Graphing and linearisation of data

Raw experimental data plotted exactly as a physical law predicts often comes out as a curve rather than a straight line — inverse-square laws, quadratic relationships, and exponential decays are all common in physics and none of them produce straight lines when plotted directly. Curves are harder to work with than straight lines for a very practical reason: the human eye is excellent at judging whether points lie on a straight line and at spotting outliers or systematic curvature in the scatter, but far worse at judging these things against a curved reference. Fitting parameters are also easier to extract from a line — a slope and an intercept — than from the several parameters that typically define a curve.

Linearisation exploits this by re-plotting a transformed combination of the measured variables, chosen specifically so that the underlying physical law predicts a straight line in the new coordinates. If theory predicts y = kx², plotting y against x² (rather than against x) produces a straight line through the origin with slope k — the transformation does not change the underlying physics at all, it only changes which combination of variables is placed on each axis. If theory predicts exponential decay, y = A·e^(−λx), plotting the natural logarithm of y against x produces a straight line with slope −λ and intercept ln(A), since taking the logarithm of both sides gives ln(y) = ln(A) − λx, an equation linear in x. Power-law relationships y = Ax^n are similarly linearised by plotting log(y) against log(x), which gives a line of slope n and intercept log(A).

The choice of which transformation to apply is not arbitrary; it must follow directly from the specific functional form the underlying physics predicts. Applying an unjustified transformation can force almost any data into looking roughly linear, which defeats the entire purpose — linearisation is a tool for testing and extracting parameters from a specific predicted relationship, not for making arbitrary data look tidy. Once genuinely linearised, a straight-line fit (see least-squares fitting) gives both the best-fit values of the physical parameters and a clear visual and statistical check of whether the data actually supports the assumed relationship in the first place.