Problem — Systematic error versus random error

Problem

Thermometer A always reads $+0{,}5\ ^\circ\text{C}$ above the true value. Thermometer B fluctuates randomly with $\sigma = 0{,}5\ ^\circ\text{C}$. For each, classify the type of error and discuss what happens when averaging $N$ readings. Which of the two is more insidious to discover?

Solution

Thermometer A — systematic error. The offset is constant: every reading is shifted by the same amount $+0{,}5\ ^\circ\text{C}$. Averaging $N$ readings preserves the offset: $\bar{T}_A = T_\text{true} + 0{,}5\ ^\circ\text{C}$ The error does not decrease as the number of measurements increases, because it is not due to fluctuations but to incorrect calibration.

Thermometer B — random error. The readings fluctuate around the true value with standard deviation $\sigma = 0{,}5\ ^\circ\text{C}$. Averaging $N$ independent readings, the standard deviation of the mean decreases: $\sigma_{\bar{T}} = \frac{\sigma}{\sqrt{N}} \xrightarrow[N\to\infty]{} 0$ So the random error can be reduced as much as desired by repeating the measurement.

Which is more insidious? Thermometer A, with the systematic error. Averaging many readings, the result appears stable and precise (small spread), but it is always wrong by the same amount: repetition does not reveal the defect. It can only be discovered through calibration or comparison with a reference standard. The random error of B, by contrast, shows up immediately as spread and is controlled by averaging.

$\ev{\text{A} = \text{systematic (does not average away)}\ ;\quad \text{B} = \text{random (reduces as }\sigma/\sqrt{N})\ ;\quad \text{more insidious} = \text{A}}$

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Topics: Metodo sperimentale Concepts: Cifre significative e incertezza Skills: Propagazione degli errori