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# 2.3 - Lookingatdata:relationships Correlation tells us...

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Looking at data: relationships   Least-squares regression

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Correlation tells us about strength (scatter) and direction of the linear relationship between two quantitative variables. In addition, we would like to have a numerical description of how both variables vary together. And we would like to make predictions based on that numerical description. But which line best describes our data?
Unlike correlation, this requires that we have an explanatory variable and a response variable.

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A straight line has equation Given observed data , find and that minimize Method of least squares

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The regression line The least-squares regression line is the unique line such that the sum of the squared vertical distances between the data points and the line is the smallest possible.
Solution: Least squares regression line:

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Software output intercept slope R 2 intercept slope
Properties of least squares regression line Slope = change in y for each unit increase in x The line need not pass through the observed data points, but it always passes through . An increase of one standard deviation in x corresponds to a change of r standard deviation in y

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The distinction between explanatory and response  variables is crucial in regression. Regression examines  the distance of all points from the line   in the  y  direction  only.
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2.3 - Lookingatdata:relationships Correlation tells us...

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