IBChemistryHLRevision NotesToolsTool 3: MathematicsProcessing Uncertainties in Chemistry

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First exams 2025

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Processing Uncertainties in Chemistry (HL IB Chemistry)

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Processing Uncertainties in Chemistry

What is uncertainty?

  • Uncertainty is 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 is grouped into three main types:

    • Absolute uncertainty

      • The actual amount by which the quantity is uncertain
      • e.g.if v = 5.0 ± 0.1 cm, the absolute uncertainty in v is 0.1 cm

    • Fractional uncertainty

      • The absolute uncertainty divided by the quantity itself
      • e.g.if v = 5.0 ± 0.1 cm, the fractional uncertainty in v is fraction numerator 0.1 over denominator 5.0 end fraction = begin mathsize 14px style 1 over 50 end style

    • Percentage uncertainty

      • The ratio of the expanded uncertainty to the measured quantity on a scale relative to 100%
      • This is calculated using the following formula:

Percentage uncertainty = fraction numerator uncertainty over denominator measured space value end fraction cross times 100

How to calculate absolute, fractional and percentage uncertainty

uncertainty-in-burette-reading

 

The key pieces of information from this burette reading are the smallest division and the reading

  • The uncertainties in this reading are:
    • Absolute
      • Uncertainty = begin mathsize 14px style fraction numerator 0.1 over denominator 2 end fraction end style = 0.05 cm3 
      • Reading = 19.6 ± 0.05 cm3 
    • Fractional
      • Uncertainty = uncertainty over value = fraction numerator 0.1 over denominator 19.6 end fraction = 1 over 196 cm3 
    • Percentage
      • Uncertainty = uncertainty over value cross times 100 = fraction numerator 0.1 over denominator 19.6 end fraction cross times 100 = 0.5%
      • Reading = 19.6 ± 0.5% cm3 

Propagating uncertainties in processed data

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

Adding or subtracting measurements

  • When you are adding or subtracting two measurements then you add together the absolute measurement uncertainties
  • For example,
    • Using a balance to measure the initial and final mass of a container
    • Using a thermometer for the measurement of the temperature at the start and the end
    • Using a burette to find the initial reading and final reading
  • In all of these examples, you have to read the instrument twice to obtain the quantity
    • If each time you read the instrument the measurement is 'out' by the stated uncertainty, then your final quantity is potentially 'out' by twice the uncertainty

Multiplying or dividing measurements

  • When you multiply or divide experimental measurements then you add together the percentage uncertainties
  • You can then calculate the absolute uncertainty from the sum of the percentage uncertainties

 

Exponential measurements (HL only)

  • When experimental measurements are raised to a power, you multiply the fractional or percentage uncertainty by the power

The coefficient of determination, R2

  • The coefficient of determination is a measure of fit that can be applied to lines and curves on graphs
  • The coefficient of determination is written as R2 
  • It is used to evaluate the fit of a trend line / curve:
    • R2 = 0
      • The dependent variable cannot be predicted from the independent variable. 
      • R² is usually greater than or equal to zero
    • R2 between 0 and 1
      • The dependent variable can be predicted from the independent variable, although the degree of success depends on the value of R2 
      • The closer to 1, the better the fit of the trend line / curve
    • R2 = 1
      • The dependent variable can be predicted from the independent variable
      • The trend line / curve is perfect 
      • Note: This does not guarantee that the trend line / curve is a good model for the relationship between the dependent and independent variables
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Richard

Author: Richard

Richard has taught Chemistry for over 15 years as well as working as a science tutor, examiner, content creator and author. He wasn’t the greatest at exams and only discovered how to revise in his final year at university. That knowledge made him want to help students learn how to revise, challenge them to think about what they actually know and hopefully succeed; so here he is, happily, at SME.