Measurements and their Errors
Measurements and their errors is the A-level Physics topic covering SI units, significant figures, and the precision, accuracy and uncertainty attached to every reading. It opens the AQA 7407/7408 course and underpins every required practical, since raw data and calculated results must always be reported with an appropriate uncertainty.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Identify the resolution of the measuring instrument, then state a single reading's absolute uncertainty as half that resolution.
- Repeat the measurement several times, calculate the mean, and find the absolute uncertainty from half the range of the readings if this is larger than the resolution-based uncertainty.
- Convert absolute uncertainty to percentage uncertainty using (absolute uncertainty / measured value) x 100, so uncertainties in different quantities can be compared and combined.
- Combine uncertainties correctly: add absolute uncertainties for a sum or difference, add percentage uncertainties for a product or quotient, and multiply the percentage uncertainty by the power for a term raised to a power.
- State the final calculated result to the same number of significant figures as the least precise measurement used, together with its absolute uncertainty.
- Distinguish systematic errors (which shift every reading in the same direction and must be corrected, not averaged away) from random errors (which scatter readings and are reduced by repeating and averaging).
Worked example
A student measures the diameter of a small ball bearing with a micrometer of resolution 0.01 mm, obtaining a mean diameter of 6.20 mm from several readings. Calculate the volume of the ball bearing, using V = (4/3) x pi x r^3, and express the answer with its absolute uncertainty.
- Find the absolute uncertainty in the diameter: half the micrometer's resolution = 0.01 / 2 = 0.005 mm.
- Convert to a percentage uncertainty: (0.005 / 6.20) x 100 = 0.081%. This is also the percentage uncertainty in the radius, since dividing by the constant 2 does not change a percentage.
- Since V depends on r^3, multiply the percentage uncertainty by 3: percentage uncertainty in V = 3 x 0.081% = 0.24%.
- Calculate the radius and volume: r = 3.10 mm, so V = (4/3) x pi x (3.10)^3 = 124.8 mm^3.
- Convert the percentage uncertainty in V to an absolute uncertainty: 124.8 x 0.0024 = 0.3 mm^3.
- Final answer: V = (124.8 plus or minus 0.3) mm^3.
Practice questions
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Q1State the SI base unit of length.Show answer
Answer: metre (m)
Q2Write 0.0000045 m in standard form.Show answer
Answer: 4.5 x 10^-6 m
Q3A ruler has a resolution of 1 mm. State the absolute uncertainty in a single length reading made with it.Show answer
Answer: 0.5 mm (half the smallest scale division)
Q4A length is measured as 20.0 cm with an absolute uncertainty of 0.2 cm. Calculate the percentage uncertainty.Show answer
Answer: 1.0% ( (0.2/20.0) x 100 )
Q5A quantity has a value of 15.0 units with a percentage uncertainty of 4.0%. State the absolute uncertainty.Show answer
Answer: 0.6 units (15.0 x 0.04)
Q6A cube has sides of length 4.00 mm, each with an absolute uncertainty of 0.05 mm. Calculate the percentage uncertainty in its volume.Show answer
Answer: 3.75% (side: (0.05/4.00) x 100 = 1.25%; volume is proportional to side^3, so 3 x 1.25% = 3.75%)
Exam-style questions
Written in the style of a A Level Science exam paper, with a full mark scheme.
A student measures the length of a pencil as 14.6 cm, using a ruler of resolution 1 mm. Calculate the percentage uncertainty in this measurement.
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A wire has a measured diameter of 0.84 mm, found using a micrometer of resolution 0.01 mm. Calculate the cross-sectional area of the wire, using A = pi x d^2 / 4, and state the percentage uncertainty in this area.
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A rectangular metal sheet has a measured length of 25.0 cm (absolute uncertainty 0.2 cm) and a measured width of 10.0 cm (absolute uncertainty 0.2 cm). Calculate the area of the sheet and express your answer with its absolute uncertainty, to an appropriate number of significant figures.
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Free printable worksheet
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