Squaring the vertical residuals keeps positive and negative deviations from canceling in the fitting criterion. Least squares therefore evaluates how far observed outcomes lie above or below their predicted values while selecting one slope and intercept for the line. This makes the fitted result reflect the overall scatter around the line rather than simply adding signed differences.
The slope communicates how the predicted outcome changes as the predictor changes, while the intercept supplies the line’s starting value on the fitted scale. Together, these two quantities specify the line used for estimation. Interpreting them should remain tied to the variables and data range shown in the scatterplot, not treated as proof of causation.
Points that sit far above or below the fitted line have large vertical residuals, meaning their observed values differ substantially from the line’s predicted values. Reviewing these observations helps analysts identify unusual cases and decide whether they deserve closer attention during interpretation. This check complements the overall summary by showing where individual data points do not follow the common pattern.
To apply the method, first display the two quantitative variables in a scatterplot and identify which variable will be predicted. Calculate the least-squares slope and intercept, then use the resulting equation to obtain predicted values. Comparing those predictions with observations through vertical residuals helps assess how closely the data follow the fitted linear pattern.
Researchers turn to this approach when they need both a concise description of association and an estimate of an outcome from a predictor. In statistics, it can summarize a scatterplot, evaluate whether observations follow a linear trend, and flag unusual points. These uses support applications in biology, economics, and engineering.
A close-fitting line does not establish that changing the predictor causes the outcome to change. It summarizes the observed association and can support estimation within the data being examined. Statistical interpretation therefore requires separating what the line shows about the relationship from any causal explanation, especially when using the result to discuss real-world systems.