Measurement Errors and Standards Questions and Answers

Q: The standard deviation of the mean of \( n \) readings is smaller than that of a single reading by a factor of

Correct answer: \( \sqrt{n} \)\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}. \] The square root is what makes averaging a case of diminishing returns: four readings halve the uncertainty, but a hundred are needed to reduce it tenfold. Beyond a point, effort is better spent on removing systematic error, which averaging cannot touch.

Q: Probable error of a single reading is related to its standard deviation by a factor of about

Correct answer: 0.6745The 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.

Q: Ten readings of a quantity give a mean of 100.0 and a standard deviation of 0.5. The standard deviation of the mean is nearest

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.

Q: A resistance is found from a voltage known to 1 per cent and a current known to 2 per cent. The resistance is uncertain by about

Correct answer: 2.2 per centFor 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.

Q: A primary standard of a quantity is one that

Correct answer: realises the unit from its definitionA 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.

21.

The standard deviation of the mean of \( n \) readings is smaller than that of a single reading by a factor of

A.unity
B.\( \sqrt{n} \)
C.\( n \) itself
D.\( n^2 \)

Answer: B

Correct answer: \( \sqrt{n} \)

\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}. \] The square root is what makes averaging a case of diminishing returns: four readings halve the uncertainty, but a hundred are needed to reduce it tenfold. Beyond a point, effort is better spent on removing systematic error, which averaging cannot touch.

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22.

Probable error of a single reading is related to its standard deviation by a factor of about

A.2.0
B.0.3333
C.0.6745
D.1.0

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.

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23.

Ten readings of a quantity give a mean of 100.0 and a standard deviation of 0.5. The standard deviation of the mean is nearest

A.0.50
B.5.00
C.0.05
D.0.16

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.

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24.

When two measured quantities are added, their absolute uncertainties combine

A.in quadrature, as the root of the sum of squares
B.by simple addition of the two absolute uncertainties
C.by subtracting the smaller from the larger
D.not at all, the sum carrying no uncertainty

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.

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25.

When two measured quantities are multiplied, it is their

A.relative uncertainties that simply subtract
B.relative uncertainties that combine in quadrature
C.absolute uncertainties that combine in quadrature
D.absolute uncertainties that simply add together

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.

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26.

A resistance is found from a voltage known to 1 per cent and a current known to 2 per cent. The resistance is uncertain by about

A.1.0 per cent
B.0.5 per cent
C.2.2 per cent
D.3.0 per cent

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.

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27.

A primary standard of a quantity is one that

A.is used for the everyday calibration of workshop instruments
B.is kept in a laboratory and calibrated against a working standard
C.is carried between laboratories for comparison purposes
D.realises the unit from its definition

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.

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28.

A secondary standard is one that

A.is calibrated against a primary standard and held in a laboratory
B.realises the unit directly from its definition
C.is used for the routine calibration of production instruments
D.is built into every instrument as a reference

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.

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29.

A working standard is one that

A.is carried between national laboratories for intercomparison
B.is used for the routine calibration of instruments
C.realises the unit from its definition without reference to any other
D.is kept sealed and never used except to check a secondary standard

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.

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30.

Traceability of a measurement means

A.the ability to repeat the measurement at a later date
B.the instrument having been bought from an accredited supplier
C.an unbroken chain of comparisons back to a national standard
D.a record of every person who has used the instrument

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.

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