# Regression and the sum of residuals

The answer to the last part provided is The sum of square of residuals is minimum for points lying on the regression line and so cannot be less than 8.8 for any other line.

Can somebody please explain what this means?

It is almost evident that these points doesn't lie on the given regression line. So if I were to provide a much more accurate regression line, won't the square of sum much more smaller?

• I cant read anything in that photo Commented May 21, 2017 at 23:15
• Is it okay now? Commented May 21, 2017 at 23:29
• It gives you only seven of the eight pairs and asks you to find the eight. The line $y = 15.1803 - 0.7049x$ is the least squares line for the seven given points. The correct line for all eight points is different. Commented May 21, 2017 at 23:54

The sum of square of residuals is minimum for points lying on the regression line and so cannot be less than $8.8$ for any other line.

This is misleadingly stated. It says "for points lying on the regression line". What it ought to say is that for the line whose slope was specified slope and intercept, the sum of squares of residuals is smaller than it is for any other slope or other intercept.

Note that only seven of the eight pairs are given. You are asked to find the eighth pair.

• Yes exactly I was confused at this point too. How are the points lying on the line when they themselves asked us to calculate the deviation? Commented May 21, 2017 at 23:48
• I'll be back tomorrow . . . . . . . Commented May 21, 2017 at 23:56

One way to look at the result line we get from a linear regression is that this is the line we get by minimizing the sum of squared residuals of the points (to visualize, it is the sum of squared vertical distance of points to the regression line).

Thus any line other than the regression line will not have a smaller sum of squared residues.

EDIT

I feel you might misunderstand what is a regression line - so a regression line is not a line that you give arbitrary $a$, $b$ parameters to it. Instead, it is a line that you calculate you parameter so that the sum of squared residual is the smallest out of all lines. Hope this would help.

• But what if I provide an equation which is basically a much more accurate regression line than the one provided? Commented May 21, 2017 at 23:42
• @FaiqRaees : They're saying there isn't any that is more accurate. Commented May 21, 2017 at 23:44
• @FaiqRaees define "accurate" - if you mean smaller sum of squared residuals, then you cannot, because the regression line you get always have the smallest such sum. Commented May 21, 2017 at 23:44
• @MichaelHardy Is that an assumption I have to make? Because I used the formulas and I got a much more accurate regression line $y=-0.7049180..x+15.18032787$ Commented May 21, 2017 at 23:46
• @FaiqRaees : It appears that your proposed more accurate line is correct for the seven pairs given. But whatever the correct numbers, what the book ought to say is that for the least squares line, the sum of squares of residuals is smaller than for all other lines. However, notice that it says "Seven of the pairs". That means it's not giving you all of them. There are more than seven. Commented May 21, 2017 at 23:51