Regression Lines. Today’s Aim: To learn the method for calculating the most accurate Line of Best...
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Transcript of Regression Lines. Today’s Aim: To learn the method for calculating the most accurate Line of Best...
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