Regression Constant Calculator

Formula

The formula to calculate the regression constant (a) is:

\[ a = \frac{(\Sigma Y \cdot \Sigma X^2 - \Sigma X \cdot \Sigma XY)}{(n \cdot \Sigma X^2 - (\Sigma X)^2)} \]

Where:

a is the regression constant
\(\Sigma Y\) is the sum of Y values
\(\Sigma X\) is the sum of X values
\(\Sigma XY\) is the sum of the products of X and Y values
\(\Sigma X^2\) is the sum of X values squared
n is the number of data points

What is a Regression Constant?

The regression constant (a) is the y-intercept of the linear regression line. It represents the point where the regression line crosses the Y-axis. In the context of a linear regression equation \(y = ax + b\), the constant is the value of \(y\) when \(x\) equals zero. It is a crucial part of the linear regression model as it provides a starting point for the predicted relationship between the independent variable (\(x\)) and the dependent variable (\(y\)).

Example Calculation

Example 1:

Sum of Y values (ΣY): 150
Sum of X values (ΣX): 100
Sum of XY values (ΣXY): 2000
Sum of X² values (ΣX²): 1200
Number of Data Points (n): 10

Step 1: Calculate the numerator:

\[ \Sigma Y \cdot \Sigma X^2 - \Sigma X \cdot \Sigma XY = 150 \cdot 1200 - 100 \cdot 2000 = 180000 - 200000 = -20000 \]

Step 2: Calculate the denominator:

\[ n \cdot \Sigma X^2 - (\Sigma X)^2 = 10 \cdot 1200 - 100^2 = 12000 - 10000 = 2000 \]

Step 3: Calculate the regression constant:

\[ a = \frac{-20000}{2000} = -10 \]

Example 2:

Sum of Y values (ΣY): 180
Sum of X values (ΣX): 120
Sum of XY values (ΣXY): 2400
Sum of X² values (ΣX²): 1500
Number of Data Points (n): 12

Step 1: Calculate the numerator:

\[ \Sigma Y \cdot \Sigma X^2 - \Sigma X \cdot \Sigma XY = 180 \cdot 1500 - 120 \cdot 2400 = 270000 - 288000 = -18000 \]

Step 2: Calculate the denominator:

\[ n \cdot \Sigma X^2 - (\Sigma X)^2 = 12 \cdot 1500 - 120^2 = 18000 - 14400 = 3600 \]

Step 3: Calculate the regression constant:

\[ a = \frac{-18000}{3600} = -5 \]