Data Session 7, Part D: Fitting Lines to Data

Session 7, Part D:
Fitting Lines to Data

In This Part: Trend Lines | Error | The SSE | More Lines | Summary

Can we do better? Recall that for the 24 people in this study, the mean arm span is 175.5 cm and the mean height is 174.8 cm. Note that the mean arm span is .7 cm longer than the mean height. This suggests that we might try the line Height = Arm Span - .7 to describe the trend in our bivariate data. Let's see how this line compares with the previous models.

Here is the scatter plot of the data and a graph of the line YL = X - .7:

Here is the table for the line YL = X - .7:

Person #

Arm Span (X)

Height (Y)

YL = X - .7

Error = Y - YL

(Error)²
=
(Y - YL)²

1	156	162	155.3	6.7	44.89
2	157	160	156.3	3.7	13.69
3	159	162	158.3	3.7	13.69
4	160	155	159.3	-4.3	18.49
5	161	160	160.3	-0.3	0.09
6	161	162	160.3	1.7	2.89
7	162	170	161.3	8.7	75.69
8	165	166	164.3	1.7	2.89
9	170	170	169.3	0.7	0.49
10	170	167	169.3	-2.3	5.29
11	173	185	172.3	12.7	161.29
12	173	176	172.3	3.7	13.69
13	177	173	176.3	-3.3	10.89
14	177	176	176.3	-0.3	0.09
15	178	178	177.3	0.7	0.49
16	184	180	183.3	-3.3	10.89
17	188	188	187.3	0.7	0.49
18	188	187	187.3	-0.3	0.09
19	188	182	187.3	-5.3	28.09
20	188	181	187.3	-6.3	39.69
21	188	192	187.3	4.7	22.09
22	194	193	193.3	-0.3	0.09
23	196	184	195.3	-11.3	127.69
24	200	186	199.3	-13.30	176.89

For this line, the sum of squared errors is 770.56, which makes it a slightly better model than the line YL = X - 1 (whose SSE was 772).

Problem D8

Here are the three lines we've considered, plus two new ones:

a.	Judging on the basis of the SSE, which is the best line? Which is the worst?
b.	What other ways could we change the line equation in an attempt to further reduce the SSE?
c.	Is it possible to reduce the SSE to 0? Why or why not?

We have examined several lines that have yielded different SSEs. The lines, however, had one thing in common: they all had a slope of 1, so they were all parallel. Keep in mind that the slope of a line is often described as the ratio of rise to run. The formula for slope is: slope = (change in Y) / (change in X). Now, let's investigate a line with a different slope to describe the trend in the data.

One such line, with slope 0.75, passes through (164, 164) and (188, 182) and near many of the other data points; its equation is YL = 0.75X + 41. Let's compare this line to line YL = X - .7, which is the best fit we have found so far.

Note that these two lines are not parallel since they have different slopes.

Here is the scatter plot of the 24 people and the graph of the lines YL = .75X + 41 and YL = X - .7:

Here is the table to find the SSE for the line YL = .75X + 41:

Person #

Arm Span (X)

Height (Y)

YL = .75X + 41

Error = Y - YL

(Error)²
=
(Y - YL)²

1	156	162	1558	4	16
2	157	160	158.75	1.25	1.5625
3	159	162	160.25	1.75	3.0625
4	160	155	161	-6	36
5	161	160	161.75	-1.75	3.0625
6	161	162	161.75	0.25	0.0625
7	162	170	162.5	7.5	56.25
8	165	166	164.75	1.25	1.5625
9	170	170	168.5	1.5	2.25
10	170	167	168.5	1.5	2.25
11	173	185	170.75	14.25	203.0625
12	173	176	170.75	5.25	27.5625
13	177	173	173.75	-0.75	0.5625
14	177	176	173.75	2.25	5.0625
15	178	178	174.5	3.5	12.25
16	184	180	179	1	1
17	188	188	182	6	36
18	188	187	182	5	25
19	188	182	182	0	0
20	188	181	182	-1	1
21	188	192	182	10	100
22	194	193	186.5	6.5	42.25
23	196	184	188	-4	16
24	200	186	191	-5	25

The SSE for the line YL = .75X + 41 is 616.8 (as compared to 770.56). So this new line, with its different slope, turns out to be a better fit for the data set. Note 3

Next > Part D (Continued): Summary

Session 7: Index | Notes | Solutions | Video