IBPhysicsDPHLRevision NotesToolsTool 3: MathematicsProcessing Uncertainties

Processing Uncertainties (DP IB Physics): Revision Note

Katie M

Written by: Katie M

Reviewed by: Caroline Carroll

Updated on

DP Physics: HLDP Physics: SL

Processing uncertainties

What is uncertainty?

  • Uncertainty is a quantitative indication of the quality of the result 

    • It is the difference between the actual reading taken (caused by the equipment or techniques used) and the true value

    • It is a range of values around a measurement within which the true value is expected to lie, and is an estimate

  • Uncertainties are not the same as errors

    • Errors arise from equipment or practical techniques that cause a reading to be different from the true value

  • Uncertainties in measurements are recorded as a range (±) to an appropriate level of precision

Table showing different uncertainties

 

Uncertainty

in a reading

± half the smallest division

in a measurement

at least ±1 smallest division

in repeated data

half the range
i.e. ± ½ (largest - smallest value)

in digital readings

± the last significant digit
(unless otherwise quoted)

Types of uncertainty

  • Uncertainty can be expressed in one of three main forms

    • Absolute uncertainty

      • The actual amount by which the measurement is uncertain

      • e.g.if L = 5.0 ± 0.1 cm, the absolute uncertainty in L is increment L = 0.1 cm

    • Fractional uncertainty

      • The uncertainty in the measurement expressed as a fraction

      • e.g.if L = 5.0 ± 0.1 cm, the fractional uncertainty in L is fraction numerator increment L over denominator L end fraction space equals space fraction numerator 0.1 over denominator 5.0 end fraction space equals space 1 over 50

    • Percentage uncertainty

      • The uncertainty in the measurement expressed as a percentage

      • e.g.if L = 5.0 ± 0.1 cm, the percentage uncertainty in L is fraction numerator 0.1 over denominator 5.0 end fraction cross times 100 = 2%

      • In general, it is calculated using the following formula:

percentage space uncertainty space equals space fraction numerator uncertainty over denominator measured space value end fraction cross times 100 percent sign

How to calculate absolute, fractional and percentage uncertainty

Analogue milliammeter showing a reading of 1.6 milliamperes, with a scale from 0 to 15, and labelled "MILLIAMPERES (mA)"

  • The uncertainties in this reading are:

    • Absolute

      • Uncertainty = fraction numerator 0.2 over denominator 2 end fraction = 0.1 mA

      • Reading = 1.6 ± 0.1 mA

    • Fractional

      • Uncertainty = uncertainty over value space equals space fraction numerator 0.1 over denominator 1.6 end fraction space equals space 1 over 16

      • Reading = 1.6 mA ± 1 over 16

    • Percentage

      • Uncertainty = uncertainty over value cross times 100 space equals space fraction numerator 0.1 over denominator 1.6 end fraction cross times 100 space equals space 6.2 percent sign

      • Reading = 1.6 mA ± 6%

Propagating uncertainties in processed data

  • Uncertainty propagates in different ways depending on the type of calculation involved

  • When combining uncertainties, the rules are as follows:

Operation

Example 

Propagation Rule

Addition & Subtraction

y space equals space a plus-or-minus b

increment y space equals space increment a plus increment b

The sum of the absolute uncertainties

Multiplication & Division

y space equals space a cross times b or y space equals space a over b

fraction numerator increment y over denominator y end fraction space equals space fraction numerator increment a over denominator a end fraction plus fraction numerator increment b over denominator b end fraction

The sum of the fractional uncertainties

Power

y space equals space a to the power of n

fraction numerator increment y over denominator y end fraction space equals space n open parentheses fraction numerator increment a over denominator a end fraction close parentheses

The magnitude of n times the fractional uncertainty

 

Adding or subtracting measurements

  • Add together the absolute uncertainties

Combining Uncertainties (1), downloadable AS & A Level Physics revision notes

Multiplying or dividing measurements

  • Add the percentage or fractional uncertainties

Combining Uncertainties (2), downloadable AS & A Level Physics revision notes

Measurements raised to a power

  • Multiply the percentage uncertainty by the power

Combining Uncertainties (3), downloadable AS & A Level Physics revision notes

Examiner Tips and Tricks

Remember:

  • Absolute uncertainties (denoted by Δ) have the same units as the quantity

  • Percentage uncertainties have no units

  • The uncertainty in constants, such as π, is taken to be zero

Uncertainties in trigonometric and logarithmic functions will not be tested in the exam, so just remember these rules and you’ll be fine!

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    Author
    Reviewer
    Katie M

    Author: Katie M

    Expertise: Physics Content Creator

    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.

    Caroline Carroll

    Reviewer: Caroline Carroll

    Expertise: Physics & Chemistry Subject Lead

    Caroline graduated from the University of Nottingham with a degree in Chemistry and Molecular Physics. She spent several years working as an Industrial Chemist in the automotive industry before retraining to teach. Caroline has over 12 years of experience teaching GCSE and A-level chemistry and physics. She is passionate about creating high-quality resources to help students achieve their full potential.