What Is the Least Squares Method?

The "least squares" method is a form of mathematical regression analysis used to determine the line of best fit for a set of data, providing a visual demonstration of the relationship between the data points. Each point of data represents the relationship between a known independent variable and an unknown dependent variable.

What Does the Least Squares Method Tell You?

The least squares method provides the overall rationale for the placement of the line of best fit among the data points being studied. The most common application of this method, which is sometimes referred to as "linear" or "ordinary", aims to create a straight line that minimizes the sum of the squares of the errors that are generated by the results of the associated equations, such as the squared residuals resulting from differences in the observed value, and the value anticipated, based on that model.

This method of regression analysis begins with a set of data points to be plotted on an x- and y-axis graph. An analyst using the least squares method will generate a line of best fit that explains the potential relationship between independent and dependent variables.

In regression analysis, dependent variables are illustrated on the vertical y-axis, while independent variables are illustrated on the horizontal x-axis. These designations will form the equation for the line of best fit, which is determined from the least squares method.

In contrast to a linear problem, a non-linear least squares problem has no closed solution and is generally solved by iteration. The discovery of the least squares method is attributed to Carl Friedrich Gauss, who discovered the method in 1795.

Key Takeaways

  • The least squares method is a statistical procedure to find the best fit for a set of data points by minimizing the sum of the offsets or residuals of points from the plotted curve.
  • Least squares regression is used to predict the behavior of dependent variables.

Example of the Least Squares Method

An example of the least squares method is an analyst who wishes to test the relationship between a company’s stock returns, and the returns of the index for which the stock is a component. In this example, the analyst seeks to test the dependence of the stock returns on the index returns. To achieve this, all of the returns are plotted on a chart. The index returns are then designated as the independent variable, and the stock returns are the dependent variable. The line of best fit provides the analyst with coefficients explaining the level of dependence.

The Line of Best Fit Equation

The line of best fit determined from the least squares method has an equation that tells the story of the relationship between the data points. Line of best fit equations may be determined by computer software models, which include a summary of outputs for analysis, where the coefficients and summary outputs explain the dependence of the variables being tested.

Least Squares Regression Line

If the data shows a leaner relationship between two variables, the line that best fits this linear relationship is known as a least squares regression line, which minimizes the vertical distance from the data points to the regression line. The term “least squares” is used because it is the smallest sum of squares of errors, which is also called the "variance".