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Data Fitting & Linearisation (IPhO)

IPhO experimental lesson on fitting and linearisation: picking transforms, interpreting slope/intercept, and writing parameter conclusions.

  • International Physics Olympiad preparation

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Good fitting is mostly bookkeeping: choose a plot that matches the model, label it correctly, extract slope and intercept with uncertainties, then translate back to physical parameters.

Prerequisites (quick refresh)

1. Definitions (Must Know)

  • Model: a relationship you expect, written with parameters (for example y = ax + b).
  • Linearisation: transforming variables so the model becomes Y = mX + c.
  • Gradient (slope): m = Δ Y/Δ X from the best-fit line.
  • Intercept: c at X = 0 on the chosen plot.
  • Residual: difference between data and model, rᵢ = Yᵢ - (mXᵢ + c).
  • Error bars: graphical representation of uncertainty in each point.

2. Key Ideas (What Earns Marks)

Marks usually come from these statements
  • Write the starting model and show how you rearranged it to Y = mX + c.
  • Label axes with the transformed quantities and units.
  • Draw a best-fit line that balances scatter, not point-to-point joins.
  • Quote m and c with uncertainties (often using max and min acceptable lines).
  • Convert m and c into the physical parameter(s), including uncertainty propagation.
  • Comment on fit quality using scatter or residual trend, not just “looks linear”.

3. Detailed Explanations

A. Picking the right plot

Common patterns:

  • Power law y = kxⁿ: take logs, ln y = ln k + n ln x.
  • Exponential y = Ae^bx: ln y = ln A + bx.
  • Inverse y = a + b/x: plot y against 1/x.
  • Square law y = ax²: plot y against x².

The best choice is the one that makes parameters appear as a slope or intercept with clear units.

B. Uncertainties under transformations

If you transform axis values, you should also transform uncertainties.

Useful transform rules (small-uncertainty approximations)

If Y = ln y, then Δ Y ≈ Δ y/y.

If X = 1/x, then Δ X ≈ Δ x/x².

If X = x², then Δ X ≈ 2x Δ x.

C. Getting gradient and its uncertainty

A standard IPhO-style method:

  • draw one best-fit line through the error bars, balancing points above and below
  • draw the steepest and shallowest lines still consistent with the error bars

Then m = m_best, Δ m ≈ (mₘₐₓ-mₘᵢₙ)/2. You can do the same for c.

D. Converting slope to parameters

If a parameter is a function of the gradient, propagate uncertainty using relative uncertainties. Example: if g = 4π²/s then Δ g/g = Δ s/s.

4. Common Mistakes

  • Axes labels missing units after a transform.
  • Mixing log ₁₀ and ln between derivation and calculation.
  • Forgetting to transform uncertainties when plotting transformed variables.
  • Forcing the best-fit line through the origin without a physical reason.
  • Taking two data points to find a gradient instead of using the full fit.
  • Reporting a parameter but not stating whether it came from slope or intercept.

5. Exam Tips

  • Put the linearised form directly under the graph: “Plot Y against X so Y = mX + c”.
  • Choose two well-separated points on the best-fit line (not raw data) to calculate m.
  • Quote m with units of “vertical axis unit per horizontal axis unit”.
  • A short fit-quality sentence helps: random scatter is fine, systematic curvature suggests a wrong model.

6. Worked Examples

Example 1: Pendulum, find g

Model: T = 2π square root of (L/g) ⇒ T² = (4π²/g)L. Plot T² (s²) against L (m). The slope s has units s² m⁻¹ and s = 4π²/g ⇒ g = 4π²/s.

If your graph gives s = 4.02 ± 0.08 s² m⁻¹ then g ≈ 39.48/4.02 = 9.82 m s⁻², Δ g ≈ 9.82 × 0.08/4.02 ≈ 0.20. So g = (9.82 ± 0.20) m s⁻².

Example 2: Inverse-square check

Suppose theory predicts I ∝ 1/r². Plot I against X = 1/r². If the graph is straight within uncertainty and the intercept is consistent with zero, the model is supported.

If your fitted line is I = (2.50 ± 0.10)X + (0.03 ± 0.05) in suitable units, the intercept is consistent with zero, and the slope is the proportionality constant with about a 4 percent uncertainty.

7. Mind Stretchers

  • If you take logs, are the error bars symmetric in ln y even when they were symmetric in y?
  • When would you prefer to fit the original non-linear model instead of linearising?
  • A residual plot shows a U-shape trend. What does that imply about the model?

8. Practice

  1. You expect y = a + b/x. What should you plot to get a straight line, and what are slope and intercept?
  2. For y = kxⁿ, you plot ln y against ln x and get slope 1.85 ± 0.10. State n and its uncertainty.
  3. Your best-fit slope is m = 0.620 with mₘᵢₙ = 0.590 and mₘₐₓ = 0.650. Find Δ m and the percentage uncertainty in m.
  4. A fitted gradient is used to compute k = m/2. How does Δ k/k compare to Δ m/m?

Back to IPhO Experimental Skills Hub

Answer key and feedback

  • Plot y against 1/x: intercept a, slope b.
  • Symmetric errors in y are generally asymmetric after taking ln y; use transformed upper and lower bounds or fit in the original variables.
  • Prefer a direct nonlinear fit when linearisation distorts uncertainties, creates correlated errors or overweights part of the range.
  • A U-shaped residual trend is evidence of model curvature left unexplained, not random scatter about an adequate model.
  • Since k = m/2 differs only by an exact constant, Delta k/k = Δ m/m.
Syllabus and review details

No official syllabus alignment is listed for this lesson.