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

Errors and Uncertainties

Learning Objective

Understand and explain the effects of systematic errors (including zero errors) and random errors in measurements, and be able to evaluate and combine uncertainties.

Key Definitions

• Measurement error: The difference between the measured value and the true value.
• Uncertainty: A quantitative estimate of the range within which the true value is expected to lie.
• Systematic error: A reproducible inaccuracy that consistently shifts measurements in the same direction.
• Random error: A statistical fluctuation that causes measurements to scatter about the true value.
• Zero error: A type of systematic error that occurs when an instrument does not read zero when it should.

Systematic Errors

Systematic errors affect the accuracy of a measurement. They arise from:

1. Instrument calibration faults (e.g., worn scale, mis‑aligned sensor).
2. Zero errors – the instrument reading is offset by a constant amount \$e_0\$.
3. Environmental influences that are not accounted for (e.g., temperature drift).
4. Procedural biases (e.g., consistently reading the top of a meniscus).

Because the error is the same for each reading, the spread of repeated measurements does not reveal it. It must be identified and corrected.

Zero Error Example

If a vernier caliper reads \$0.02\ \text{mm}\$ when the jaws are closed, the zero error \$e0 = +0.02\ \text{mm}\$. The true measurement \$L{\text{true}}\$ is obtained from the observed reading \$L_{\text{obs}}\$ by

\$L{\text{true}} = L{\text{obs}} - e_0.\$

Random Errors

Random errors affect the precision of a measurement. They arise from unpredictable fluctuations such as:

• Human reaction time when using stop‑watches.
• Electronic noise in sensors.
• Variations in environmental conditions during the experiment.

Random errors can be reduced by:

1. Taking multiple independent measurements.
2. Increasing the measurement interval (e.g., timing over many periods).
3. Improving experimental technique.

Quantifying Random Uncertainty

For a set of \$n\$ measurements \$x_i\$, the mean \$\bar{x}\$ and standard deviation \$s\$ are

\$\$\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 is

\$u_{\bar{x}} = \frac{s}{\sqrt{n}}.\$

Combining Uncertainties

When a result \$Q\$ depends on several measured quantities, the combined (propagated) uncertainty is obtained using the partial‑derivative method:

\$\$uQ = \sqrt{\left(\frac{\partial Q}{\partial x}\,ux\right)^2 +

\left(\frac{\partial Q}{\partial y}\,u_y\right)^2 +

\dots }.\$\$

Systematic and random components are combined separately and then added in quadrature:

\$u{\text{total}} = \sqrt{u{\text{random}}^{\,2}+u_{\text{systematic}}^{\,2}}.\$

Comparison of Systematic and Random Errors

Practical Example: Timing a Pendulum

Suppose a student measures the period of a pendulum by timing 20 complete oscillations with a stop‑watch. The recorded time is \$T_{20}=40.2\ \text{s}\$.

• Zero error of the stop‑watch: \$e_0 = -0.02\ \text{s}\$ (the watch starts \$0.02\ \text{s}\$ early).
• Corrected total time: \$T{20}^{\text{corr}} = 40.2\ \text{s} - e0 = 40.22\ \text{s}\$.
• Period: \$T = \dfrac{T_{20}^{\text{corr}}}{20} = 2.011\ \text{s}\$.

Random uncertainty from repeat measurements (e.g., \$n=5\$ trials) yields a standard deviation \$s_T = 0.012\ \text{s}\$, giving

\$u{\text{random}} = \frac{sT}{\sqrt{5}} = 0.005\ \text{s}.\$

If the stop‑watch calibration contributes a systematic uncertainty of \$u_{\text{sys}} = 0.003\ \text{s}\$, the total uncertainty is

\$u_{\text{total}} = \sqrt{(0.005)^2 + (0.003)^2} \approx 0.006\ \text{s}.\$

Result: \$T = 2.011 \pm 0.006\ \text{s}\$.

Key Take‑aways

• Systematic errors shift all results; they must be identified and corrected, not reduced by averaging.
• Zero errors are a common systematic error and are corrected by subtracting the offset from each reading.
• Random errors cause scatter; they are quantified by statistical methods and reduced by repeated measurements.
• Combine uncertainties using quadrature, keeping systematic and random components distinct until the final step.
• Always report a measurement with its combined uncertainty and indicate the dominant source of error.