.00889	0.00995
   8	0.995704	1.06	1.0641	0.00413	0.00468
   9	0.946385	0.87	0.8728	0.00583	0.00542
   10	0.867702	0.68	0.6814	0.03523	0.03470
   			Sum:	0.12203	0.12132