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1. Errors and Uncertainties (Cambridge AS & A Level Physics 9702 – Syllabus 1.3)

1.1 Learning objectives (syllabus 1.3)

Define measurement error, uncertainty, systematic error, random error and zero error.

Explain how systematic and random errors affect accuracy and precision.

Identify, assess and correct systematic uncertainties (including zero‑error correction).

Quantify random uncertainties using the statistical method required for AO2 (standard deviation and standard uncertainty of the mean).

Propagate uncertainties to derived quantities by:
- the partial‑derivative (quadrature) method, and
- simple addition of absolute or percentage uncertainties (AO2 requirement).

Combine systematic and random components and present results with appropriate significant figures and error bars.

1.2 Key definitions

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 \(e_0\).

1.3 Systematic errors

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.

1.3.1 Common sources

Instrument calibration faults (e.g. worn scale, sensor drift).

Zero errors – a constant offset \(e_0\) in the instrument reading.

Manufacturer tolerances or stated accuracy limits (e.g. digital voltmeter ±0.5 % + 2 mV).

Unaccounted environmental influences (temperature, humidity, magnetic fields).

Procedural bias (reading the top of a meniscus, parallax error).

1.3.2 Assessment checklist (identify, assess, correct)

Calibration check: Compare the instrument with a known standard or consult its calibration certificate.

Zero‑error verification: Close the instrument (e.g. jaws of a caliper) and record any offset.

Manufacturer’s specification: Use the quoted accuracy as a systematic uncertainty when no calibration is performed.

Environmental audit: Note temperature, humidity, magnetic fields and assess whether they could bias the reading.

Document the correction: Record the value of the systematic bias and apply it to every measurement (or quote it as a systematic component).

1.3.3 Zero‑error correction (example)

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 .

Diagram (placeholder): Vernier caliper showing a positive zero error.

1.4 Random errors

Random errors cause the measured values to scatter about the true value. Their magnitude can be reduced by statistical treatment.

1.4.1 Quantifying random uncertainty (AO2 statistical method)

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}} .

1.5 Propagation of uncertainties

1.5.1 Partial‑derivative (quadrature) method

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.

1.5.2 Simple addition of absolute or percentage uncertainties (AO2)

For many A‑Level calculations it is sufficient to combine uncertainties by direct addition:

Absolute addition (addition/subtraction of quantities): \(uQ = uA + u_B + \dots\)

Percentage addition (multiplication/division of quantities): \(\displaystyle \frac{uQ}{Q} = \frac{uA}{A} + \frac{u_B}{B} + \dots\)

Worked example – \(V = IR\)

Measured current \(I = 0.200 \pm 0.005\;\text{A}\) (absolute uncertainty 0.005 A).

Measured resistance \(R = 50.0 \pm 0.3\;\Omega\) (absolute uncertainty 0.3 Ω).

Voltage \(V = I R = 10.0\;\text{V}\).

Percentage uncertainties: \(\displaystyle \frac{uI}{I}= \frac{0.005}{0.200}=2.5\%\), \(\displaystyle \frac{uR}{R}= \frac{0.3}{50.0}=0.6\%\).

Combined percentage: \(2.5\% + 0.6\% = 3.1\%\).

Absolute uncertainty in \(V\): \(u_V = 3.1\% \times 10.0\;\text{V}=0.31\;\text{V}\).

Result reported as \(V = 10.0 \pm 0.3\;\text{V}\) (uncertainty rounded to one significant figure).

1.6 Combining systematic and random components

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}} .

1.7 Graphical representation of uncertainties

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.

Diagram (placeholder): Data point with vertical error bar representing \(u_{\text{total}}\).

1.8 Comparison of systematic and random errors

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

1.9 Worked examples

1.9.1 Example A – Timing a pendulum (mechanics)

Zero‑error correction
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: \(T = \dfrac{T_{20}^{\text{corr}}}{20}=2.011\;\text{s}\).

Random uncertainty (AO2 statistical method)
Five repeat trials give periods (s): 2.009, 2.014, 2.008, 2.012, 2.015.
\(\bar{T}=2.011\;\text{s},\; s_T = 0.012\;\text{s}\).
\(u{\text{rand}} = \dfrac{sT}{\sqrt{5}} = 0.005\;\text{s}\).

Systematic uncertainty
Manufacturer’s specification for the stopwatch: ±0.003 s (systematic).
\(u_{\text{sys}} = 0.003\;\text{s}\).

Combine uncertainties
\(u_{\text{total}} = \sqrt{(0.005)^2+(0.003)^2}=0.006\;\text{s}\).

Result with appropriate significant figures
Uncertainty quoted to one significant figure (0.006 s) → period reported as
\[
T = 2.011 \pm 0.006\;\text{s}.
\]

1.9.2 Example B – Determining resistance from voltage and current (electricity)

Measurements (systematic uncertainties from specifications)
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.

Calculate resistance
\(R = \dfrac{V}{I}= \dfrac{12.00}{0.200}=60.0\;\Omega\).

Propagation by simple percentage addition (AO2)
\(\displaystyle \frac{u_V}{V}= \frac{0.03}{12.00}=0.25\% \) (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: \(u_R = 2.3\% \times 60.0\;\Omega = 1.4\;\Omega\).

Result
\(R = 60.0 \pm 1.4\;\Omega\) (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.

1.10 Significant figures and reporting

Uncertainty is always quoted to one significant figure (or two if the first figure is “1”).

The measured value is then rounded to the same decimal place as the uncertainty.

If the uncertainty has two significant figures (e.g. 0.12), keep both and round the result accordingly.

Write the result in the form “value ± uncertainty” and always include units.

1.11 Key take‑aways

Systematic errors shift all results; they must be identified, assessed (using the checklist), corrected (e.g., zero‑error subtraction) or quoted as a systematic uncertainty.

Random errors cause scatter; they are quantified by the standard deviation and reduced by repeated measurements (standard uncertainty of the mean).

Assess systematic uncertainty from calibration certificates, manufacturer specifications, or known standards.

Propagate uncertainties:
- Partial‑derivative (quadrature) method for independent variables.
- Simple addition of absolute (addition/subtraction) or percentage (multiplication/division) uncertainties – an AO2 requirement.

Combine systematic and random components in quadrature to obtain the total uncertainty.

Present results with correct significant figures, a clear “± uncertainty” statement, and, where appropriate, error bars on graphs.

understand and explain the effects of systematic errors (including zero errors) and random errors in measurements

1. Errors and Uncertainties (Cambridge AS & A Level Physics 9702 – Syllabus 1.3)

1.1 Learning objectives (syllabus 1.3)