Calculation of the equation of the regression line

The Equation of the Regression Line

A regression line is utilised to model the relationship between two statistical variables.
This line is an optimal way to represent the possible relationship between two sets of data, often represented as Y = a + bX.
Here, Y represents the dependent variable we aim to predict or forecast, X is the independent variable used as a predictor, while a and b represent the regression coefficients.

The intercept ‘a’ is essentially the expected mean value of Y when all X variables are set to 0.
The slope ‘b’ defines the direction (either positive or negative) and steepness of the line, this is synonymous with the amount of change in Y for each unit change in X.

In simple linear regression, the coefficients ‘a’ and ‘b’ can be calculated using the following formulas:
- b = (∑(xi - mean(x)) * (yi - mean(y))) / ∑(xi - mean(x))^2
- a = mean(y) - b * mean(x)
Here, xi and yi represent the observations of variables X and Y, and mean(x) and mean(y) represent the respective means of these observations.

The equation of the regression line plays a crucial role in making predictions about the dependent variable Y based on the value of an independent variable X.
The slope ‘b’ indicates the rate of change in Y , in proportion to the change in X. A positive slope specifies that Y increases with X, and a negative slope signifies that Y decreases as X increases.
The intercept ‘a’ is the point where the regression line intersects the Y-axis. It’s the predicted value of Y when X equals zero.

For instance, a researcher might utilise the regression line to forecast how a slight increase in temperature (X) could possibly affect ice cream sales (Y).