[Maths Class Notes] on Linear Regression Formula Pdf for Exam

Ever been to a shop and have noticed how the size of an object directly affects its price as well? Well, a relation is seen when two quantities are compared and there is either an increase or decrease in the value of both of them or it can also be that one quantity increases while the other decreases and vice versa. If these two quantities are further plotted on a graph, it is observed that there is a linear relation between them.  Linear regression formula helps to define this linear relation that is present between the two quantities and how they are interdependent.

 

Linear regression is known to be the most basic and commonly used predictive analysis. In this concept, one variable is considered to be an explanatory variable, and the other variable is considered to be a dependent variable. For example, a modeller might want to relate the weights of individuals to their heights using the concept of linear regression.

 

Simple Linear Regression

 

Multiple Linear Regression

 

Logistic Regression

 

Ordinal Regression

 

Multinomial Regression

 

Discriminant Analysis

 

What is Linear Regression?

Let’s know what linear regression is. It is very important and used for easy analysis of the dependency of two variables. One variable will be considered to be an explanatory variable, while others will be considered to be a dependent variable. Linear regression is a linear method for modelling the relationship between the independent variables and dependent variables. The linearity of the learned relationship makes the interpretation very easy. Linear regression models have long been used by people as statisticians, computer scientists, etc. who tackle quantitative problems. For example, a statistician might want to relate the weights of individuals to their heights using a linear regression model. Now we know what linear regression is.

 

The Formula of Linear Regression

Let’s know what a linear regression equation is. The formula for linear regression equation is given by:

y = a + bx

 

a and b can be computed by the following formulas:

b= [frac {nsum xy – (sum x)(sum y)} {nsum x^2 – (sum x)^2}]

a= [frac {sum y – b(sum x)} {n}]

Where

x and y are the variables for which we will make the regression line.

• b =  Slope of the line.

• a =  Y-intercept of the line.

• X  = Values of the first data set.

• Y = Values of the second data set.

 

Note: The first step in finding a linear regression equation is to determine if there is a relationship between the two variables. This is often a judgment call for the researcher. You’ll also need a list of your data in an x–y format (i.e. two columns of data – independent and dependent variables).

 

Simple Linear Regression Formula Plotting

Table 1. Example data.

 

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The concept of linear regression consists of finding the best-fitting straight line through the given points. The best-fitting line is known as a regression line. The black diagonal line in the figure given below (Figure 2) is the regression line and consists of the predicted score on Y for each possible value of the variable X. The lines in the figure given above, the vertical lines from the points to the regression line, represent the errors of prediction. As you can see, the red point is actually very near the regression line; we can see its error of prediction is small. By contrast, the yellow point we can see is much higher than the regression line and therefore its error of prediction is large.

 

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The black line given in the figure consists of the predictions, the points that are the actual data, and the vertical lines between the points and the black line represent errors of prediction.

 

Properties of Linear Regression

For the regression line where the regression parameters b0 and b1 are defined, the properties are given as below:

• The line reduces the sum of squared differences between observed values and predicted values.

• The regression line passes through the mean of X and Y variable values.

• The regression constant (b0) is equal to the y-intercept of the linear regression.

• The regression coefficient (b0) is the slope of the regression line which is equal to the average change in the dependent variable (Y) for a unit change in the independent variable (X).

 

What is Linear Regression Used for?

Linear regression is used for: 

• The concept of studying engine performance from test data in automobiles.

• Linear regression can be used in market research studies and customer survey results analysis.

• Linear regression can be used in observational astronomy commonly enough. A number of statistical tools and methods can be used in astronomical data analysis, and there are entire libraries in languages like Python meant to do data analysis in astrophysics.

• Linear regression can also be used to analyze the marketing effectiveness, pricing, and promotions on sales of a product.

 

Questions to be Solved

Question 1) Find out the linear regression equation from the given set of data.

 

Solution:

 

Using the simple linear regression formula,

 

b= [frac {nsum xy – (sum x)(sum y)} {nsum x^2 – (sum x)^2}] ,

 

b= [frac {4times 145-18times 26} {4times 102 -324}] , Value of b is equal to 1.33.

 

Now using the simple linear regression formula to calculate the value of a=[frac {sum y-b(sum x)} {n}]= [frac {26 – 1.33times 18} {4}] = 0.515

 

Putting the values of a and b in the equation, y = a + bx

Answer: y = 0.515 + 1.33x.

 

Standard Error in Linear Regression Formula:

The standard error that is seen about the regression line can be defined as the measure of the average proportion that the regression equation over- or under-predicts. This standard error is denoted by SE. The higher the coefficient of the determination being involved, the lower the standard error and hence, a more accurate result will be available.