Graphical Analysis and Curve Fitting

Graphical analysis is the primary tool for extracting physical relationships from data.

Plotting conventions

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

Linearisation — Many physical laws become linear with the right transformation:

Least-squares fit — For $y = mx + c$, minimise:

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

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

import numpy as np
x = np.array([1, 2, 3, 4, 5])
y = np.array([2.1, 3.9, 6.2, 7.8, 10.1])
m, c = np.polyfit(x, y, 1)
print(f"Slope = {m:.2f}, Intercept = {c:.2f}")

Key insight: If your data doesn't linearise as expected, the assumed model may be wrong. The graph tells you this before any statistical test does.

Common pitfall: Fitting a curve to data proves little by itself — enough parameters fit anything. Linearize so theory predicts a straight line, then let the residuals judge: structure in the residuals means the model, not the data, is wrong.

Least-Squares Regression

A best-fit line $y = mx + b$ minimises the sum of squared residuals:

$$S = \sum_{i=1}^{n} (y_i - mx_i - b)^2$$

Setting $\partial S/\partial m = 0$ and $\partial S/\partial b = 0$ yields slope and intercept.

The correlation coefficient $r^{2}$ quantifies fit quality (1 = perfect, 0 = none).

A poor fit may indicate systematic errors or an incorrect theoretical model.