Line of Best Fit

Date: 07/08/2003 at 20:42:36
From: Steph
Subject: I am looking to find the slope of my line

I am looking for help in how to manually find the slope and intercept 
of this problem. 

(X variables)
Volume, mL:
1.0, 2.0, 3.0, 4.0, 5.0, 6.0

(y variables)
Mass, grams:
2.96,5.05,7.03,8.92,10.94,13.04

Thank you for your time.

Date: 07/08/2003 at 21:09:55
From: Doctor Jaffee
Subject: Re: I am looking to find the slope of my line

Hi Steph,

First of all, these points don't really form a line, although they are 
pretty close. So you don't really want the equation of the line that 
connects these points, because it doesn't exist. What you really want 
is the equation of a line that comes close to the points, also known 
as "the line of best fit" or the "regression line."

Manually, there are several approaches you can take. If you connect 
the first point and the last point, you should have a line that is 
pretty close. You can calculate the slope using these two points, then 
use the point-slope equation to get the equation of the line. You 
should notice that some of the points are slightly above the line and 
some are slightly below. If you can draw a line through two other 
points that will make your line even closer, do that.

Of course, the best way to do it is to use statistics software like 
Fathom that will figure it out for you, or use the regression line 
utility on your graphing calculator. Check the manual for an 
explanation of how to use that utility.

If you calculate the vertical distance from each point to the line, 
square each of them, then add up the squares, the line that has the 
best fit is the one with the smallest sum. The computer programs and 
calculators calculate that number to determine the equation of the 
line.

I hope this explanation helps.  Give the problem a try and if you want 
to check your solution with me or if you have difficulties or other 
questions write back to me and I'll try to help you some more.

Good luck,

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/08/2003 at 21:09:58
From: Doctor Ian
Subject: Re: I am looking to find the slope of my line

Hi Steph,

I'm not sure what you mean when you say that 'the line does not
intercept'. Note that in the real world, data points will rarely, if
ever, fall exactly on a line together. This may be caused by
imperfections in the experimental apparatus, errors in measurement, or
just the complexity of the situation. 

So in the real world, what we're usually looking for is the line that
gives the 'best' fit to the data. As you might imagine, there are
various ways to define 'best', and each one has an associated 
technique. 

One of the simplest things you can do is to find the slope between
each adjacent set of points.  For example, between (1.0, 2.96) and
(2.0, 5.05), the slope is 

  5.05 - 2.96   2.09
  ----------- = ---- = 2.09
  2.0  - 1.0    1.0

If you do this for the other pairs, you'll get a collection of slopes,
and you could take the average of these.  

Another thing you can do is plot the points on graph paper, and use a
ruler to draw the line that seems to go through the 'middle' of the
points, i.e., so that some of the points are above the line, while
others are below.  

On the other hand, if this represents real data that someone really
cares about, you'd want to do something more sophisticated, like a
least-squares analysis. This has the same effect as finding the line
such that the sum of the squares of the distances from each point to
the line is minimized, which is one way of defining 'best'. 

Does this help?  Write back if you'd like to talk more about this, or
anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/