Least Squares Regression
Facts and r2
AP Statistics
The coefficient of determination, r2
• The fraction of the variation in the values
of y that is explained by least squares
regression of y on x.
Facts about least squares regression:
• Fact 1: The distinction between explanatory and response variables
is essential . If we reverse the roles of the variables x and y, we get
a different least squares regression line.
• Fact 2: There is a close relationship between correlation and the
slope of the least-squares regression line. A change of one
standard deviation in x corresponds to a change of r standard
deviations in y.
• Fact 3: The least squares regression line always passes through the
point containing the mean of x and y. ( x , y )
• Fact 4: The correlation r describes the strength of a straight line
relationship. In the regression setting, the description takes a
specific form: the square of the correlation r2 is the fraction of the
variation in the values of y that is explained by the least squares
regression of y on x.
3.45 p. 166-167 Beavers and Beetles
Ecologists sometimes find rather strange relationships in our environment. One study
seems to show that beavers benefit beetles. The researchers laid out 23 circular plots,
each four meters in diameter, in an area where beavers were cutting down cottonwood
trees. In each plot they counted the number of stumps from trees ct down by beavers
and the number of clusters of beetle larvae. Here are the data:
• Stumps: 2
• Beetle: 10
•
•
•
•
•
•
•
•
2
30
1
12
3
24
3
36
4
40
3
43
1
11
2
27
5
56
Larvae
Stumps: 1
3
2
1
2
2
1
1
4
1
Beetle
18
40 25
8
21
14 16
6
54
9
Larvae:
Stumps: 2
1
4
Beetle 13
14 50
Larvae:
Make a scatterplot that shows how the number of beaver-caused stumps influences
the number of beetle larvae clusters. What does your plot show?
FIRST OF ALL. WHAT IS THE EXPLANATORY VARIABLE? The beaver caused stumps
THE RESPONSE VARIABLE? The number of beetle larvae clusters.
The plot shows a positive linear association.
3.45 p. 166-167 Beavers and Beetles CONTINUED
Ecologists sometimes find rather strange relationships in our environment. One study
seems to show that beavers benefit beetles. The researchers laid out 23 circular plots,
each four meters in diameter, in an area where beavers were cutting down cottonwood
trees. In each plot they counted the number of stumps from trees ct down by beavers
and the number of clusters of beetle larvae. Here are the data:
• b. Find the least squares regression line and draw it on your plot.
• The least squares regression line is
• predicted number of beetle larvae clusters = -1.286 + 11.89 (
number of stumps)
• Or in terms of variables
• C. What percent of the observed variation in beetle larvae counts
can be explained by straight line dependence on stump counts?
• The straight line relationship explains r2 = 83.9 % of the variation
between beetle larvae.