Problem — True or false on experimental errors

Problem

True or false, with justification. (a) A random error is eliminated by repeating the measurement many times. (b) A systematic error is eliminated by repeating the measurement many times. (c) The mean of many independent measurements reduces the random error by a factor $1/\sqrt{N}$. (d) An uncalibrated balance produces a systematic error. (e) The sensitivity of an instrument coincides with its precision.

Solution

(a) FALSE. By repeating the measurement and averaging, the random error is reduced but not eliminated. The spread of values around the mean shrinks as $1/\sqrt{N}$, but a finite residual uncertainty always remains: with finite $N$ it never vanishes entirely.

(b) FALSE. A systematic error shifts every measurement always in the same direction and by the same amount. Averaging many trials, this offset stays identical: the mean is affected by the same error as a single measurement. Systematics are fought by recalibrating the instrument or changing method, not by repeating.

(c) TRUE. For $N$ independent measurements with random error, the uncertainty on the mean (standard error of the mean) is $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{N}}$ so the random error on the mean value indeed scales as $1/\sqrt{N}$.

(d) TRUE. An uncalibrated balance (shifted zero) adds the same offset to every weighing: it is the typical example of a systematic offset error. All measurements come out biased in the same direction.

(e) FALSE. Sensitivity is the smallest change in the quantity that the instrument can detect (resolution); precision is repeatability, i.e. how little repeated measurements of the same quantity scatter. These are distinct concepts: an instrument can be sensitive but not very precise, or vice versa.

$\ev{\text{(a) F}\quad \text{(b) F}\quad \text{(c) V}\quad \text{(d) V}\quad \text{(e) F}}$

Links

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