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### Course: AP®︎/College Statistics>Unit 5

Lesson 4: Least-squares regression

# Calculating the equation of a regression line

We determine the correlation coefficient for bivariate data, which helps understand the relationship between variables. We then build the equation for the least squares line, using standard deviations and the correlation coefficient. The regression line equation y hat = mx + b is calculated.

## Want to join the conversation?

• What video is he referring to in the beginning?
• Why for a least-squares regression line I'm definitely going to have the sample mean of x and y on the line?
• At ,why regeression line must go through the point (mean of x,mean of y)?
• Why do we not use x hat in the equation of the least regression line?
y hat = m (x) + b?
• A hat over a variable in statistics means that it is a predicted value. In general, the explanatory variable is on the x-axis and the response variable is on the y-axis. The response variable can be predicted based on the explanatory variable. The response variable is not exact, while the explanatory variable is exact. This is why the response variable (y) is written with a hat.
• For those who don't get it.

Goal is to find regression line that best fits the data point. He shows formula to get the correlation coefficient, but they have already done all the calculation to get the best correlation coefficient. They have also provided x,y mean and stddev.

Now the way they derive the y=mx+b.

First they use the Xmean and Ymean as reference. The Ymean is NOT the y intercept. And then he draws 1 stddev lines for x and y axis. Then he shows that rise over run, which is slope, is equal to Sy/Sx. But the r also factors into this calculation. Therefore m = r*Sy/Sx. But we still have to find y intercept.

We know for a fact that for the regression line function, we have Xmean and Ymean as part of its points or at its intersection. So we substitute the m, Xmean, Ymean, and then get Y intercept.

Honestly it's pretty smart. Wouldn't have thought about it and was going to skip this video. But glad I spent time to understand it.

This has applications in machine learning and AI - FYI.
• Why is r always between -1 and 1?

I know that this question has been asked before but the answers are either too technical or too naive. Could someone please provide an answer that is mathematical in nature but can be understood by someone who have ok but not strong mathematical foundation.

• The number and the sign are talking about two different things. If the scatterplot dots fit the line exactly, they will have a correlation of 100% and therefore an r value of 1.00 However, r may be positive or negative depending on the slope of the "line of best fit". So, a scatterplot with points that are halfway between random and a perfect line (with slope 1) would have an r of 0.50, because there is a 50% correlation and because the slope is positive.
• In later videos we see another formula for calculating m, which is m = (X_bar*Y_bar - XY_bar) / X_bar^2 - X^2_bar, which is derived by taking the partial derivatives of the square errors function with respect to m and b. and here we see another formula m = r*Sy/Sx. can someone please say if there is any relationship between these two?
• I am still quite confused. Why is m=r(Sy/Sx)? I think r is just to measure the strength of the correlation, no? What is r doing in this formula? Thanks for your help in advance!