Residuals and Chi-Squared

Analyzes normalized residuals and chi-squared to assess compatibility between data and model.

Conceptual context

An experimental fit does not end when a curve appears close to the points. The physical question is whether differences between each $y_i$ and its $y_model_i$ are compatible with the stated $\sigma_{i}$. The $r_i$ preserves the sign of that difference and helps locate bias; the $z_i$ expresses separation on a comparable scale. The $chi2$ combines all discrepancies, while the $chi2nu$ accounts for how much independent information remains after fitting parameters. Local, graphical, and global readings together separate reasonable fluctuations from systematic trends, misestimated uncertainties, or an incomplete physical model.

🟢 Essential level

Each experimental point tells a small physical story. The $y_i$ says what happened; the $y_model_i$ says what the model expected.

Their signed separation is the $r_i$. A positive value means the datum lies above the prediction, while a negative value means it lies below. Yet one physical unit can be enormous in a precise experiment and negligible in a noisy one. Dividing by the $\sigma_{i}$ gives the $z_i$, which measures separation in uncertainty units. A plot against the $x_i$ should look like an unstructured cloud around zero. The $chi2$ summarizes the set, but it does not preserve order or sign. Therefore it must never be interpreted alone without also examining the $r_i$ pattern and the physical assumptions behind the uncertainties. When the $chi2nu$ is close to one, the observed scatter matches the stated $\sigma_{i}$; much larger values suggest an incomplete model or underestimated uncertainty, while values far below one usually indicate overestimated uncertainty rather than an exceptionally good fit.

🔵 Formal level

Basic formulas

For each experimental condition identified by $x_i$, the $y_i$ is compared with the corresponding $y_model_i$. The adopted convention is observed minus predicted:

Experimental residual

The $r_i$ keeps the units of the measured variable. To compare points having different precision, introduce the $\sigma_{i}$:

Normalized residual

The $z_i$ is dimensionless. Its sign retains the direction of discrepancy, and its magnitude gives separation in uncertainty units. The globally weighted discrepancy is formed by summing squared contributions:

Chi-squared

Real student questions

Does a zero $r_i$ prove the model is correct? No. It shows agreement at one point only.

Why square the values? Squaring prevents cancellation between signs and penalizes large discrepancies strongly.

Is reduced $chi2$ equal to one perfect? No. It is a conditional reference depending on error assumptions and $\nu$.

Can a very small value be suspicious? Yes. It may indicate overestimated $\sigma_{i}$, correlations, or an overly flexible fit.

What should I inspect first? The $z_i$ plot, because it preserves position and pattern, then the $chi2nu$ summary.

Final synthesis

Residuals show where and in which direction a model fails; normalization shows whether the discrepancy matters relative to uncertainty. Chi-squared summarizes the set, and its reduced form incorporates effective information. Physical compatibility emerges from combining numbers, graphical pattern, units, assumptions, and model complexity.