Regression Coefficients
Regression coefficients are the quantities by which the variables in a regression equation are multiplied. The most commonly used type of regression is linear regression. The aim of linear regression is to find the regression coefficients that produce the bestfitted line.
The regression coefficients in linear regression help in predicting the value of an unknown variable using a known variable. In this article, we will learn more about regression coefficients, their formulas as well as see certain associated examples so as to find the bestfitted regression line.
1.  What are Regression Coefficients? 
2.  Formula for Regression Coefficients 
3.  How to Find Regression Coefficients? 
4.  Regression Coefficients Interpretation 
5.  FAQs on Regression Coefficients 
What are Regression Coefficients?
Regression coefficients can be defined as estimates of some unknown parameters to describe the relationship between a predictor variable and the corresponding response. In other words, regression coefficients are used to predict the value of an unknown variable using a known variable. Linear regression is used to quantify how a unit change in an independent variable causes an effect in the dependent variable by determining the equation of the bestfitted straight line. This process is known as regression analysis.
Formula for Regression Coefficients
The goal of linear regression is to find the equation of the straight line that best describes the relationship between two or more variables. For example, suppose a simple regression equation is given by y = 7x  3, then 7 is the coefficient, x is the predictor and 3 is the constant term. Suppose the equation of the bestfitted line is given by Y = aX + b then, the regression coefficients formula is given as follows:
a = \(\frac{n(\sum xy)(\sum x)(\sum y)}{n(\sum x^{2})(\sum x)^{2}}\)
b = \(\frac{(\sum y)(\sum x^{2})(\sum x)(\sum xy)}{n(\sum x^{2})(\sum x)^{2}}\)
here, n refers to the number of data points in the given data sets.
How to Find Regression Coefficients?
Before determining the regression coefficients to find the bestfitted line, it is necessary to check whether the variables follow a linear relationship or not. This can be done by using the correlation coefficient and interpreting the corresponding value. Given below are the steps to find the regression coefficients for regression analysis.
 To find the coefficient of X use the formula a = \(\frac{n(\sum xy)(\sum x)(\sum y)}{n(\sum x^{2})(\sum x)^{2}}\).
 To find the constant term the formula is b = \(\frac{(\sum y)(\sum x^{2})(\sum x)(\sum xy)}{n(\sum x^{2})(\sum x)^{2}}\).
 Now input the regression coefficients in the equation Y = aX + b.
 A scatter plot can also be made so as to visually depict the regression line as shown below.
Regression Coefficients Interpretation
It is necessary to understand the nature of the regression coefficient as this helps to make certain predictions about the unknown variable. It helps to check to what extent a dependent variable will change with a unit change in the independent variable. Given below are the regression coefficients interpretation.
 If the sign of the coefficients is positive it implies that there is a direct relationship between the variables. This means that if the independent variable increases (or decreases) then the dependent variable also increases (or decreases).
 If the sign of the coefficients is negative it means that if the independent variable increases then the dependent variable decreases and vice versa. This means it is an indirect relationship.
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Important Notes on Regression Coefficients
 Regression coefficients are values that are used in a regression equation to estimate the predictor variable and its response.
 The most commonly used type of regression is linear regression. The equation of the bestfitted line is given by Y = aX + b.
 By using formulas, the values of the regression coefficient can be determined so as to get the regression line for the given variables.
Examples on Regression Coefficients

Example 1: Find the regression coefficients for the following data:
Age Glucose Level 43 99 21 65 25 79 42 75 57 87 59 81 Solution:
Age (x) Glucose Level (y) xy x^{2} y^{2} 43 99 4257 1849 9801 21 65 1365 441 4225 25 79 1975 625 6241 42 75 3150 1764 5625 57 87 4959 3249 7569 59 81 4779 3481 6561 Total = 247 486 20485 11409 40022 The formula for finding the regression coefficients are as follows:
a = \(\frac{n(\sum xy)(\sum x)(\sum y)}{n(\sum x^{2})(\sum x)^{2}}\)
= 0.39
b = \(\frac{(\sum y)(\sum x^{2})(\sum x)(\sum xy)}{n(\sum x^{2})(\sum x)^{2}}\)
= 65.14
The regression equation is Y = 0.39X + 65.14
Answer: a = 0.39 and b = 65.14

Example 2: Find the regression line for the following data.
A B 6.25 4.03 6.5 4.02 6.5 4.02 6 4.04 6.25 4.03 6.25 4.03 Solution:
X Y XY X^{2} Y^{2} 6.25 4.03 25.19 39.06 16.24 6.5 4.02 26.13 42.25 16.16 6.5 4.02 26.13 42.25 16.16 6 4.04 24.24 36 16.32 6.25 4.03 25.19 39.06 16.24 6.25 4.03 25.19 39.06 16.24 Total = 37.75 24.17 152.06 237.69 97.37 The formula for finding the regression coefficients are as follows:
a = \(\frac{n(\sum xy)(\sum x)(\sum y)}{n(\sum x^{2})(\sum x)^{2}}\)
= 0.04
b = \(\frac{(\sum y)(\sum x^{2})(\sum x)(\sum xy)}{n(\sum x^{2})(\sum x)^{2}}\)
= 4.28
The regression equation is Y = 0.04X + 4.28
Answer: Regression equation is Y = 0.04X + 4.28

Example 3: Plot the graph for the following data if the regression coefficients are given as a = 0.07 and b = 68.63
X Y 130 55 135 56 140 62 142 63 147 63 156 51 Solution: The regression coefficients are given as a = 0.07 and b = 68.63
Thus, the regression line is Y = 0.07X + 68.63
Thus, the scatter plot can be drawn as follows:
FAQs on Regression Coefficients
What are Regression Coefficients in Statistics?
In statistics, regression coefficients can be defined as multipliers for variables. They are used in regression equations to estimate the value of the unknown parameters using the known parameters.
What are Regression Coefficients Independent of?
Regression coefficients are independent of the change of scale as well as the origin of the plot.
What is the Formula for Regression Coefficients?
The formula for regression coefficients is given as a = \(\frac{n(\sum xy)(\sum x)(\sum y)}{n(\sum x^{2})(\sum x)^{2}}\) and b = \(\frac{(\sum y)(\sum x^{2})(\sum x)(\sum xy)}{n(\sum x^{2})(\sum x)^{2}}\).
How are Regression Coefficients used in a Linear Regression Line?
The equation of a linear regression line is given as Y = aX + b, where a and b are the regression coefficients.
How to Interpret Regression Coefficients?
If the value of the regression coefficients is positive then it means that the variables have a direct relationship while negative regression coefficients imply that the variables have an indirect relationship.
How to Calculate Regression Coefficients?
The steps to calculate the regression coefficients are as follows:
 Substitute values to find a (coefficient of X).
 Substitute values for b (constant term).
 Put the values of these regression coefficients in the linear equation Y = aX + b.
What Do Regression Coefficients Tell Us?
Regression Coefficients tell us how much a dependent variable changes with a unit change in the independent variables.
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