Digital Lesson
Linear Models and
Scatter Plots
A scatter plot represents data graphically using
points plotted on a rectangular coordinate system.
y
Example: (x, y)
(1, – 4)
(2, – 2)
4
2
(3, – 1)
x
(4, 0)
(5, 2)
2
4
6
–2
(6, 4)
–4
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2
Example:
The average salary S (in millions of dollars) for professional
baseball players from 1996 through 2002 is shown in the table.
Let t = 6 represent the year 1996. Draw the scatter plot.
Year
1996
1997
1998
1999
2000
2001
2002
Salary, S
1.1
1.3
1.4
1.6
1.8
2.1
2.3
Salary (in millions of dollars)
S
2
1
t
4
8
Year (6  1996)
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3
Graphing Utility: Draw the scatter plot that describes the data.
Year
1996
1997
1998
1999
2000
2001
2002
Salary, S
1.1
1.3
1.4
1.6
1.8
2.1
2.3
Stat Edit Menu:
Stat Plot Menu:
Zoom Menu:
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4
In a collection of ordered pairs (x, y), if y tends to increase as x
increases, the collection has a positive correlation.
If y tends to decrease as x increases, the collection has a negative
correlation.
y
y
y
x
x
positive correlation
negative correlation
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no correlation
x
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Finding a linear model to represent the relationship described by a
scatter pot is called fitting the line to data.
Example:
The table and scatter plot for the average
salary S (in millions of dollars) for
professional baseball players from 1996
through 2002 is shown. Let t = 6 represent
the year 1996.
Year
Salary, S
1996
1.1
1997
1.3
1998
1.4
1999
1.6
2000
1.8
2001
2.1
2002
2.3
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2
S
1
Find the
equation of
the line.
4
8
t
Example continues.
6
Example continued:
Year
1996
1997
1998
1999
2000
2001
2002
Salary, S
1.1
1.3
1.4
1.6
1.8
2.1
2.3
2
S
(12, 2.3)
1
(6, 1.1)
4
8
t
The equation of this line is S = 0.2t – 0.1.
This line approximates the data.
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7
Graphing Utility: Find a linear model that describes the data.
Year
1996
1997
1998
1999
2000
2001
2002
Salary, S
1.1
1.3
1.4
1.6
1.8
2.1
2.3
Stat Edit:
Stat Menu:
A linear model for this data is S = 0.2t – 0.14. This equation is
very close to the equation found using two data points.
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The correlation coefficient (or r-value) of the data gives the measure of
how well the model fits the data.
The closer |r| is to 1, the better the points can be described by a line.
2
0
r = 0.99
strong positive
correlation
52.6
13
0
0
52.3
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r = – 0.93
negative
correlation
r = 0.66
weak correlation
100
13
50
90
50
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