Measurements and uncertainties — OCR A-Level Physics
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Measurements and uncertainties explained
Systematic errors shift every reading by the same amount or in the same proportion, so they affect accuracy but not precision.
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A zero error is a systematic error where an instrument does not read zero when it should; for example, a top-pan balance reading 0.02 g with nothing on it. Random errors vary unpredictably between readings, often due to fluctuations or reading uncertainty, and affect precision. Repeating measurements and averaging reduces random errors, while systematic errors require calibration or correction. When analysing data, look for a consistent offset (systematic) or scatter around a mean (random).
(b) precision and accuracy
Precision describes how close repeated measurements are to each other, often indicated by the spread or uncertainty. Accuracy describes how close a measurement is to the true value. A set of readings can be precise but not accurate if they cluster around a wrong value, for example due to a systematic error. Conversely, readings can be accurate on average but imprecise if they are scattered. In experimental work, precision is improved by careful technique and repeated readings, while accuracy is improved by calibration and eliminating systematic errors. When evaluating data, comment on both the spread (precision) and the closeness to the true value (accuracy).
(c) absolute and percentage uncertainties when data are combined by addition, subtraction, multiplication, division and raising to powers
When combining measurements, uncertainties propagate. For addition and subtraction, add absolute uncertainties: if z = x + y or z = x − y, then Δz = Δx + Δy. For multiplication and division, add percentage uncertainties: if z = xy or z = x/y, then %Δz = %Δx + %Δy. For powers, multiply the percentage uncertainty by the power: if z = xⁿ, then %Δz = n × %Δx. Absolute uncertainty is the uncertainty in the same units as the quantity; percentage uncertainty is (absolute uncertainty / value) × 100%. Always convert to the required form at the end.
(d) graphical treatment of errors and uncertainties; line of best fit; worst line; absolute and percentage uncertainties; percentage difference.
Graphical treatment means plotting data with error bars, then drawing a line of best fit through the points so that the trend is represented fairly. A worst line is the steepest or shallowest line that still passes through every error bar; the spread between worst lines gives the uncertainty in the gradient or intercept. Absolute uncertainty carries the same unit as the quantity, for example 0.5 cm, while percentage uncertainty is the absolute uncertainty divided by the measured value, multiplied by 100, for example 0.5 cm ÷ 20.0 cm × 100 = 2.5%. Percentage difference compares an experimental value with an accepted value: difference ÷ accepted value × 100. Always label axes with quantity and unit, choose sensible scales, and state the uncertainty from the graph rather than guessing.
Your focus
- Distinguish between systematic and random errors in measurements.
- Identify zero errors and describe how to correct them.
- Explain how repeated measurements reduce random error.
Show all 12 objectives
- Define precision and accuracy in the context of measurements.
- Distinguish between precise and accurate data using examples.
- Evaluate experimental data in terms of precision and accuracy.
- Apply the rules for combining absolute uncertainties in addition and subtraction.
- Apply the rules for combining percentage uncertainties in multiplication and division.
- Calculate the percentage uncertainty for a quantity raised to a power.
- Draw error bars and a line of best fit on a graph of experimental data.
- Identify worst lines and use them to estimate the uncertainty in a gradient or intercept.
- Calculate absolute uncertainty, percentage uncertainty and percentage difference correctly.
Measurements and uncertainties exam tips
Marking Points
- Systematic errors cause all readings to be shifted by a consistent amount or proportion.
- Zero errors are systematic errors where the instrument does not read zero correctly.
- Random errors cause unpredictable variation between repeated readings.
- Averaging repeated readings reduces the effect of random errors.
- Systematic errors cannot be reduced by averaging; they require calibration or correction.
- Precision is the closeness of repeated measurements to each other.
- Accuracy is the closeness of a measurement to the true value.
- Precise readings can be inaccurate if there is a systematic error.
- Accurate readings can be imprecise if there is large random scatter.
- Precision is often quantified by the spread or uncertainty of repeated readings.
- For addition and subtraction, absolute uncertainties add.
- For multiplication and division, percentage uncertainties add.
- For raising to a power, percentage uncertainty is multiplied by the power.
- Percentage uncertainty = (absolute uncertainty / measured value) × 100%.
- Final uncertainty should be expressed in the form requested, either absolute or percentage.
- Error bars represent the absolute uncertainty in each plotted point and must be drawn to the scale of the axes.
- The line of best fit balances points above and below it and should pass through as many error bars as possible.
- A worst line is the steepest or shallowest straight line consistent with all error bars, giving the range of possible gradients.
- Absolute uncertainty has the same unit as the measurement; percentage uncertainty = (absolute uncertainty ÷ measured value) × 100.
- Percentage difference = (difference between experimental and accepted value ÷ accepted value) × 100.
- Gradient uncertainty can be estimated as half the difference between the steepest and shallowest worst-line gradients.
Examiner Tips
- 💡Identify whether an error shifts all readings in the same direction (systematic) or scatters them (random).
- 💡For a zero error, state the correction: subtract the zero reading from all measurements.
- 💡Use repeated readings and a mean to reduce random error, and quote the uncertainty.
- 💡When describing an improvement, match it to the type of error: calibration for systematic, averaging for random.
- 💡When asked to compare, state whether the data are precise, accurate, both or neither, and justify with reference to spread and true value.
- 💡Use the terms correctly: precision for repeatability, accuracy for closeness to true value.
- 💡If a graph shows a straight line with a non-zero intercept, comment on accuracy and suggest a systematic error.
- 💡Quote uncertainties to support statements about precision.
- 💡Write down the rule you are using before substituting numbers to avoid mixing rules.
- 💡Convert all uncertainties to the same type (absolute or percentage) before combining.
- 💡For powers, remember to multiply the percentage uncertainty by the exponent.
- 💡Check that the final uncertainty has the same units as the quantity if absolute, or is a percentage if percentage.
- 💡Label each axis with the quantity symbol, quantity name and unit, for example time t / s, before plotting.
- 💡When estimating gradient uncertainty, calculate the steepest and shallowest gradients from the worst lines and halve their difference.
- 💡Check that any percentage calculation uses consistent units and quote the result to a sensible number of significant figures.
Common Mistakes
- Thinking that repeating measurements removes a systematic error; repetition only reduces random error, while systematic error needs calibration or correction.
- Confusing zero error with random error; a zero error is a fixed offset and is systematic.
- Believing that systematic errors affect precision; they mainly affect accuracy, while random errors affect precision.
- Assuming all errors are random; check for a consistent offset that indicates a systematic error.
- Using 'accurate' and 'precise' interchangeably; they describe different things: closeness to true value versus closeness of repeats.
- Assuming precise data must be accurate; a systematic error can make precise data inaccurate.
- Thinking that accuracy can be improved by taking more readings; accuracy requires calibration or correction of systematic errors.
- Confusing precision with the number of decimal places; precision relates to the spread of repeated measurements, not just the format.
- Adding absolute uncertainties for multiplication or division; you must add percentage uncertainties instead.
- Forgetting to multiply the percentage uncertainty by the power when raising to a power.
- Mixing absolute and percentage uncertainties without converting; always convert to a consistent form before combining.
- Subtracting uncertainties when subtracting quantities; absolute uncertainties always add, even for subtraction.
- Drawing error bars only in one direction: error bars must extend both above and below each point by the stated absolute uncertainty.
- Confusing percentage uncertainty with percentage difference: percentage uncertainty uses the measured value as the denominator, while percentage difference uses the accepted value.
- Choosing a line of best fit that passes through every point exactly: real data scatter, so the line should balance the scatter rather than join points.
- Forgetting to convert units before calculating a percentage: both quantities in the fraction must have the same unit.