Best Line Fit: Least Squares
Date: 03/23/97 at 18:27:36 From: Mike Litrownik Subject: Best Line Fit: Least Squares Hi there. I'm a freshman in high school and in math now we're plotting data points and trying to find a line of best fit. We figured out that by using our calculators and using a linear regression it will give us the exact line of best fit. I know it uses something called least squares fitting or something like that to determine the line. My question is: What is least squares fitting? How exactly does it do it? Also, with cubic and quartic regressions and logistic and other regressions...how exactly do they work? Any help would be appreciated. Thanks.
Date: 03/23/97 at 19:46:32 From: Doctor Steven Subject: Re: Best Line Fit: Least Squares Imagine you're trying to fit a line through and you want it to be as close as possible to the points. One way to do this is to minimize the sum of the distances of the points from the line. But this might lead to the line passing through some points while other points are very far away from the line. So instead what least squares fitting does is tries to get every point as equally close to the line as it can get it. The way to do this is to make sure that no point is far from the average distance away from the line. In statistical terms this means you are trying to minimize the standard deviation of the distances of the points from the line. We use squares of numbers to figure out the standard deviation. If a point is really far away from the line that really increases the standard deviation since the distance away from the line is squared which makes its value much bigger. On the other hand if the distance between the point and the line is small, then squaring it makes it even smaller. (.5^2 = .25) So we get this effect of really disliking points far away from the line and really liking points close to the line. This creates a fit that gives each point close to an average distance away from the line. The actual equations used to get this line are pretty complicated but if you continue to take courses in math in college you'll probably encounter it at the freshman level. Cubic and quartic regressions rely on cubic and quartic equations which model the data points. The trick is to pick three or four points at random and then create a cubic or quartic polynomial to fit them and hope it fits the rest of the data points. Calculators would probably do several of these and then choose the best fit out of the bunch. Logistic regression use logarithmic expressions to fit a set of data points in much the same way as cubic and quartic regressions do. Hope this helps. -Doctor Steven, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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