Uncertainty Quick Reference
Free resource by Flowxiom — Edexcel A-level Physics
Not everything. Just what’s on the paper. High-frequency topics only — covering ~80% of exam marks.
Every paper — pure format marks — memorise and collect them.
Key Concepts
| Term | Meaning |
|---|---|
| Random error | Scatter randomly above/below the true value → reduced by averaging repeated measurements |
| Systematic error | Always offset in the same direction → cannot be reduced by repetition; must change the instrument or experimental design |
| Accuracy | How close a measurement is to the true value — affected by systematic error |
| Precision | How close repeated measurements are to each other — affected by random error |
⚠️ Most common mistake: claiming that repetition reduces systematic error — it does NOT.
Estimating Uncertainty
| Situation | Method |
|---|---|
| Single reading | \(\Delta x = \pm\) half the smallest scale division (or one full division — follow question context) |
| Repeated readings | \(\Delta x = \dfrac{x_{max} – x_{min}}{2}\) |
Propagation Rules
Addition / Subtraction → Add absolute uncertainties
\[y = a + b \quad \text{or} \quad y = a – b\]
\[\Delta y = \Delta a + \Delta b\]
⚠️ When subtracting quantities, you still ADD their uncertainties.
Example: temperature difference \(\Delta\theta = \theta_2 – \theta_1\), uncertainty \(= \Delta\theta_1 + \Delta\theta_2\)
Multiplication / Division → Add percentage uncertainties
\[y = ab \quad \text{or} \quad y = \frac{a}{b}\]
\[\%U_y = \%U_a + \%U_b \quad \text{where} \quad \%U_a = \frac{\Delta a}{a} \times 100\%\]
Powers → Multiply percentage uncertainty by the index
\[y = a^n\]
\[\%U_y = n \times \%U_a\]
Example: volume \(V = \frac{4}{3}\pi r^3\) — if \(r\) has 2% uncertainty, then \(V\) has \(3 \times 2\% = 6\%\) uncertainty.
Constants carry no uncertainty
Known constants such as \(\pi\) and \(g\) contribute zero uncertainty.
Summary Table
| Operation | Which type | Rule |
|---|---|---|
| Add / Subtract | Absolute | Add directly |
| Multiply / Divide | Percentage | Add directly |
| Power \(a^n\) | Percentage | Multiply by index \(n\) |
| Mean | Absolute | Range ÷ 2 |
Answer Format
Correct format:
\[L = 1.50 \pm 0.02 \text{ m}\]
Common errors:
- ❌ \(L = 1.5 \pm 0.02\) m — decimal places of value and uncertainty don’t match
- ❌ \(L = 1.50 \pm 0.023\) m — uncertainty given to too many significant figures
- ❌ \(L = 1.504 \pm 0.02\) m — result is more precise than the uncertainty
Rules:
Uncertainty: 1 significant figure (occasionally 2)
Decimal places of result must match those of the uncertainty
Graph Uncertainties (WPH13 & WPH16)
Error bars: Plot error bars on the graph to show the uncertainty range for each data point.
Max-min gradient method:
Draw all error bars
Draw the steepest possible line through all error bars → maximum gradient \(k_{max}\)
Draw the shallowest possible line → minimum gradient \(k_{min}\)
Uncertainty in gradient: \(\Delta k = \dfrac{k_{max} – k_{min}}{2}\)
Judging Experimental Success
\[|\text{experimental value} – \text{theoretical value}| < \text{total uncertainty}\]
If the condition is satisfied, write:
“The result is consistent with the theoretical value within experimental uncertainty.”
Practice Questions
Q1. \(m = 50.0 \pm 0.1\text{ g}\), \(V = 20.0 \pm 0.5\text{ cm}^3\). Find the percentage uncertainty in density \(\rho = m/V\).
Answer
\(\%U_m = 0.1/50.0 \times 100\% = 0.2\%\)
\(\%U_V = 0.5/20.0 \times 100\% = 2.5\%\)
\(\%U_\rho = 0.2\% + 2.5\% = \mathbf{2.7\%}\)
Q2. Formula \(\rho = m/d^3\), \(\%U_m = 2\%\), \(\%U_d = 1\%\). Find \(\%U_\rho\).
Answer
\(\%U_{d^3} = 3 \times 1\% = 3\%\)
\(\%U_\rho = 2\% + 3\% = \mathbf{5\%}\)
Q3. 20 oscillations timed as \(t = 30.0 \pm 0.2\text{ s}\). Find the absolute uncertainty in the period \(T\).
Answer
\(T = 30.0/20 = 1.50\text{ s}\)
\(\Delta T = 0.2/20 = \mathbf{0.01\text{ s}}\)
Write as: \(T = 1.50 \pm 0.01\text{ s}\)
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