Learning Objectives
3 objectivesBy the end of this note, you should be able to:
- Understand and explain the effects of systematic errors (including zero errors) and random errors in measurements.
- Understand the distinction between precision and accuracy.
- Assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties.
Systematic and Random Errors
A systematic error shifts every reading by the same fixed amount or in the same direction, so repeating the measurement does not reduce it. A random error causes readings to scatter unpredictably above and below the true value, so repeating and averaging reduces its effect.
Systematic errors arise from faults in the apparatus or the procedure. Common causes include a non-zero reading on an instrument before use (a zero error), a wrongly calibrated scale, parallax viewing from a fixed wrong angle, or a thermometer with a damaged bulb. Because the bias is constant, every reading is offset in the same direction, so the mean of many readings still differs from the true value.
A zero error is corrected by subtracting the zero reading from every measurement. Calibration errors are corrected by checking the instrument against a known standard before use.
Random errors arise from unpredictable fluctuations in conditions or in observer judgement. Examples include small temperature changes during an experiment, vibrations of a balance, and the difficulty of judging the exact position of a needle on a scale. Each repeated reading varies slightly in both size and direction. Random errors are reduced by taking many repeat readings and calculating the mean, and by using instruments with finer resolution.
MisconceptionRepeating a measurement many times does not improve a systematic error. The mean is shifted by the same fixed amount as every individual reading. Only random scatter cancels out on averaging. Exam cue: if asked how to reduce a systematic error, never write “repeat and average”.
Precision and Accuracy
Accuracy describes how close a measurement is to the true value, while precision describes how close repeated measurements are to each other. The two ideas are independent: a set of readings can be precise but inaccurate, or accurate on average but imprecise.
| Feature | Accuracy | Precision |
|---|---|---|
| Definition | Closeness of a measurement to the true value | Closeness of repeated readings to each other |
| Affected by | Systematic errors | Random errors and instrument resolution |
| Improved by | Calibration and removal of zero errors | Finer-resolution instruments and repeated readings |
| Distinguishing feature | Concerns the mean of readings | Concerns the spread of readings |
A measurement may also be described as precise if it is recorded to many significant figures, since this reflects the smallest division of the instrument. A digital balance reading 24.567 g is more precise than one reading 24.6 g, but it is only accurate if it is also correctly calibrated.
Examiner InsightCambridge mark schemes accept “close to true value” for accuracy and “close to each other” or “small spread” for precision. Avoid vague language such as “exact” or “correct”. Exam cue: state both ideas in opposite terms when asked to distinguish them.
Combining Uncertainties in Derived Quantities
The uncertainty in a derived quantity is found by combining the uncertainties of the measured values, using simple addition rules that depend on whether the quantities are added, multiplied, or raised to a power.
Key Equations
Sum or difference of quantities:
$$Z=A\pm B⇒\Delta Z=\Delta A+\Delta B$$
Product or quotient of quantities:
$$Z=\frac{A\times B}{C}⇒\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}+\frac{\Delta C}{C}$$
Power relationship:
$$Z={A}^{n}⇒\frac{\Delta Z}{Z}=|n|\times \frac{\Delta A}{A}$$
Percentage uncertainty:
$$\text{percentage uncertainty}=\frac{\Delta A}{A}\times 100\%$$
Variables: $Z$ is the derived quantity; $A$, $B$, $C$ are measured quantities; $\Delta A$, $\Delta B$, $\Delta C$, $\Delta Z$ are absolute uncertainties; $n$ is a power (which may be a fraction).
For sums and differences, add the absolute uncertainties. For products, quotients, and powers, add the percentage (or fractional) uncertainties. Once the percentage uncertainty in $Z$ is known, the absolute uncertainty is recovered by multiplying by $Z$.
The absolute uncertainty in a single reading is normally taken as the smallest division of the instrument, or half the smallest division for an analogue scale read at both ends. For a set of repeated readings, the uncertainty is half the range of the values.
A final answer is written as $Z=(\text{value})\pm \Delta Z$, where the value and the uncertainty are quoted to the same decimal place. The uncertainty itself is given to one significant figure.
Worked Example: Uncertainty in the Density of a Cylinder
A student measures a solid metal cylinder. The mass is $m=25.0\pm 0.1 g$, the diameter is $d=1.20\pm 0.02 cm$, and the length is $L=5.00\pm 0.05 cm$. Determine the density and its absolute uncertainty.
Equation used — the volume of a cylinder gives density as
$$\rho =\frac{m}{V}=\frac{4m}{\pi {d}^{2}L}$$
Given (after converting to SI units):
$$m=25.0\times {10}^{-3} kg, \Delta m=0.1\times {10}^{-3} kg$$
$$d=1.20\times {10}^{-2} m, \Delta d=0.02\times {10}^{-2} m$$
$$L=5.00\times {10}^{-2} m, \Delta L=0.05\times {10}^{-2} m$$
Working — percentage uncertainty in each measurement
$$\frac{\Delta m}{m}\times 100\%=\frac{0.1}{25.0}\times 100\%=0.40\%$$
$$\frac{\Delta d}{d}\times 100\%=\frac{0.02}{1.20}\times 100\%=1.67\%$$
$$\frac{\Delta L}{L}\times 100\%=\frac{0.05}{5.00}\times 100\%=1.00\%$$
Combining percentage uncertainties (note the power of 2 on $d$):
$$\frac{\Delta \rho }{\rho }\times 100\%=0.40\%+2\times 1.67\%+1.00\%=4.74\%$$
Calculating the density:
$$\rho =\frac{4\times 25.0\times {10}^{-3}}{\pi \times (1.20\times {10}^{-2}{)}^{2}\times 5.00\times {10}^{-2}}$$
$$\rho =4421 kg {m}^{-3}$$
Absolute uncertainty:
$$\Delta \rho =4.74\%\times 4421=209 kg {m}^{-3}$$
$$\rho =(4400\pm 200) kg {m}^{-3}$$
The diameter contributes the largest share of the uncertainty because it is squared in the formula. Reducing $\Delta d$ would improve the precision of the density most effectively.
MisconceptionWhen a quantity is raised to a power, the percentage uncertainty is multiplied by that power, not by the square or square root. For ${d}^{2}$, multiply $\Delta d/d$ by 2. Exam cue: identify every power in the equation before adding percentage uncertainties.
QUICK RECAP
Key Points
- Systematic error: same direction, fixed magnitude, not reduced by averaging.
- Zero error: instrument reads non-zero when input is zero; subtract from each reading.
- Random error: unpredictable scatter; reduced by repeats and averaging.
- Accuracy: closeness of (mean) reading to the true value.
- Precision: closeness of repeated readings to each other.
- Precise readings can still be inaccurate.
- For sums and differences: add absolute uncertainties.
- For products and quotients: add percentage uncertainties.
- For ${A}^{n}$: percentage uncertainty in $A$ is multiplied by $n$.
- Absolute uncertainty of one reading: smallest scale division (analogue) or last digit (digital).
- Absolute uncertainty of repeated readings: half the range.
- Quote uncertainty to one significant figure.
- Match value and uncertainty to the same decimal place.
CAN I…? PROGRESS CHECK
Self-Assessment
- Define systematic and random errors with one example of each.
- Identify zero error from an instrument reading and apply the correction.
- State one method to reduce a systematic error and one to reduce a random error.
- Distinguish accuracy from precision using a target-board scenario.
- Recognise when a set of readings is precise but not accurate.
- Calculate the absolute uncertainty in a sum or difference of two quantities.
- Calculate the percentage uncertainty in a product, quotient, or power.
- Combine percentage uncertainties to find the absolute uncertainty in a derived quantity.
- Quote a final result correctly as value ± absolute uncertainty.
- Identify which measured quantity dominates the uncertainty in a derived result.