What quantity does the method of least squares minimize?
The method of least squares is used to analyze bivariant data, where a scatter plot and a correlation coefficient of the data indicate that the variables 𝑥 and 𝑦 are linearly related. Our first step in analyzing such data is to try and model the relationship with a line of best fit. However, attempting to construct the line of best fit by eye can lead to inaccuracies in our results. And so the least squares regression line allows us to find the exact line of best fit. It aims to find the line whose overall average distance from all of the data points is minimized.
And so the least squares regression line in effect minimizes the sum of the squared differences of the points from the line. In other words, it minimizes the sum of the squares of the residuals. And so the answer to the question “What quantity does it minimize?” is the sum of the squares of the residuals.