Use of the Regression Line

Introductory Overview

The regression line, also known as a line of best fit, is a key tool for understanding and predicting relationships in data sets.
Regression lines can be used to identify and project patterns, estimate relevant values outside the data set, and indicate statistical relationships between variables.

A regression line is a straight line that best represents the data on a scatter plot.
It can be represented as: y = ax + b where y is the predicted score, a is the gradient of the line, x is the input score, and b is the y-intercept.
The gradient (a) of the regression line denotes the effect that the x-variable has on the y-variable.
The y-intercept (b) shows the predicted value when x equals zero.

Regression lines have a strong application in prediction or forecasting, where they can be used to estimate one variable based on the other.
By locating an x-value on the x-axis, moving to the regression line, and then across to the y-axis, you can find a ‘predicted’ y-value, and vice versa.

The stronger the correlation, the closer the data points will lie to the regression line, and the more accurately it will predict.
Positive gradients denote a positive correlation (as x increases, so does y), whereas negative gradients signify a negative correlation (as x increases, y decreases).
A zero gradient would suggest no correlation between variables.
The residuals (distances between the data points and the regression line) provide an indication of the accuracy of the predictions.
Any data not fitting the pattern (outliers) can greatly affect the regression line and the correlation coefficient.

A regression line is only reliable for predicting within the range of values included in the data set.
The relationship between variables assumes to be linear. It won’t accurately represent the data if the relationship is not linear.
As the saying goes, correlation does not imply causation. Even if your variables correlate, it doesn’t mean that changes in one directly cause changes in the other.
The prediction from a regression line will likely be less accurate the further you go from the range of the original data, known as extrapolation.

For example, using a regression line plotted from historical data of sales and advertising expenditures, a company might estimate future sales based on planned advertising spend.