Measurement, Units, Uncertainty and Estimation
This domain tests the practical mathematics of measuring: choosing a unit and an instrument, converting between units, reading an instrument to the correct precision, and reporting a result with a realistic uncertainty.
SI units and prefixes
Base units used here include the metre (m), kilogram (kg), second (s), ampere (A), kelvin (K) and mole (mol); derived units include the newton (N), joule (J), watt (W), pascal (Pa) and volt (V). Prefixes: k = 10³, c = 10⁻², m = 10⁻³, µ = 10⁻⁶, n = 10⁻⁹. Convert by multiplying by the power: 2.5 km = 2 500 m, and 250 mL = 0.250 L = 250 cm³.
Precision and accuracy
Precision is how close repeated measurements are to each other; accuracy is how close a measurement is to the accepted value.
- Random error scatters results in both directions and averaging reduces it.
- Systematic error shifts every result the same way, as with a badly zeroed instrument or a worn ruler; averaging does not help.
Reading an instrument
Read at eye level to avoid parallax, and for water in a narrow tube read the bottom of the meniscus. Quote the smallest graduation and estimate one digit beyond it. A ruler marked in 1 mm gives readings such as 42.3 ± 0.5 mm, never 42.30 mm.
Worked example
A student reads the volume in a graduated cylinder whose smallest division is 1 mL and obtains 42.5 mL at eye level. The accepted volume is 43.0 mL. Uncertainty ≈ ±0.5 mL, half the smallest division. Percentage error = |42.5 − 43.0| ÷ 43.0 × 100 = 1.2 %. The result is reported as 42.5 ± 0.5 mL, about 1 % below the accepted value.
Significant figures
In calculations the answer carries the precision of the least precise input: 12.3 cm × 2.0 cm = 25 cm², not 24.6 cm².
Traps
- Writing more decimals than the instrument can justify.
- Equating precision with accuracy: a very repeatable result can be consistently wrong.
- Mixing units before multiplying or dividing, for example kilometres with metres.
- Reading the top of the meniscus or viewing the scale at an angle.
- Treating a systematic error as something averaging will fix.