Why the Fitted Line Passes Through the Mean Point

Introduction

A fitted regression line summarizes how a response variable changes with an explanatory variable. One precise fact about the least-squares line is that it always passes through the point whose coordinates are the two sample means, \((\bar{x},\bar{y})\). This is not an extra condition imposed after the line is fitted; it follows from the equations that define the least-squares line.

The property is useful as a check on calculations and as a way to understand the intercept. It also has limits: the mean point need not be one of the observed data points, and the fact that the line passes through it does not mean the line fits every observation closely.

The Intercept Equation Gives the Result

For paired quantitative data with variation in \(x\), the usual least-squares regression line has slope \(b_1\) and intercept

Substitute \(x=\bar{x}\) into the fitted-line equation. The fitted response is

Therefore, \((\bar{x},\bar{y})\) is on the fitted line. The slope terms cancel: the intercept is positioned exactly so that the line’s fitted response at the mean explanatory value is the mean response.

The equation for the intercept is part of the least-squares equations. A short derivation shows why it holds. For a proposed intercept and slope, the sum of squared residuals is

At the least-squares fit, this quantity is minimized. Holding the fitted slope fixed and differentiating with respect to the intercept gives

At a minimizing intercept, this derivative is zero. Thus the residuals \(e_i=y_i-(b_0+b_1x_i)\) sum to zero:

Divide by \(n\) and use the definitions of the sample means:

This establishes the intercept equation and, by substitution, the mean-point property. This algebraic result concerns the least-squares fit to the observed pairs. It does not require a normally distributed response or a perfect linear relationship. For the usual uniquely calculated slope, the \(x\)-values must have nonzero variation.

Worked Examples

What the Property Does and Does Not Say

The mean point gives a reference location for the fitted line: when the explanatory variable equals its sample mean, the fitted response equals the sample mean response. In that sense, the intercept adjusts the vertical position of the line to balance the fitted values around the response mean. This is a mathematical feature of the least-squares equations, not evidence by itself that the relationship is strong or that the line is a good practical summary.

The zero-sum residual result provides another route to the same idea. Since the residuals sum to zero, the sum of observed responses equals the sum of fitted responses. The fitted values lie on a line, and their average is the line’s value at \(\bar{x}\):

This equality concerns averages. Positive and negative residuals can still be large, and the data may show a curved pattern or substantial scatter. The mean-point property alone cannot establish that the linear form is appropriate or that predictions are accurate.

The property can also help identify arithmetic errors. After calculating a least-squares line, substitute \(\bar{x}\) into the equation and check that the result equals \(\bar{y}\), up to rounding. If it does not, inspect the intercept calculation, the sample means, and any prematurely rounded coefficients. With rounded slope and intercept, a small discrepancy can arise because the reported coefficients are only approximations.

Common Mistake

A common mistake is to say, “The regression line passes through the average observation, so one of the data points must lie at \((\bar{x},\bar{y})\).” That reasoning confuses the coordinate-wise averages with an observed pair. The mean \(x\)-value may come from averaging all the \(x\)-values, and the mean \(y\)-value from averaging all the \(y\)-values; those two averages need not belong to the same individual.

The correction is to state exactly what the identity guarantees: the fitted value at \(x=\bar{x}\) is \(\bar{y}\). It does not guarantee a zero residual for any particular observation, nor does it make every fitted value equal to its observed response.