Regression Lines

Today’s Aim:To learn the method

for calculating the most accurate Line of

Best Fit for a set of data

Make a Scatterplot of the following data:

X Y

23 81

25 80

27 90

Lets guess where the Line of Best

Fit should go

Now we want to measure the distance between the actual Y values for each point and the predicted Y

value on our possible Line of Best Fit

Now, lets try with a different line…

We can also measure with numbers the vertical

distances between the Scatterplot points and

the Line of Best Fit

Actual y

values:

81

80

90

81

80

90Predicted y

values:

79.1

83.6

88.2

79.1

83.6

88.2

Difference in y

values:

.9

3.6

1.8

.9

3.6

1.8

.9

3.6

1.8

6.3

For the first possible Line of Best Fit, the sum of the vertical

distances (errors) was 6.3

81

80

90

79.6

83

85.2

.4

3

4.8

.4

3

4.8

8.2

The sum of the vertical distances

(errors) on the second possible line was 8.2.

The correct Line of Best Fit is called a Regression Line.

A Regression Line is the line that makes the sum

of the squares of the vertical distances

(errors) of the data points from the line as

small as possible.

To Calculate the Error:

Error = actual y value - predicted y value

Note: If the predicted value is larger than the actual value, the error will be a negative number. This is why we square the errors - to turn them into positive numbers.

For example…

X YPredicted Y values (Line A)

Vertical Distances

(errors)

Distances Squared

3 7 7.2 - 0.2 .04

4 9 9.6 - 0.6 .036

7 12 9.5 2.5 6.25

SUM:6.35

X YPredicted Y values (Line B)

Vertical Distances

(errors)

Distances Squared

3 7 7.5 - 0.5 .25

4 9 9.2 - 0.2 .04

7 12 11.3 .7 .49

SUM:.78