Topic 11.1: Measurement and data processing – Uncertainties and errors in measurements and results
All measurement has a limit of precision and accuracy, and this must be taken into account when evaluating experimental results.
• Understanding: Qualitative data includes all non-numerical information obtained from observations not from measurement.
▪ Qualitative data: non-numerical information obtained from observations obtained from measurement
• Understanding: Quantitative data are obtained from measurements, and are always associated with random errors/uncertainties,
determined by the apparatus, and by human limitations such as reaction times.
▪ Quantitative data: numerical information obtained from measurements and are associated with random and systematic errors
• Understanding: Propagation of random errors in data processing shows the impact of the uncertainties on the final result.
▪ Random errors: associated with uncontrolled variables in an experiment and cannot be eliminated
• Causes minor deviation in all direction (accuracy will be affected)
• Source of uncertainty calculation through propagation of random errors
• Understanding: Experimental design and procedure usually lead to systematic errors in measurement, which cause a deviation in
a particular direction.
▪ Systematic errors: associated with a flaw in the experimental design or instrument
• Causes deviation into a particular direction (all values will always be greater or less than)
• Understanding: Repeat trials and measurements will reduce random errors but not systematic errors.
▪ Repeating trials in measurement will reduce random errors but not systematic errors
▪ This is because random errors impact precision; repeat trials increase precision but not accuracy
• Applications and skills: Distinction between random errors and systematic errors.
• Applications and skills: Discussion of ways to reduce uncertainties in an experiment.
Random errors Systematic errors
Impacts Accuracy Precision
Cause Uncontrolled variable in an experiment Experimental design and procedure
Solution Repeat trials Instrumentation error: use apparatus that is not faulty
Use more precise apparatus Personal error: minimize human error
Experimental methodology error: minimize side
reactions, evaporations, and ensure insulation
• Applications and skills: Record uncertainties in all measurements as a range (+) to an appropriate precision.
▪ All uncertainty values are recorded as a range (+) to an appropriate precision
• Absolute uncertainty: margin of uncertainty associated with the result (same smallest decimal point)
• Relative uncertainty: ratio comparing the size of the uncertainty to the size of measured data (same significant figure)
• Applications and skills: Propagation of uncertainties in processed data, including the use of percentage uncertainties.
Operation involving addition (subtraction) Operation involving multiplication (division)
Type of uncertainty Absolute uncertainty Relative uncertainty
Involves Smallest number of decimal places Smallest number of significant figures
𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦
▪ Percentage uncertainty: × 100
𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑑𝑎𝑡𝑎
▪ Calculation of uncertainties: √𝐴2 + 𝐵 2
• Applications and skills: Discussion of systematic errors in all experimental work, their impact on the results and how they can be
reduced.
▪ Types of systematic error: instrumentation error, methodology error, personal errors
• Impact on result: affects precision of the given data
• Solution: adopting better experimental design for improved accuracy
• Applications and skills: Estimation of whether a particular source of error is likely to have a major or minor effect on the final result.
• Applications and skills: Calculation of percentage error when the experimental result can be compared with a theoretical or
accepted result.
𝑙𝑖𝑡𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒
▪ Percentage error: × 100
𝑙𝑖𝑡𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒
, • Applications and skills: Distinction between accuracy and precision in evaluating results.
▪ Precision: closeness between independent test results of the same experiment; the smaller
the extent of random error, the more precise the experiment
• Repeating the experiment improves precision but not accuracy
▪ Accuracy: closeness of the independent test results to the true value of measurand
(impacted by systematic error)
• Nature of science: Making quantitative measurements with replicates to ensure reliability—
precision, accuracy, systematic, and random errors must be interpreted through replication.
• International-mindedness: As a result of collaboration between seven international organizations, including IUPAC, the International
Standards Organization (ISO) published the Guide to the Expression of Uncertainty in Measurement in 1995. This has been widely adopted
in most countries and has been translated into several languages.
• Utilization: Crash of the Mars Climate Orbiter spacecraft.
• Utilization: Original results from CERN regarding the speed of neutrinos were flawed.
• Guidance: The number of significant figures in a result is based on the figures given in the data. When adding or subtracting, the final
answer should be given to the least number of decimal places. When multiplying or dividing the final answer is given to the least number of
significant figures.
• Guidance: Note that the data value must be recorded to the same precision as the random error.
• Guidance: SI units should be used throughout the programme.
All measurement has a limit of precision and accuracy, and this must be taken into account when evaluating experimental results.
• Understanding: Qualitative data includes all non-numerical information obtained from observations not from measurement.
▪ Qualitative data: non-numerical information obtained from observations obtained from measurement
• Understanding: Quantitative data are obtained from measurements, and are always associated with random errors/uncertainties,
determined by the apparatus, and by human limitations such as reaction times.
▪ Quantitative data: numerical information obtained from measurements and are associated with random and systematic errors
• Understanding: Propagation of random errors in data processing shows the impact of the uncertainties on the final result.
▪ Random errors: associated with uncontrolled variables in an experiment and cannot be eliminated
• Causes minor deviation in all direction (accuracy will be affected)
• Source of uncertainty calculation through propagation of random errors
• Understanding: Experimental design and procedure usually lead to systematic errors in measurement, which cause a deviation in
a particular direction.
▪ Systematic errors: associated with a flaw in the experimental design or instrument
• Causes deviation into a particular direction (all values will always be greater or less than)
• Understanding: Repeat trials and measurements will reduce random errors but not systematic errors.
▪ Repeating trials in measurement will reduce random errors but not systematic errors
▪ This is because random errors impact precision; repeat trials increase precision but not accuracy
• Applications and skills: Distinction between random errors and systematic errors.
• Applications and skills: Discussion of ways to reduce uncertainties in an experiment.
Random errors Systematic errors
Impacts Accuracy Precision
Cause Uncontrolled variable in an experiment Experimental design and procedure
Solution Repeat trials Instrumentation error: use apparatus that is not faulty
Use more precise apparatus Personal error: minimize human error
Experimental methodology error: minimize side
reactions, evaporations, and ensure insulation
• Applications and skills: Record uncertainties in all measurements as a range (+) to an appropriate precision.
▪ All uncertainty values are recorded as a range (+) to an appropriate precision
• Absolute uncertainty: margin of uncertainty associated with the result (same smallest decimal point)
• Relative uncertainty: ratio comparing the size of the uncertainty to the size of measured data (same significant figure)
• Applications and skills: Propagation of uncertainties in processed data, including the use of percentage uncertainties.
Operation involving addition (subtraction) Operation involving multiplication (division)
Type of uncertainty Absolute uncertainty Relative uncertainty
Involves Smallest number of decimal places Smallest number of significant figures
𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦
▪ Percentage uncertainty: × 100
𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑑𝑎𝑡𝑎
▪ Calculation of uncertainties: √𝐴2 + 𝐵 2
• Applications and skills: Discussion of systematic errors in all experimental work, their impact on the results and how they can be
reduced.
▪ Types of systematic error: instrumentation error, methodology error, personal errors
• Impact on result: affects precision of the given data
• Solution: adopting better experimental design for improved accuracy
• Applications and skills: Estimation of whether a particular source of error is likely to have a major or minor effect on the final result.
• Applications and skills: Calculation of percentage error when the experimental result can be compared with a theoretical or
accepted result.
𝑙𝑖𝑡𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒−𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒
▪ Percentage error: × 100
𝑙𝑖𝑡𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒
, • Applications and skills: Distinction between accuracy and precision in evaluating results.
▪ Precision: closeness between independent test results of the same experiment; the smaller
the extent of random error, the more precise the experiment
• Repeating the experiment improves precision but not accuracy
▪ Accuracy: closeness of the independent test results to the true value of measurand
(impacted by systematic error)
• Nature of science: Making quantitative measurements with replicates to ensure reliability—
precision, accuracy, systematic, and random errors must be interpreted through replication.
• International-mindedness: As a result of collaboration between seven international organizations, including IUPAC, the International
Standards Organization (ISO) published the Guide to the Expression of Uncertainty in Measurement in 1995. This has been widely adopted
in most countries and has been translated into several languages.
• Utilization: Crash of the Mars Climate Orbiter spacecraft.
• Utilization: Original results from CERN regarding the speed of neutrinos were flawed.
• Guidance: The number of significant figures in a result is based on the figures given in the data. When adding or subtracting, the final
answer should be given to the least number of decimal places. When multiplying or dividing the final answer is given to the least number of
significant figures.
• Guidance: Note that the data value must be recorded to the same precision as the random error.
• Guidance: SI units should be used throughout the programme.
