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If you have a set of (x,y) pairs that you collect from observation…

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If you have a set...
If you have a set of (x,y) pairs that you collect from observation (ex. no of years performing a certain job versus salary paid to an employee), then after you plot these (x.y) pairs , you want to create a straight line that most closely goes by those (x,y) pairs in such a way that the sum of the squared distances from each (x,y) point to the line ,measured vertically, is the smallest possible. Does this make sense. Please tell me
Submitted: 8 years ago.Category: Math Homework
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7/15/2010
Math Tutor or Teacher: Ryan, Engineer replied 8 years ago
Ryan
Ryan, Engineer
Category: Math Homework
Satisfied Customers: 9,184
Experience: B.S. in Civil Engineering
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Hi,

Thank you for your question.

Yes, what you are describing makes perfect sense. It is called linear regression, and the process of calculating such a line is included in many (if not all) basic statistics books.

Does that answer your question for now, or would you like to have the details of the calculation process as well?

Thanks,

Ryan
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Customer reply replied 8 years ago
No, I need as much information as you can give me please.
Math Tutor or Teacher: Ryan, Engineer replied 8 years ago
Hi,

Ok, no problem.

To begin with, this can be done much more quickly with a spreadsheet application such as Excel. But if you were to do it by hand, these are the steps to follow.

First, you need to calculate the following totals:

n (the total number of pairs in your data set)

∑x (the sum of all of the x values)

∑y (the sum of all of the y values)

∑xy (the sum of all of the products of each pair)

∑x² (the sum of the squares of all of the x values)

∑y² (the sum of the squares of all of the y values)

Having done that, the next number to calculate is the "correlation coefficient". This number will give some idea of how closely the x and y values are related. This coefficient will have a value between -1 and 1. The closer the value is to +1, the stronger the relationship. If the coefficient is closer to the -1 side, a linear regression calculation will not provide any useful information.

The calculation for the correlation coefficient, r, is:

r = [ n(∑xy) - (∑x)(∑y) ] / √[ [ n(∑x²) - (∑x)² ][ n(∑y²) - (∑y)² ] ]

Assuming that the coefficient is acceptable, the rest of the procedure is as follows:

The regression line that best fits the data has the form:

y' = a + bx

where:

a = [ (∑y)(∑x²) - (∑x)(∑xy) ] / [ n(∑x²) - (∑x)² ]

and

b = [ n(∑xy) - (∑x)(∑y) ] / [ n(∑x²) - (∑x)² ]

It looks more complicated than it is. Once you have the sums calculated in the beginning step, the rest of it is just a matter of putting the right number in the right place and finishing the calculation. And, as I said, this can be done very easily with a spreadsheet.

Please let me know if you need more information about this.

Thanks,

Ryan
Ryan
Ryan, Engineer
Category: Math Homework
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Customer reply replied 8 years ago
Thanks Ryan, I'm pretty sure I'll need more help with this subject in the near future. If there is a way I could get more help from you please let me know.
Math Tutor or Teacher: Ryan, Engineer replied 8 years ago
Hi,

Thank you for accepting my answer.

If you want to make sure that I get your questions in the future, just type "For Ryan" in the subject line or at the beginning of your question.

Thanks,

Ryan
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