By Eye Best Fit of a Straight Line Graph
STRAIGHT LINE GRAPHS Page 7 of 8
8.
Doing it by Hand.
Here we deal with the practical aspect of doing line graphs by hand (manually).
10
9
8
7
6
5
4
3
2
1 y
(4.2,3.0)
(2.0,2.2)
(6.0,5.2)
(9.7,9.0)
(7.5,8.0) x
8.1.
Suppose your data gives you the points shown in Figure
The task is to fit a STRAIGHT line that best represents these points. This is called “ fit straight line the best fit straight line is one in which the points are “
”. The best balanced ” around that straight line. This means the “average total distance” of points on one side of the line is about the same as “average total distance” on the other side of the line.
-2 -1
-1
0
1 2 3 4 5
Figure 8.1
7 8 9 10
Data Points
Here we do it “by eye” using a clear (transparent) ruler, as shown in Figure 8.2. Lines A and C are obviously not good choices. Line A has too many points below and to the right of it.
Line C is just the opposite: It has too many points above and to the left of it. Line B seems to be the best fit.
Figure 8.2
Seeking BEST FIT straight line y
R ul er A
5
4
3
2
1
-2 -1
-1
0
10
9
8
7
6
(2.0,2.2)
1 2 3 4 5
(7.5,8.0)
(6.0,5.2)
(4.2,3.0)
7 8 9 10
(9.7,9.0) x y
5
4
3
2
1
-2 -1
0
-1
10
9
8
7
6
(4.2,3.0)
(2.0,2.2)
1 2 3 4 5
R ul er
(7.5,8.0)
(6.0,5.2)
7 8 9 10
B
(9.7,9.0) x y
5
4
3
2
1
-2 -1
-1
0
10
9
8
7
6
(2.0,2.2)
1 2 3 4 5
(7.5,8.0)
(6.0,5.2)
(4.2,3.0)
7 8 9 10
R ul er
C
(9.7,9.0) x
2
1
5
4
3
-2 -1
-1
0
10
9
8
7
6
Now we must find the slope of this “ best fit straight line ” shown in Figure 8.3. y B
We no longer need the actual experimental points since this BEST
FIT STRAIGHT LINE represents their behavior. But let’s remove
(8,8.2) their labels and shrink them to remind us what this line is about.
To get the slope of this “ best fit straight line ” we must first select two convenient points on it. Let’s choose the points
(x
1
,y
1
) = (0,-1) and (x
2
,y
2
) = (8,8.2). They require the least guess work to get their values. Well, on second thought (7,7) looks better. But ....
1 2 3 4 5
(0,-1)
7 8 9 10 x
Figure 8.3
Next, let’s construct the RISE and RUN TRIANGLE to calculate the slope.
© Hulan E. Jack Jr. 2004,2011 Straight Lines I
040912 110911

STRAIGHT LINE GRAPHS Page 8 of 8 y
B
5
4
3
2
1
-2 -1
-1
0
10
9
8
7
6
(8,8.2)
=8.2+1=9.2 yunits
1 2 3 4 5
(0,-1)
7 8 9 10 x
Run = x -x =8-0=8 xunits
2 1


9, 2
8




2
1


2

1.15
1 y = mx+b = 1.15x + (-1) =1.15x -1 yunits .
Figure 8.4
.
© Hulan E. Jack Jr. 2004,2011 Straight Lines I
040912 110911