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/
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