Best fitting line
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F ITTING A L INE TO D ATA. 8. 6. 4. 2. –8. –6. –4. –2. 0. 2. 4. 6. –2. –4. –6. –8. There are several ways to find the best-fitting line for a given set of data points. In this lesson, you will use a graphical approach.

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Best fitting line

FITTING A LINE TO DATA

8

6

4

2

–8

–6

–4

–2

0

2

4

6

–2

–4

–6

–8

There are several ways to find the best-fitting line for a given set of data points. In this lesson, you will use a graphical approach.

Usually, there is no single line that passes through all the data points, so you try to find the line that best fits the data. This is called the best-fitting line.

best-fitting line.


Best fitting line

250

DISCUS THROWS

Approximating a Best-Fitting Line

240

230

220

210

200

190

180

Distance (ft)

170

160

150

140

130

120

110

100

0

8

16

24

32

40

48

56

64

72

80

88

96

104

Years since 1900

The winning Olympic discus throws from 1908 to 1996 are plotted in the graph. Approximate the best-fitting line for these throws.

Write an equation of your line.


Best fitting line

(96, 230)

250

Approximating a Best-Fitting Line

240

230

220

(8, 138)

210

(96, 230).

200

190

180

Distance (ft)

170

160

150

140

(8, 138)

130

120

110

100

0

8

16

24

32

40

48

56

64

72

80

88

96

104

Years since 1900

SOLUTION

Find two points that lie on the best-fitting line, such as (8, 138) and (96, 230).

Find the slope of the line through these points.


Best fitting line

(96, 230)

250

Approximating a Best-Fitting Line

y2–y1

92

88

230–138

240

=

1.05

230–138

m =

92

88

1.05

=

=

=

96–8

x2–x1

230

96–8

220

210

200

190

180

Distance (ft)

170

160

150

140

(8, 138)

130

120

An equation of the best-fitting line isy = 1.05x + 129.6.

110

In most years, the winner of the discus throw was able to throw the discus farther than the previous winner.

100

0

8

16

24

32

40

48

56

64

72

80

88

96

104

Years since 1900

y = mx+b

Write slope intercept form.

Substitute 1.05 for m, 8 for x,

138 for y.

138= (1.05)(8) + b

138 = 8.4 + b

y = mx+b

Simplify.

Solve for b.

129.6 =b


Best fitting line

DETERMINING THE CORRELATION OF X AND Y

In this scatter plot, x and yhave a positive correlation, which means that the points can be approximated by a line with a positive slope.


Best fitting line

DETERMINING THE CORRELATION OF X AND Y

In this scatter plot, x and y have a negative correlation, which means that the points can be approximated by a line with a negative slope.


Best fitting line

DETERMINING THE CORRELATION OF X AND Y

In this scatter plot, x and y have relatively no correlation, which means that the points cannot be approximated by a line.


Best fitting line

TYPES OF CORRELATION

DETERMINING THE CORRELATION OF X AND Y

Positive Correlation

Negative Correlation

No Correlation


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