The method of least squares is a statistical technique used to approximate the relationship between two or more variables by minimizing the sum of the squares of the differences between observed and predicted values. When fitting a straight line to a set of data points, the method of least squares is commonly used to find the line that best fits the data.

Here’s how the method of least squares is applied to fitting a straight line:

1. Model Selection: Assume that the relationship between the independent variable $x$ and the dependent variable
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   y is linear, given by the equation of a straight line: y=mx+b
   $$

   m is the slope of the line andb is the y-intercept.

2. Error Calculation: For each data point (xi​,yi​), calculate the vertical distance (residual) between the observed
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   yi​ value and the predicted value on the line. The residual is given by: ei​=yi​−(mxi​+b)

3. Sum of Squared Residuals: Square each residual and sum them up to obtain the sum of squared residuals (SSR):SSR=∑i=1n​ei2​
4. Minimization: The goal is to minimize the SSR by adjusting the parameters $m$ and
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   b. This is typically done using calculus or numerical optimization techniques. In the case of fitting a straight line, the minimization problem can be solved analytically.

5. Parameter Estimation: The values of $m$ and
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   b that minimize the SSR are estimated as: m=∑i=1n​(xi​−xˉ)2∑i=1n​(xi​−xˉ)(yi​−yˉ​)​ b=yˉ​−mxˉ

   where xˉ and yˉ​ are the means of the x and  y values, respectively.

6. Line of Best Fit: Once the parameters $m$ and
   $$

   b are estimated, the equation of the line of best fit is determined.

The resulting straight line is the best linear approximation to the given data set in terms of minimizing the sum of squared differences between the observed and predicted y values.

This method is widely used in various fields, including statistics, economics, engineering, and physics, for analyzing data and modeling linear relationships between variables. It provides a simple and effective way to find the “best-fit” line for a given set of data points.