Least Squares Method Definition


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 data set, providing a visual demonstration of the relationship between the data points. Each data point represents the relationship between a known independent variable and an unknown dependent variable.

What does the method of least squares tell you?

The least squares method provides the overall rationale for placing the line of best fit among the data points studied. The most common application of this method, sometimes called “linear” or “ordinary”, is to create a straight line that minimizes the sum of the squares of the errors generated by the results of the associated equations, such as as the resulting squared residues differences in the observed value and the expected value, based on this model.

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

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

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

Key points to remember

  • 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 on the plotted curve.
  • Least squares regression is used to predict the behavior of the dependent variables.

Example of the method of least squares

An example of the least squares method is an analyst who wants 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 stock returns on index returns. To achieve this, all returns are plotted on a graph. Index returns are then referred to as the independent variable, and stock returns are the dependent variable. The line of best fit provides the analyst with coefficients explaining the level of dependence.

The line of the 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. The line of best fit equations can be determined by computer software models, which include a summary of the results to be analyzed, where the coefficients and summary results explain the dependence of the variables tested.

Least squares regression line

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

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