Practise › Questions › Uncertainty and error
Uncertainty and error questions
Every measurement comes with an uncertainty, an interval the true value is likely to lie in. This topic covers how to find that interval, how to make it smaller, and how to carry it through a calculation to the final answer.
19 original questions · 60 marks · the uncertainty and error notes · Measurements
Every question here is written for this library rather than taken from a past paper. Write your answer out before opening a mark scheme: the schemes award marks point by point, and the marks are easier to see when you have something of your own to compare against.
Distinguish between random errors and systematic errors.
Mark scheme
Random errors scatter readings unpredictably above and below the true value, and are reduced by repeating and averaging (1). Systematic errors shift all readings the same way, a zero error for example, and are not reduced by repeating (1).A length is measured as 2.50 ± 0.05 m. Calculate the percentage uncertainty.
Mark scheme
Percentage uncertainty = (0.05/2.50) × 100 (1)
= 2.0% (1)Distinguish between the precision and the accuracy of a set of measurements.
Mark scheme
Precision is how close repeated readings are to each other, that is how small the scatter is (1). Accuracy is how close readings are to the true value (1). Measurements can be precise but inaccurate if a systematic error is present.State what is meant by the resolution of an instrument, and give the resolution of a digital stopwatch that displays times to 0.01 s.
Mark scheme
Resolution is the smallest change in the quantity that the instrument can display (1). For this stopwatch the resolution is 0.01 s (1).A student's repeated measurements of a resistance agree closely with each other. A second student, using a different meter, obtains values that agree closely with the first student's. State the term that describes each of these two observations.
Mark scheme
The first student's results are repeatable: the same experimenter with the same equipment gets consistent results (1). The agreement between the two students, using different equipment, shows the results are reproducible (1).Two lengths, 3.0 ± 0.1 m and 5.0 ± 0.2 m, are added. State the total length and its absolute uncertainty.
Mark scheme
When adding, absolute uncertainties add (1)
Total = 8.0 ± 0.3 m (1)A rectangle has length 10.0 ± 0.2 cm and width 5.0 ± 0.1 cm. Calculate its area and the absolute uncertainty in the area.
Mark scheme
Percentage uncertainties: length 2%, width 2% (1)
For a product they add, so the area uncertainty is 4% (1)
Area = 10.0 × 5.0 = 50 cm2 (1)
Uncertainty = 4% of 50 = ±2 cm2 (1)A length of 45 mm is measured with a ruler of 1 mm resolution, giving a reading uncertainty of ±0.5 mm. Calculate the percentage uncertainty.
Mark scheme
Uncertainty is ±0.5 mm on a reading of 45 mm (1)
Percentage uncertainty = (0.5/45) × 100 (1)
= 1.1% (1)Four repeated readings of a length are 20.1, 20.3, 20.2 and 20.4 mm. Calculate the mean and estimate its uncertainty from the range.
With its jaws fully closed, a micrometer reads −0.03 mm. With a wire between the jaws it reads 0.36 mm. (a) Name the type of error the closed-jaw reading reveals. (b) Determine the diameter of the wire. (c) Explain why taking repeat readings would not remove this error.
To find the extension of a rubber cord, a student measures its stretched length as 482 ± 1 mm and its unstretched length as 476 ± 1 mm. (a) State the extension and its absolute uncertainty. (b) Calculate the percentage uncertainty in the extension. (c) Suggest one change to the experiment that would reduce this percentage uncertainty.
A student determines g from g = 4π2l/T2 using a simple pendulum, with l = 0.650 ± 0.002 m and T = 1.62 ± 0.02 s. Calculate the percentage uncertainty in the value of g.
The radius of a circle is 4.0 ± 0.1 cm. Calculate the area and its absolute uncertainty (area = πr2).
A density is found from ρ = m/V, with m = 0.500 ± 0.005 kg and V = 2.0 × 10−4 ± 0.1 × 10−4 m3. Calculate the density and its absolute uncertainty.
State how you would reduce (a) a random error and (b) a systematic error in an experiment.
A student investigating a falling ball plots a graph with error bars on every point. The best-fit line has gradient 0.204 s2 m−1. The steepest and shallowest lines that still pass through all the error bars have gradients 0.212 s2 m−1 and 0.196 s2 m−1. Determine the absolute and percentage uncertainty in the gradient, and quote the gradient with its uncertainty.
A supplier claims that a batch of ball bearings has diameter 5.00 mm with a tolerance of ±1%. A student measures one bearing four times with a micrometer, reading 5.11, 5.09, 5.07 and 5.09 mm. Deduce whether the bearing meets the supplier's claim.
A student measures g by timing a ball falling a height h and using g = 2h/t2, with h = 1.250 ± 0.005 m and t = 0.505 ± 0.005 s. Determine the percentage uncertainty each measurement contributes to g, and state which single measurement most needs improving.
A student varies the length l of a simple pendulum, measures the period T for each length, and plots T2 on the y-axis against l on the x-axis, with error bars on the T2 values. The period is given by T2 = 4π2l/g. Describe fully how the student should use the graph to determine a value for g and its percentage uncertainty.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
Practise uncertainty and error one question at a time
The player marks nothing for you. It shows one question, waits, then shows the scheme so you can mark yourself, and brings a question back sooner when it went badly.