A Dataset That Lies Exactly on a Line

When Every Point Falls on One Line

A scatterplot sometimes shows a striking pattern: every observed point lies exactly on the same straight line. In that case, a least-squares regression line can reproduce every response in the dataset with no error. Working through this special case makes the relationship between fitted values, residuals, and squared error especially clear.

The key word is observed. A dataset can lie exactly on a line without showing that the same relationship holds for every individual in a larger population, or that changing the predictor causes a change in the response. Here we study the arithmetic of fitting a line to the data at hand.

For a simple linear regression fit, the fitted response at predictor value \(x_i\) is \(\hat{y}_i=\hat{\beta}_0+\hat{\beta}_1x_i\), where \(\hat{\beta}_0\) and \(\hat{\beta}_1\) are the estimated intercept and slope. The residual for observation \(i\) is \(e_i=y_i-\hat{y}_i\). The residual sum of squares, also called the sum of squared errors, is \(\mathrm{SSE}=\sum_{i=1}^{n}e_i^2\).

To calculate a simple regression line from observed data, the predictor values must vary: at least two \(x_i\) values must be distinct. The calculations below use that condition. They do not require a probability model for the observations; the focus is on the fitted line and its errors within the dataset.

Why an Exact Line Gives Zero Squared Error

If \(y_i=a+bx_i\) for each observation, the line with intercept \(a\) and slope \(b\) predicts \(a+bx_i\) at \(x_i\). Therefore it predicts \(y_i\) exactly. Least squares chooses the intercept and slope to minimize the sum of squared residuals. Since squared residuals cannot be negative, no candidate line can have an SSE below zero. The exact line attains that lower bound.

For calculation, the simple least-squares slope and intercept are

When a dataset is exactly on a line and its \(x\) values vary, these formulas recover that line's slope and intercept. The examples show the calculation rather than relying only on the visual pattern.

Checking the Calculation Observation by Observation

A useful verification is to substitute every observed predictor into the fitted equation and compare the result with the recorded response. If the observations really do follow one line, each comparison matches. Checking individual values guards against a mistaken coefficient or arithmetic error that could be hidden by reporting only a slope and intercept.

A Short Justification of the Least-Squares Result

The conclusion that the exact line is a least-squares fit follows directly from the definition of the objective. For any candidate intercept \(c\) and slope \(d\), its residual at observation \(i\) is \(y_i-(c+dx_i)\). Its SSE is the sum of the squares of those residuals, so it must be at least zero. If \(y_i=a+bx_i\) for every \(i\), choosing \(c=a\) and \(d=b\) makes every residual zero and gives SSE zero. Thus that line reaches the smallest possible SSE.

If at least two observed predictor values are distinct, this exact-fitting line is also the only line that can give SSE zero. To see why, an SSE of zero is a sum of nonnegative squared residuals. It can equal zero only if every residual equals zero. Any zero-SSE line must therefore pass through every observed point. Two points with distinct \(x\) values determine one straight line, so any such line must have the same slope and intercept as the exact line.

What a Zero SSE Does and Does Not Say

A zero residual sum of squares means there is no discrepancy between observed and fitted responses at the predictor values in this dataset. Because every squared residual is nonnegative, a sum of zero also confirms that each residual is zero individually; positive and negative errors cannot cancel inside a sum of squares.

The squared errors use squared response units. For example, if responses are measured in millimeters, residuals are in millimeters and SSE is in square millimeters. In an exactly linear dataset the numerical total is zero regardless of the response units, but stating the units helps clarify what quantity is being calculated.

A perfect fit to a finite set of observations is not by itself proof of a deterministic law. The pattern might hold only over the observed range, and measurements rounded before recording can affect whether the displayed values lie exactly on a line. Nor does an exact line establish that changing \(x\) causes \(y\) to change. Regression here summarizes the observed association and fits a line; causal claims require appropriate evidence and design.

Common Mistakes

• Claiming the SSE is merely small. If every observed response equals its fitted value, each residual is exactly zero, so the SSE is exactly zero.
• Adding residuals before squaring and calling that the SSE. The SSE is the sum of squared residuals, \(\sum e_i^2\), not the square of their sum. Here both happen to be zero, but they are different calculations in general.
• Thinking positive and negative residuals cancel in the SSE. Squaring makes each contribution nonnegative. Cancellation cannot produce an SSE of zero unless every residual is zero.
• Generalizing the perfect fit beyond the data. An exact fit at the observed predictor values does not guarantee exact predictions for unobserved cases or establish a causal relationship.