How to Read a Linear Regression Output

Linear regression is one of the most widely used statistical techniques in data analysis. It models the relationship between a dependent variable (Y) and one or more independent variables (X) by fitting a linear equation to observed data. The result is a straight line that best predicts Y from X.

Understanding the output of a linear regression is critical for interpreting data in fields ranging from economics and social sciences to engineering and healthcare. This guide walks through each element of a typical regression output and explains what it tells you about your data.

The core output of simple linear regression is the equation y = mx + b. The slope (m) describes how much Y changes for each one-unit increase in X. A positive slope means Y increases as X increases; a negative slope means Y decreases as X increases. The y-intercept (b) is the value of Y when X is zero.

For example, if you are modelling house prices based on square footage and get the equation y = 250x + 50,000, it means each additional square foot adds $250 to the price, and a 0 sq ft house would theoretically cost $50,000 (the intercept captures factors not in the model).

Pearson r measures the strength and direction of the linear relationship between X and Y, ranging from -1 to +1. A value of +1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no linear relationship. Values above 0.7 or below -0.7 are generally considered strong correlations.

Correlation does not imply causation – a high r value means the variables move together, but it does not prove that X causes Y. There may be a third variable (a confounder) driving both, or the relationship may be coincidental.

R² tells you how well the regression model fits your data. It represents the proportion of variance in Y that is explained by X, ranging from 0 to 1. An R² of 0.85 means 85% of the variation in Y is explained by the model, leaving 15% unexplained.

In many real-world contexts, especially with human behaviour data, R² values of 0.3-0.5 are considered good. In controlled scientific experiments, you would expect higher values. A low R² does not necessarily mean the model is useless – it may just mean other important factors are not included in the model.

Always check for outliers, which can significantly skew your regression line. Visualising your data with a scatter plot before running regression is essential. Also verify that the relationship is approximately linear – if the data follows a curve, linear regression will give misleading results.

Be cautious about extrapolating beyond your data range. A model built on X values between 10 and 100 may not hold for X = 200. Also check for homoscedasticity (constant variance of residuals) – if the spread of residuals changes with X, your model may be unreliable.