| Measurement error | The difference between the measured value and the true (or accepted) value. |
| Uncertainty | A quantitative estimate of the interval within which the true value is expected to lie. |
| Systematic error | A reproducible inaccuracy that shifts all measurements in the same direction (affects accuracy). |
| Random error | Statistical fluctuations that cause scatter about the true value (affects precision). |
| Zero error | A systematic error where an instrument does not read zero when it should; the reading is offset by a constant \(e_0\). |
Systematic errors are constant (or slowly varying) biases. They are not revealed by the spread of repeated readings and must be identified, assessed and corrected.
If a vernier caliper reads \(+0.02\;\text{mm}\) when the jaws are closed, the zero error is \(e0 = +0.02\;\text{mm}\). For any observed reading \(L{\text{obs}}\)
\[
L{\text{true}} = L{\text{obs}} - e_0 .
\]
Random errors cause the measured values to scatter about the true value. Their magnitude can be reduced by statistical treatment.
For \(n\) independent measurements \(x_i\):
\[
\bar{x}= \frac{1}{n}\sum{i=1}^{n}xi ,\qquad
s = \sqrt{\frac{1}{n-1}\sum{i=1}^{n}(xi-\bar{x})^{2}} .
\]
The standard uncertainty of the mean (the random component required for AO2) is
\[
u_{\text{rand}} = \frac{s}{\sqrt{n}} .
\]
When a result \(Q\) depends on measured quantities \(x, y, \dots\) that are independent, the combined uncertainty is
\[
uQ = \sqrt{\left(\frac{\partial Q}{\partial x}\,ux\right)^{2}
+\left(\frac{\partial Q}{\partial y}\,u_y\right)^{2}
+\dots } .
\]
Note: The syllabus assumes the variables are independent; covariances are therefore omitted.
For many A‑Level calculations it is sufficient to combine uncertainties by direct addition:
Worked example – \(V = IR\)
Systematic and random uncertainties are kept separate until the final step. The total uncertainty is obtained by quadrature:
\[
u{\text{total}} = \sqrt{u{\text{rand}}^{\,2}+u_{\text{sys}}^{\,2}} .
\]
Uncertainties are often shown as error bars on data points. An error bar extends a distance equal to the absolute uncertainty above and below (or to the left and right of) the plotted value.
| Aspect | Systematic error | Random error | Typical AO2 requirement |
|---|---|---|---|
| Cause | Calibration fault, zero error, manufacturer tolerance, environmental bias | Statistical fluctuations, human reaction time, electronic noise | Identify source; state correction or uncertainty |
| Effect on data | Shifts all results in the same direction → affects accuracy | Scatter about the mean → affects precision | Quantify and combine appropriately |
| Detection | Comparison with standards, calibration certificates, zero‑error check | Statistical analysis (standard deviation, repeatability) | Provide evidence of detection |
| Reduction strategy | Calibrate, apply zero‑error correction, use better equipment | Increase number of measurements, longer measurement intervals, improve technique | Explain how uncertainty was reduced |
| Uncertainty contribution | Added as a constant offset or quoted as a systematic component \(u_{\text{sys}}\) | Evaluated from repeatability (standard deviation) → \(u_{\text{rand}}\) | Combine by quadrature |
Recorded time for 20 oscillations: \(T_{20}=40.20\;\text{s}\).
Stopwatch zero error: \(e_0 = -0.02\;\text{s}\) (starts early).
Corrected total time: \(T{20}^{\text{corr}} = T{20} - e_0 = 40.22\;\text{s}\).
Period: \(T = \dfrac{T_{20}^{\text{corr}}}{20}=2.011\;\text{s}\).
Five repeat trials give periods (s): 2.009, 2.014, 2.008, 2.012, 2.015.
\(\bar{T}=2.011\;\text{s},\; s_T = 0.012\;\text{s}\).
\(u{\text{rand}} = \dfrac{sT}{\sqrt{5}} = 0.005\;\text{s}\).
Manufacturer’s specification for the stopwatch: ±0.003 s (systematic).
\(u_{\text{sys}} = 0.003\;\text{s}\).
\(u_{\text{total}} = \sqrt{(0.005)^2+(0.003)^2}=0.006\;\text{s}\).
Uncertainty quoted to one significant figure (0.006 s) → period reported as
\[
T = 2.011 \pm 0.006\;\text{s}.
\]
Voltage: \(V = 12.00 \pm 0.03\;\text{V}\) (digital voltmeter, ±0.25 % + 0.01 V).
Current: \(I = 0.200 \pm 0.004\;\text{A}\) (ammeter, ±2 % of reading).
Both uncertainties are taken as systematic components.
\(R = \dfrac{V}{I}= \dfrac{12.00}{0.200}=60.0\;\Omega\).
\(\displaystyle \frac{u_V}{V}= \frac{0.03}{12.00}=0.25\% \) (rounded to one sig‑fig).
\(\displaystyle \frac{u_I}{I}= \frac{0.004}{0.200}=2.0\% \).
Combined percentage: \(0.25\%+2.0\%=2.3\%\).
Absolute uncertainty: \(u_R = 2.3\% \times 60.0\;\Omega = 1.4\;\Omega\).
\(R = 60.0 \pm 1.4\;\Omega\) (uncertainty to one significant figure, resistance to the same decimal place).
On a \(V\)‑vs‑\(I\) graph the error bar would be ±0.03 V vertically and ±0.004 A horizontally.
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