Plotting, Error Bars & Significant Figures (IPhO Experimental)

Graph marks in IPhO are “high leverage”: a clean plot can reveal the parameter in one glance, while a messy plot forces the examiner to hunt for meaning. This lesson is a checklist for turning raw data into a defensible graph quickly.

1. Definitions (Must Know)

• Independent variable: the quantity you control or set (usually on the horizontal axis).
• Dependent variable: the quantity you measure in response (usually on the vertical axis).
• Scale: how much of the paper/axis corresponds to a change in value.
• Error bar: a drawn range representing uncertainty in a plotted point (horizontal and/or vertical).
• Best-fit line: a single line that represents the trend; it is not a join-the-dots polyline.
• Significant figures (s.f.): digits that carry meaning about precision. You should not report more precision than the data supports.
• Consistent precision: in a final result Q ± Δ Q, the value and its uncertainty must be rounded to the same decimal place.

2. Key Ideas (What Earns Marks)

3. Detailed Explanations

A. Choosing a scale quickly

Aim for your data to occupy about 70 to 90 percent of the plot width and height.

Good habits:

• use “round” step sizes (1, 2, 5, 10, 20, 50 …)
• avoid awkward scales like 3.7 per large square
• keep at least one major tick beyond the largest point when convenient

B. Axis labels that prevent lost marks

Write labels like:

• L / m, T / s, V / V, I / A, 1/r² / m⁻²

If you plotted a transform, label the transform, not the raw symbol. Example:

• ln I (dimensionless) vs ln r (dimensionless)
• T² / s² vs L / m

C. Error bars

If uncertainty is dominated by resolution, error bars are often the same size for all points. If uncertainty is dominated by scatter, the size may vary.

Common exam-friendly rule:

• if an instrument resolution is δ, use Δ x ≈ δ/2 unless a different rule is specified
• if you have repeats, use the scatter estimate you stated, and apply it consistently

D. Best-fit line vs “line through all points”

The best-fit line should balance points above and below, reflecting scatter. If the data show curvature, forcing a straight line is a modelling error, not “bad luck”.

E. Gradient uncertainty (max/min acceptable lines)

Draw:

1. one best-fit line
2. one steepest acceptable line
3. one shallowest acceptable line

Then

This method is quick and defensible under timed conditions.

4. Common Mistakes

• Missing units on one or both axes.
• Plotting the wrong quantity: you derived Y = mX + c but plotted raw y vs x.
• Using too small a scale (points bunched up, slope uncertainty explodes).
• Forcing the line through the origin without a physical reason.
• Computing gradient from two raw data points (instead of two points on the best-fit line).
• Reporting a gradient without units (gradients always have units).
• Rounding inconsistently: writing m = 0.623 ± 0.1.

5. Exam Tips

1. Before you draw anything, write the linear form: “Plot Y against X so Y = mX + c.”
2. Put a tiny triangle on your two gradient points so the examiner sees what you used.
3. If the intercept should be zero physically, still check whether the fitted intercept is consistent with zero within uncertainty.
4. If you must take logs, keep track of base: use ln consistently unless told otherwise.

6. Worked Examples (use Toggle)

7. Mind Stretchers (use Toggle)

8. Practice

1. You expect y = a + b/x. Write the linear form and state what you should plot on each axis.
2. A graph gives mₘᵢₙ = 1.20 and mₘₐₓ = 1.36. Find Δ m and the percentage uncertainty in m if m_best = 1.28.
3. You plot Y vs X and the line has a nonzero intercept, but theory expects c = 0. List two reasons this can happen (one systematic, one modelling).

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Answer key and feedback

• Option B makes error bars more nearly uniform only when its transformation makes the measurement uncertainty approximately constant across the plotted range. Check this from propagated uncertainties; visual neatness alone is not a reason to transform.