In 1885, Francis Galton plotted the heights of 928 adult children against their parents and drew a straight line through the mess. That line - the first modern regression - showed him that tall parents produced tall children, but not as tall as expected. He called it "regression to mediocrity." You are now working with the same logic, just with better software and higher stakes.
What a Regression Line Actually Is
A linear regression model takes your data and finds the single straight line that minimizes the total squared distance between itself and every data point. The result is an equation: Y = b0 + b1*X. Here, b0 is the intercept - where your line crosses the Y axis when X equals zero - and b1 is the slope, the expected change in Y for every one-unit increase in X.
The algorithm that computes b0 and b1 is called Ordinary Least Squares (OLS). Think of OLS as a photographer adjusting the camera position until the blurriness score across every subject in the room hits its lowest possible value. Each data point is a subject; the blurriness score is the squared residual.
The residual is the difference between what your model predicted and what actually happened. A model with small residuals is not necessarily a good model - that comes later - but a model with large residuals is definitely telling you something is wrong.
Key Point: The slope coefficient b1 does not say "X causes Y." It says "in your dataset, a one-unit difference in X is associated with a b1-unit difference in Y, holding all other measured variables constant." Causation requires an experiment, not a coefficient.
Reading the Output
Most statistical software returns a table after running a regression. The columns you need to understand are the coefficient estimate, the standard error, the t-statistic, and the p-value.
The standard error measures how much your coefficient estimate would vary if you repeated the study on a fresh sample drawn from the same population. A small standard error means your estimate is precise. The t-statistic is the ratio of the coefficient to its standard error - it tells you how many standard errors away from zero your estimate sits. The p-value converts that t-statistic into a probability: assuming the true coefficient is zero, how likely is it that you would observe a t-statistic at least this extreme?
The R-squared value summarizes how much of the total variance in Y your model accounts for. An R-squared of 0.72 means your predictors explain 72% of the variation in the outcome. The remaining 28% lives in the residuals - unexplained noise, unmeasured variables, or genuine randomness.
The Honest Limitation
Galton's regression line fit his data beautifully. It also meant nothing outside the 19th-century English gentry he was studying. Every regression model is a creature of its training data. You apply it to new situations at your own risk, and good analysts quantify that risk rather than ignore it.
The line through your scatterplot is not a law of nature. It is a summary of a specific dataset collected at a specific time in specific conditions. When those conditions change, the line can drift, bend, or break entirely. You are not discovering truth; you are estimating it.