Least Square Method

Least square method refers to a statistical technique used to understand relationships between two variables, make predictions, and summarize data. The technique is implemented by finding the best-fitting line through a set of data points. Here, the best fit line is the line drawn across the scatter plot to show the relationship between the variables. 

The method can be visualized on a scatter plot. Imagine a graph with data points in x and y. The process begins by analyzing each data set to find the residual value, which is the difference between the actual y value and the predicted y value. After finding the residual, we need to square them and add them all up. We try to make the sum as small as possible to find the best fit line. It is commonly used in regression analysis to find the relationship between the dependent variable and independent variables. 

The least square method formula to find the slope and the intercept is given below: 

Slope (m) = \(n\sum xy \space - \space (\sum x)(\sum y) \over n\sum x^2 \space - \space (\sum x)^2\)

Intercept (c) = ȳ - mx̄ where \(x̄ = {\sum x \over n}\) and \(ȳ = {\sum y \over n}\)

\(c = {\sum y \space- \space m (\sum x) \over n}\)

Here,

• n is the total number of data points
   
• x is the independent variable and y is the dependent variable
   
• Σ is the sum of the values
   
• m is the slope 
   
• c is the y-intercept 
  
   

We follow a certain method to calculate the least squares. Here, we shall analyze the method step-by-step:

Step 1: Consider the independent variable values as x_i and the dependent variable as y_i  

Step 2: Finding the average values of x_i and y_i as x̄ and ȳ.

Step 3: Let’s say the line of best fit is y = mx + c. Here, c is the intercept of the line on the y-axis and m is the slope 

Step 4: So, the slope \(m ={ \sum {[(x_i - \bar x)(y_i - \bar y)]} \over {\sum (x_i - x̄)^2}}\)

Step 5: Then the intercept(c) = ȳ - mx̄

So, the best fit line is y = mx + c

The least square method works by minimizing the differences between the actual data and the predicted value on the line.  Now let’s see how the least square method graph looks like: 
  

The data points are marked in red points. The x-axis represents the independent variable, and the y-axis represents the dependent variable. The line of best fit should minimize the vertical distances (residuals) from all data points to the line. This shows the method can be used to obtain the equation of the best fit line.

The least square method is considered as the best way to find the line of best fit, but also it has some disadvantages. Here are some of the pros and cons of the least square method. 
 

The least square method is used in various fields. It is mostly used to predict stock prices and analyze scientific data. Here, we'll be looking at some real-life applications of the least square method:

• To predict the future trends in stock market, investors use the least square method by analyzing the historical stock price.

• Least square method is used for weather forecasting by analyzing the past data.

• In medical research, the least square method is used to determine life expectancy based on the lifestyle. For example, the effects of alcoholism on life expectancy or predicting a patient's blood pressure based on age.