IBChemistrySLRevision NotesToolsTool 3: MathematicsProcessing Uncertainties in Chemistry

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First teaching 2023

First exams 2025

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Processing Uncertainties in Chemistry (SL IB Chemistry)

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Richard

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Richard

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Chemistry

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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