OCR Specification focus:
‘Determine percentage uncertainties when data are combined by multiplication, division, or powers.’
When physical quantities are combined through multiplication, division, or powers, their uncertainties combine differently. Understanding how to determine these percentage uncertainties ensures experimental results remain valid.
Combining Uncertainties in Multiplication, Division, and Powers
In experimental physics, every measurement carries an uncertainty — a range within which the true value is expected to lie. When these measurements are used to calculate derived quantities, it is crucial to propagate the uncertainties correctly. The OCR specification emphasises the ability to determine percentage uncertainties in cases where data are combined by multiplication, division, or powers, ensuring that final results accurately reflect measurement reliability.
Understanding Percentage Uncertainty
Before examining how uncertainties behave under different mathematical operations, it is essential to understand what a percentage uncertainty represents.
Percentage uncertainty: The ratio of the absolute uncertainty to the measured value, multiplied by 100, expressing the uncertainty as a percentage of the measured quantity.
For example, if a quantity has an absolute uncertainty of ±0.2 and a value of 5.0, the percentage uncertainty is (0.2 ÷ 5.0) × 100 = 4%.
Percentage uncertainty provides a dimensionless measure of confidence in the result, making it ideal for combining uncertainties in multiplicative operations.
Illustration of a vernier caliper showing a measurement with an explicit ± resolution. This highlights how an instrument’s smallest division leads to an absolute and then percentage uncertainty, which you add for multiplication/division. The diagram is uncluttered and suitable for OCR-level discussion of uncertainty propagation. Source.
Uncertainty Rules for Multiplication and Division
When two or more quantities are multiplied or divided, their percentage uncertainties are added together to find the percentage uncertainty in the final result. This arises because proportional (relative) errors in the quantities combine linearly during these operations.
EQUATION
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Multiplication/Division Rule:
Percentage uncertainty in result = Sum of percentage uncertainties in the quantities involved.
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For instance, if a calculation involves multiplying measured quantities A and B, each with known percentage uncertainties, the total uncertainty reflects the combined proportional effects of both.
When calculating a derived quantity such as density (ρ = m/V) or force (F = ma), uncertainties from mass, volume, and acceleration each contribute to the final uncertainty through this additive rule.
Practical Process for Multiplication or Division
To apply this rule correctly:
Step 1: Determine the percentage uncertainty of each measured quantity.
Step 2: Add all percentage uncertainties for quantities being multiplied or divided.
Step 3: The total gives the percentage uncertainty in the final result.
Step 4: If required, convert this percentage back to an absolute uncertainty using the calculated value.
This procedure ensures consistent treatment of uncertainties and aligns with OCR A-Level Physics expectations for experimental data handling.
Uncertainty in Powers and Exponents
When a measured quantity is raised to a power, its percentage uncertainty is multiplied by that power. This principle also applies when roots are taken (a power less than one).
Straight lines on a log–log graph represent power laws, with the slope equal to the exponent n. Because Q ∝ Aⁿ, the percentage uncertainty in Q scales by n times that of A. This visual ties directly to the OCR power rule for uncertainty propagation. Source.
EQUATION
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Power Rule:
If Q = Aⁿ, then percentage uncertainty in Q = |n| × (percentage uncertainty in A).
A = measured quantity with uncertainty
n = power or exponent applied to A
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For example, if a measurement of radius r has a 2% uncertainty, then an area A = πr² will have a percentage uncertainty of 2 × 2% = 4%, because the exponent on the radius is 2. This rule reflects how errors scale non-linearly when quantities are squared, cubed, or subjected to roots.
Combining Rules: A Summary of Relationships
The three major rules for uncertainty propagation in this subsubtopic can be summarised conceptually:
Addition/Subtraction: Absolute uncertainties add.
Multiplication/Division: Percentage uncertainties add.
Powers: Multiply the percentage uncertainty by the exponent.
These distinctions ensure correct uncertainty analysis across a range of physical formulae. Understanding which rule applies in each situation prevents overestimation or underestimation of uncertainty in experimental outcomes.
Applying the Concepts to Derived Quantities
Derived physical quantities, such as volume, area, density, and kinetic energy, often involve multiplicative relationships or powers. Each must include correct propagation of uncertainties to meet the homogeneity and accuracy requirements of scientific analysis.
For example:
When calculating volume from a cubed length (V = L³), the percentage uncertainty is tripled because the length is raised to the power of 3.
For density (ρ = m/V), add the percentage uncertainty in mass to the percentage uncertainty in volume to obtain the total.
For kinetic energy (Ek = ½mv²), add the percentage uncertainty in mass to twice the percentage uncertainty in velocity.
These examples demonstrate that the mathematical structure of the equation directly dictates how the uncertainties combine.
Using Uncertainty to Evaluate Experimental Quality
In practical physics, uncertainty analysis not only quantifies measurement reliability but also indicates experimental precision. A large percentage uncertainty suggests that the measurement or instrument lacks fine resolution, or that random fluctuations significantly affect readings.
To minimise total uncertainty in results involving multiplication, division, or powers:
Use instruments with smaller relative uncertainties (more precise readings).
Repeat measurements to reduce random error and refine the mean value.
Choose derived quantities that depend less on high powers or divisions, since these amplify uncertainties.
Ensure consistent unit conversions and maintain the same significant figures across all steps of a calculation.
These good practices are expected in OCR practical assessments and laboratory reports, showing awareness of uncertainty propagation and its experimental consequences.
Distinction Between Systematic and Random Effects on Uncertainty
Although this subsubtopic focuses on mathematical combination, students should remain aware that underlying systematic and random errors influence uncertainty sources.
Systematic errors shift all measurements consistently in one direction, affecting the accuracy but not the spread.
Random errors vary unpredictably, affecting the precision of repeated measurements.
Understanding the nature of these errors helps in interpreting percentage uncertainty correctly — distinguishing whether a high uncertainty arises from fluctuating readings or from instrument miscalibration.
Communicating Uncertainty in Results
When presenting final experimental results, include both the value and its associated uncertainty. The uncertainty should be rounded appropriately to one or two significant figures and expressed using the same unit or percentage form as contextually appropriate.
For quantities derived from multiplication, division, or powers:
Express final results as Q = value ± (percentage uncertainty%).
Ensure that the reported uncertainty reflects all propagated contributions.
Present the uncertainty consistently in both tables and graphs to meet OCR conventions for data presentation.
Clarity and consistency in communicating uncertainty are essential for scientific credibility and compliance with the OCR specification on data analysis and reporting.
FAQ
The number of significant figures should match the precision of your data. When you report a percentage uncertainty, it should have one or two significant figures. Reporting more implies false precision.
For example, if your uncertainty is 3.26%, it should be rounded to 3%. Likewise, the main value should be quoted to the same decimal place as the absolute uncertainty. This ensures consistency between measurements and uncertainties and avoids overstating experimental accuracy.
In multiplication or division, each variable contributes proportionally to the final result. Thus, their fractional (percentage) effects combine linearly.
In contrast, addition and subtraction involve quantities of the same unit, so their absolute uncertainties combine directly. This reflects whether the operation scales the uncertainty or simply shifts it by a fixed amount.
When you take the reciprocal, the percentage uncertainty remains the same.
For example, if a quantity X has a 3% uncertainty, then 1/X also has a 3% uncertainty. This is because both represent the same proportional deviation in the result. However, the absolute uncertainty will change since the value of 1/X differs from X, but its relative error does not.
If two quantities share a common source of uncertainty, they are correlated, and simply adding their percentage uncertainties can overestimate the total uncertainty.
In such cases:
Identify shared measurement sources (e.g. same instrument or calibration).
Combine only independent components linearly.
Treat shared uncertainties once in the total percentage.
This approach gives a more realistic measure of total uncertainty in derived results.
Percentage uncertainty is dimensionless, making it easy to compare the relative reliability of measurements with different magnitudes or units.
For instance, an uncertainty of ±0.2 m on a 1.0 m value (20%) indicates poorer precision than ±0.2 m on a 10.0 m value (2%).
Using percentage uncertainty allows fair comparison between datasets and highlights whether improvement is needed in measurement precision or instrument resolution.
Practice Questions
Question 1 (2 marks)
A student measures the radius of a small circular disc as 5.00 cm ± 0.05 cm.
Calculate the percentage uncertainty in the area of the disc.
Mark Scheme
1 mark: Recognises that area ∝ r² and hence percentage uncertainty in area = 2 × percentage uncertainty in radius.
1 mark: Calculates (0.05 ÷ 5.00) × 100 = 1%, then 2 × 1% = 2%.
Question 2 (5 marks)
In an experiment, the mass of a metal cylinder is measured as 250 g ± 2 g and its diameter as 2.50 cm ± 0.05 cm. The length of the cylinder is 10.0 cm ± 0.1 cm.
The density ρ of a cylinder is given by:
ρ = 4m / (πd²L)
(a) Determine the percentage uncertainty in the mass, diameter, and length.
(b) Calculate the overall percentage uncertainty in the density.
(c) Explain briefly which measurement contributes the greatest uncertainty to the final result and why.
Mark Scheme
(a) (3 marks total)
1 mark: Percentage uncertainty in mass = (2 ÷ 250) × 100 = 0.8%.
1 mark: Percentage uncertainty in diameter = (0.05 ÷ 2.50) × 100 = 2%.
1 mark: Percentage uncertainty in length = (0.1 ÷ 10.0) × 100 = 1%.
(b) (1 mark)
Recognises that ρ ∝ m / d²L, so total percentage uncertainty = 0.8% + (2 × 2%) + 1% = 5.8%.
(c) (1 mark)
Identifies that the diameter measurement contributes the largest uncertainty because it has the largest percentage uncertainty and it is squared in the formula, thus doubling its contribution.
