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Uncertainties and practical work
The aim of physics in studying natural phenomena is to develop explanations based on empirical evidence. Hence there is a central concern about the quality of evidence and of the explanations that are based on it. This involves an appreciation of the causes of uncertainties that can arise in practical work and how they should be dealt with, both in planning an experiment to minimise these uncertainties and in forming a valid conclusion.
There is no practical examination in this qualification and a set of practical skills has been identified as appropriate for indirect assessment. Practical skills should be developed by carrying out practical work throughout the course, for example by carrying out the Core Practicals listed in the Specification. The assessment of these skills will be through examination questions, in particular for Unit 3: Practical Skills in Physics I and Unit 6: Practical Skills in Physics II.
It is clearly important that the words used within the practical context have a precise and scientific meaning as distinct from their everyday usage. The terms used for this assessment will be those described in the publication by the Association for Science Education (ASE) entitled The Language of Measurement (ISBN 9780863574245). In adopting this terminology, it should be noted that certain terms will have a meaning different to that in the previous specification. In accordance with common practice, this qualification will adopt the Uncertainty Approach to measurement. Using this approach assumes that the measurement activity produces an interval of reasonable values together with a statement of the confidence that the true value lies within this interval.
The following Glossary is a selection of terms from the list in The Language of Measurement published by ASE (ISBN 9780863574245).
Glossary
Term Meaning and notes
Validity A measurement is valid if it measures what it is supposed to be measuring – this depends both on the method and the instruments.
True value The value that would have been obtained in an ideal measurement – with the exception of a fundamental constant the true value is considered unknowable.
Accuracy A measurement result is considered accurate if it is judged to be close to the true value. It is a quality denoting the closeness of agreement between measurement and true value – it cannot be quantified and is influenced by random and systematic errors.
Precision A quality denoting the closeness of agreement (consistency) between values obtained by repeated measurement – this is influenced only by random effects and can be expressed numerically by measures such as standard deviation. A measurement is precise if the values ‘cluster’ closely together.
Repeatability The precision obtained when measurement results are obtained by a single operator using a single method over a short timescale. A measurement is repeatable when similar results are obtained by students from the same group using the same method. Students can use the precision of their measurement results to judge this.
Reproducibility
Uncertainty
The precision obtained when measurement results are obtained by different operators using different pieces of apparatus. A measurement is reproducible when similar results are obtained by students from different groups using different methods or apparatus. This is a harder test of the quality of data.
The interval within which the true value can be considered to lie with a given level of confidence or probability – any measurement will have some uncertainty about the result, this will come from variation in the data obtained and be subject to systematic or random effects. This can be estimated by considering the instruments and the method and will usually be expressed as a range such as 20 °C ± 2 °C. The confidence will be qualitative and based on the goodness of fit of the line of best fit and the size of the percentage uncertainty.
Error The difference between the measurement result and the true value if a true value is thought to exist. This is not a mistake in the measurement. The error can be due to both systematic and random effects and an error of unknown size is a source of uncertainty.
Resolution The smallest measuring interval and the source of uncertainty in a single reading.
Significant figures (SF)
The number of SF used in recording the measurements depends on the resolution of the measuring instruments and should usually be the same as given in the instrument with the fewest SF in its reading.
c. Using the resolution of the instrument
This is used if a single reading is taken or if repeated readings have the same value. This is because there is an uncertainty in the measurement because the instrument used to take the measurement has its own limitations. If the three readings obtained above were all 64 mm then the value of the diameter being measured lies somewhere between 63.5 mm and 64.5 mm since a metre rule could easily be read to half a millimetre. In this case, the uncertainty in the diameter is 0.5 mm.
Therefore, the diameter of the metal canister is 64 mm ± 0.5 mm.
This also applies to digital instruments. An ammeter records currents to 0.1 A. A current of 0.36 A would be displayed as 0.4 A, and a current of 0.44 A would also be displayed as 0.4 A. The resolution of the instrument is 0.1 A but the uncertainty in a reading is 0.05 A.
Calculating percentage uncertainties
The percentage uncertainty in a measurement can be calculated using:
Percentage uncertainty = (Uncertainty of measurement/Measurement) × 100%
In the above example the percentage uncertainty in the diameter of the metal canister is:
Percentage uncertainty = (3/64) × 100% = 4.7%
Often the radius would be used in a calculation, for example in a calculation of volume. In this case, the percentage uncertainty for the radius of the canister is the same as its diameter, i.e. 4.7%, and not half of the percentage uncertainty. This is one reason why the percentage uncertainty in a measurement is useful.
Additionally, the value is less than 5%, which shows that the measurement is probably repeatable. Note that a percentage uncertainty would normally be quoted to 1 or 2 sf.
Compounding uncertainties
Calculations often use more than one measurement. Each measurement will have its own uncertainty, so it is necessary to combine the uncertainties for each measurement to calculate the overall uncertainty in the calculation provided all the measured quantities are independent of one another.
There are three methods of compounding uncertainties depending on whether the measurements in a calculation are raised to a power, multiplied/divided, or added/subtracted.
a. Raising a measurement to a power
If a measurement is raised to a power, for example squared or cubed, then the percentage uncertainty is multiplied by that power to give the total percentage uncertainty.
Example: A builder wants to calculate the area of a square tile. He uses a rule to measure the two adjacent sides of a square tile and obtains the following results:
Length of one side = 84 mm ± 0.5 mm Length of perpendicular side = 84 mm ± 0.5 mm
The percentage uncertainty in the length of each side of this square tile is given by:
Percentage uncertainty = (0.5/84) × 100% = 0.59 % = 0.6 %
The area of the tile A is given by A = 84 × 84 = 7100 mm^2
Note that this is to 2 sf since the measurements are to 2 sf.
The percentage uncertainty in the area of the square tile is calculated by multiplying the percentage uncertainty in the length by 2.
Percentage uncertainty in A = 2 × 0.6% = 1.2%
Therefore the uncertainty in A = 7100 × 1.2% = 85 mm^2
So A = 7100 mm^2 ± 1.2% or A = 7100 mm^2 ± 85 mm^2
b. Multiplying or dividing measurements
The total percentage uncertainty is calculated by adding together the percentage uncertainties for each measurement.
Example: A metallurgist is determining the purity of a sample of an alloy that is in the shape of a cube by determining the density of the material.
The following readings are taken:
Length of each side of the cube= 24.0 mm ± 0.5 mm Mass of cube = 48.23 g ± 0.05 g
She calculates (i) the density of the material and (ii) the percentage uncertainty in the density of the material.
(i) Density of alloy = mass / volume = mass / length^3
= (48.23 x 10−^3 kg) / (24.0 x 10-3^ m) 3 = 3490 kg m−^3
(ii) Percentage uncertainty in the length = 0.5 / 24.0 × 100% = 2.1%
Percentage uncertainty in the mass = 0.05/48.23 × 100% = 0.1%
Percentage uncertainty in density= 3 × 2.1% +0.1% = 6.4% (or 6%)
Therefore, the density of the material= 3490 kg m−^3 ± 6% or 3490 kg m−^3 ± 210 kg m−^3
Example: A student calculates the volume of a drinks can and the percentage uncertainty for the final value.
The student determines that the radius of the metal can is 33 mm with an uncertainty of 1% so the cross-sectional area A of the canister is:
A = r^2 = (33) 2 = 3.4 × 10^3 mm^2 ± 2%
Notice that the result has been expressed using scientific notation so that we can write down just two significant figures. The calculator answer (3421.1...) gives the impression of far more sf than is justified when the radius is only known to the nearest mm.
The cross-sectional area was calculated by squaring the radius. Since two quantities have in effect been multiplied together, the percentage uncertainty in the value of the cross-sectional area is found by adding the percentage uncertainty of the radius to the percentage uncertainty of the radius – doubling it.
The student measures the length L of the can = 115 mm ± 2 mm
The volume V of the can is
V = 3.4 × 10^3 mm^2 × 115 mm = 3.9 × 10^5 mm^3 = 3.9 × 10−^4 m^3
The percentage uncertainty in this value is obtained by adding together an appropriate combination of the uncertainties
Percentage uncertainty in L = (2/115) × 100% = 1. Therefore, percentage uncertainty in V = 2% + 1.7% = 3.7%
Volume V = 3.91 × 10−^4 m^3 ± 3.7% = 3.91 × 10-4^ m^3 ± 1.4 × 10-5^ m^3
