Graphical Analysis and Curve Fitting

Graphical Analysis Basics

Graphical analysis extracts physical relationships from data. If your data doesn't linearise as expected, your assumed model may be wrong.

Plotting conventions:

• Independent variable on $x$, dependent on $y$.
• Label axes with quantity, symbol, and units (e.g., Force $F$ (N)).
• Include error bars for measurement uncertainty.

Linearisation transforms non-linear laws into straight lines:

Common pitfall: Fitting a curve proves little; enough parameters fit anything. Linearise so theory predicts a straight line, then check residuals. Structure in residuals means the model is wrong.

Least-Squares Fitting

A best-fit line $y = mx + b$ minimises squared residuals using the least-squares fit formula:

$$\chi^2 = \sum_{i=1}^N \frac{(y_i - mx_i - c)^2}{\sigma_i^2}$$

Here, $y_i$ and $x_i$ are data points, and $\sigma_i$ is uncertainty.

Reduced chi-squared is defined as $\chi^{2}_\nu = \chi^{2}/(N-p)$, where $p$ is the number of fit parameters. A good fit yields $\chi^{2}_\nu \approx 1$.

The correlation coefficient $r^{2}$ quantifies fit quality, where 1 is perfect and 0 is none.

Pitfall: A poor fit or poor $r^2$ may indicate systematic errors or an incorrect theoretical model.