Processing Uncertainties (DP IB Physics: SL): Revision Note

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

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 $∆L$ = ±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 $\frac{∆L}{L}=\frac{0.1}{5.0}$ = ±0.02

  • 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 $\frac{0.1}{5.0}\times 100$ = ±2%

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

$\mathrm{percentage}\mathrm{uncertainty}=\frac{\mathrm{uncertainty}}{\mathrm{measured}\mathrm{value}}\times 100\%$

How to calculate absolute, fractional and percentage uncertainty

• The uncertainties in this reading are:

  • Absolute

    • Uncertainty = $\frac{0.2}{2}$ = 0.1 mA

    • Reading = 1.6 ± 0.1 mA

  • Fractional

    • Uncertainty = $\frac{\mathrm{uncertainty}}{\mathrm{value}}=\frac{0.1}{1.6}=\frac{1}{16}$

    • Reading = 1.6 mA ± $\frac{1}{16}$

  • Percentage

    • Uncertainty = $\frac{\mathrm{uncertainty}}{\mathrm{value}}\times 100=\frac{0.1}{1.6}\times 100=6.2\%$

    • 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:

Adding or subtracting measurements

• Add together the absolute uncertainties

Multiplying or dividing measurements

• Add the percentage or fractional uncertainties

Measurements raised to a power

• Multiply the percentage uncertainty by the power

Uncertainties and significant figures

• Measurements and processed uncertainties must be expressed to an appropriate number of significant figures

  • This is because the number of significant figures indicates how precise a measurement is

• Rules for determining the number of significant figures:

  • in an uncertainty: Absolute, percentage, and fractional uncertainties should usually be quoted to one significant figure

    • e.g. for 7.4 ± 0.2, the percentage uncertainty is ±2.703%, so this should be quoted as ±3% (1 s.f.)

    • The main exception is when the first digit is 1, then the uncertainty can be quoted to two significant figures

    • e.g. for 36.4652 ± 5%, the absolute uncertainty is ±1.82326, so this should be quoted as ± 1.8 (2 s.f.)

  • in a quantity: The number of significant figures must match the precision of the absolute uncertainty

    • The absolute uncertainty in a quantity defines how many figures are significant

    • A quantity and its uncertainty must be quoted to the same number of decimal places

    • A final calculated value must be rounded so its least significant digit aligns with the uncertainty’s place value

    • e.g. for 36.4652 ± 5%, the final result should be quoted as 36.5 ± 1.8

  • in raw data: The number of significant figures should be consistent with the resolution of the measuring instrument, e.g.

    • for a metre rule (resolution ±1 mm), measurements should be given to 1 mm, e.g. 15 ± 1 mm

    • for a pair of Vernier callipers (resolution ±0.1 mm), measurements should be given to 0.1 mm, e.g. 15.4 ± 0.1 mm

    • for a micrometer (resolution ±0.01 mm), measurements should be given to 0.01 mm, e.g. 15.42 ± 0.01 mm

  • in processed data: Calculated values should have no more significant figures than the least precise measurement

    • e.g. for the set of raw data 5.11, 5.14, 5.12, 5.10, the mean value is 5.1175, so this should be quoted to 3 significant figures, i.e. 5.12