Errors and Uncertainties (Cambridge IGCSE/A‑Level Physics 9702 – Sub‑topic 1.4)
Every measurement in physics carries an inevitable error. Understanding the nature of these errors lets you:
- interpret results correctly,
- identify the source of a problem, and
- choose the most effective remedy.
Key Terminology
- True value (X) – the exact value of a quantity (usually unknown).
- Measured value (x) – the value obtained from the experiment.
- Error – the difference between a measured value and the true value: error = x – X.
- Uncertainty (Δx) – an estimate of the range within which the true value is expected to lie.
Precision vs. Accuracy
These two aspects of measurement quality are often confused. Remember:
- Accuracy relates to AO1 – knowledge: it tells you how close the mean of a set of measurements is to the true value.
- Precision relates to AO2 – handling information: it tells you how closely repeated measurements agree with each other, irrespective of the true value.
High Accuracy & High Precision
●
●
●
--------------------> Target centre (true value)
High Precision & Low Accuracy
● ● ● (clustered but off‑centre)
Low Precision & High Accuracy
● . . . (scatter around the centre)
Low Precision & Low Accuracy
. . . . (widely scattered, not centred)
Typical dart‑board illustration of the four possible combinations.
Mathematical expressions
For n measurements \(x_i\) with mean \(\bar{x}\) and true value \(X\):
\[
\text{Accuracy error} = \bar{x} - X
\]
\[
\text{Precision (standard deviation)}\; \sigma = \sqrt{\frac{1}{n-1}\sum{i=1}^{n}(xi-\bar{x})^{2}}
\]
Comparison Table
| Aspect | What it measures | Typical cause of error | How to improve |
|---|
| Accuracy | Closeness of the mean to the true value | Systematic errors (bias) | Calibrate equipment, correct zero‑error, eliminate offsets |
| Precision | Scatter of individual readings about the mean | Random errors (noise) | Improve technique, increase number of readings, use a more stable apparatus |
Systematic Errors (including zero‑error)
- Definition: Errors that shift every measurement by (approximately) the same amount. They affect accuracy but not precision.
- Zero‑error: An instrument offset that makes the scale read a non‑zero value when the true quantity is zero (e.g., a mis‑aligned vernier caliper, a balance that reads 0.2 g when empty).
- Common sources
- Mis‑calibrated or poorly calibrated instruments.
- Parallax when reading a scale.
- Incorrect zero‑setting of a meter.
- Drift of electronic sensors over time.
- How to detect – Compare with a known standard or perform a “zero‑check” before each set of measurements.
- Remedy – Re‑calibrate, apply a correction factor, or replace the faulty instrument.
Random Errors
- Definition: Unpredictable fluctuations that cause scatter in repeated measurements; they affect precision.
- Typical sources
- Human reaction time when using a stopwatch.
- Limited resolution of the scale.
- Environmental noise (vibrations, air currents, temperature drift).
- Electrical noise in digital sensors.
- How to reduce
- Take a larger number of readings and use the mean.
- Use instruments with finer resolution.
- Stabilise the set‑up (bench‑top, shield from drafts).
- Practice a consistent technique (same start/stop method, same eye level).
Assessing Uncertainty in Derived Quantities
When a result is calculated from two or more measured quantities, the uncertainty must be propagated using the simple rules required by the syllabus.
Rule 1 – Addition or Subtraction
If \(Q = A \pm B\) then the absolute uncertainty is the sum of the absolute uncertainties:
\[
\Delta Q = \Delta A + \Delta B
\]
Rule 2 – Multiplication or Division
If \(Q = A \times B\) or \(Q = \dfrac{A}{B}\) then the percentage (relative) uncertainties add:
\[
\frac{\Delta Q}{|Q|}\times100\% = \frac{\Delta A}{|A|}\times100\% + \frac{\Delta B}{|B|}\times100\%
\]
Worked Example – Area of a Rectangle
Measured length \(L = 12.34 \pm 0.05\ \text{cm}\) and width \(W = 8.10 \pm 0.04\ \text{cm}\).
- Area \(A = L \times W = 12.34 \times 8.10 = 100.0\ \text{cm}^2\) (rounded to 3 sf).
- Percentage uncertainties:
\(\displaystyle \frac{\Delta L}{L}= \frac{0.05}{12.34}=0.0041\;(0.41\%)\)
\(\displaystyle \frac{\Delta W}{W}= \frac{0.04}{8.10}=0.0049\;(0.49\%)\)
- Total percentage uncertainty for the area (Rule 2):
\(0.41\% + 0.49\% = 0.90\%\)
- Absolute uncertainty in the area:
\(\Delta A = 0.0090 \times 100.0 = 0.9\ \text{cm}^2\)
- Result reported as:
\(\boxed{A = (100.0 \pm 0.9)\ \text{cm}^2}\)
Reporting Uncertainties
- Write the result as
value ± uncertainty (e.g., \(5.62 \pm 0.03\ \text{m s}^{-1}\)). - Both the value and its uncertainty must be quoted to the same number of decimal places (or significant figures). The uncertainty is usually given to one, at most two, significant figures.
- Round the central value to the same decimal place as the uncertainty.
- When using scientific notation, keep the uncertainty in the same exponent as the value.
Improving Measurements – Checklist
- Calibrate all instruments before use (improves accuracy).
- Check for zero‑error and correct any offset (systematic error).
- Take multiple readings; calculate the mean and standard deviation (improves precision).
- Identify and eliminate systematic biases (parallax, mis‑aligned scales, etc.).
- Use equipment with higher resolution where feasible.
- Control environmental conditions – minimise drafts, temperature swings, vibrations.
- Apply the correct uncertainty‑propagation rules (Rule 1 and Rule 2) for any derived quantity.
- Report results with the proper ± notation and appropriate significant figures.
Common Misconceptions
- “A precise measurement is automatically accurate.” – Not true; a set of consistently wrong readings is precise but inaccurate.
- “A single measurement tells you about precision.” – Precision is a statistical property; you need several readings.
- “Reducing random error will fix systematic error.” – They are independent; each requires its own corrective action.
- “Uncertainty is the same as error.” – Error is the (unknown) difference from the true value; uncertainty is our *estimate* of the possible range of that error.
Summary
Distinguishing between precision and accuracy, recognising systematic errors (including zero‑error) and random errors, and correctly propagating and reporting uncertainties are essential skills for Cambridge IGCSE/A‑Level Physics. Mastery of these concepts enables you to diagnose experimental problems, improve technique, and present results in the format expected in examinations.