Measurement Errors and Standards Questions and Answers

Answer: C

Correct answer: 0.6745

The probable error is the deviation within which half the readings fall, and for a normal distribution that is \( 0.6745\sigma \). It is an older statistic, largely displaced by the standard deviation and by stated confidence intervals, but it appears in the classical measurement literature and is worth recognising.

Answer: D

Correct answer: 0.16

\[ \sigma_{\bar{x}} = \frac{0.5}{\sqrt{10}} = \frac{0.5}{3.162} = 0.158. \] The mean is therefore far better determined than any single reading - but only against random error. If the instrument reads systematically high, all ten readings are high and this figure says nothing about it.

Answer: A

Correct answer: in quadrature, as the root of the sum of squares

\[ u = \sqrt{u_1^2 + u_2^2} \] for independent uncertainties. Simple addition would assume both errors take their maximum value in the same direction at once, which is improbable, and would overstate the combined uncertainty. Quadrature is the correct treatment when the two are genuinely independent.

Answer: B

Differentiating the product shows that the fractional uncertainties are what carry through, so a one per cent uncertainty in each gives about one and a half per cent in the product. Mixing absolute and relative quantities is the usual error.

Answer: C

Correct answer: 2.2 per cent

For a quotient the relative uncertainties combine in quadrature: \[ \sqrt{1^2 + 2^2} = \sqrt{5} = 2.24\ \mathrm{per\ cent}. \] Note how the larger contribution dominates - improving the 1 per cent measurement to 0.5 per cent would change the total only to 2.06 per cent. Effort belongs on the worst term.

Answer: D

Correct answer: realises the unit from its definition

A primary standard embodies the unit directly, by an experiment traceable to the definition rather than by comparison with anything else - a Josephson array for the volt, a quantum Hall device for the ohm. It is held at a national laboratory and is not used for routine work.

Answer: A

Correct answer: is calibrated against a primary standard and held in a laboratory

The secondary standard is the intermediate link: it is compared periodically against a primary standard and then serves as the reference within its own laboratory. The hierarchy exists so that the primary standard is disturbed as little as possible while accuracy still propagates down the chain.

Answer: B

Correct answer: is used for the routine calibration of instruments

The working standard is the bottom of the chain and the one that does the daily work, checked against a secondary standard at intervals. Because it is in constant use it is expected to drift, which is why the interval between its own calibrations is a matter of documented policy.

Answer: C

Correct answer: an unbroken chain of comparisons back to a national standard

Traceability means every link in the chain is documented, each with its own stated uncertainty, so the measurement can be related to the national realisation of the unit. Buying from a reputable supplier is not traceability; a certificate naming the standards used and their uncertainties is.

Answer: D

Correct answer: the corrections found and the uncertainty of the calibration

A certificate is a measurement result, not a verdict. It records what the instrument read against the standard at each point, the correction implied, and the uncertainty of the calibration itself - because a calibration is a measurement and carries an uncertainty like any other. A bare pass or fail conveys none of that.