Note that all content on this page is not prescriptive and is not an answer key or official statement regarding correct methodology for the Cambridge examinations. It is a compilation of common methods used in schools, with explainations and justifications based on established scientific methodology.
We assume that, as a teacher, you are familiar with the following practices:
The total uncertainty in a measurement comes from instrument error and experimental error.
Instrument error
In general, the majority of readings have a precision of half the smallest interval.
Some examples:
A ruler has 1mm intervals, so each reading is to the nearest ±0.5mm
A particular burette has 0.1ml intervals, so each reading is to the nearest ±0.1ml
A digital weighing scale displays readings to a precision of 0.01g, so each reading is to the nearest ±0.005mm*
Most measurements (e.g. length for ruler) require two readings, therefore the total uncertainty is twice of half the smallest interval (see error propagation below).
Some examples:
When measuring a length with a ruler, the length is the difference between two positions.
When measuring an interval with a pair of vernier calipers or micrometer screw gauge, you first have to zero the instrument. This is equivalent to making one measurement, and the interval is the difference between the measurement and the 'zero measurement'.
When using a burette to transfer a fluid, the amount transferred is obtained by the final amount in the burette minus the initial amount in the burette .
When measuring a time interval with a stopwatch, the time interval is the difference between the starting time and stopping time.
When using a weighing scale, the mass of the object being measured is equal to the difference between the final reading and the zero reading.
In a number of cases, the zero point is calibrated by the factor (e.g. whether the markings on a beaker are correctly printed). In that case, when filling the beaker from empty up to a certain reading, the total uncertainty is only half the smallest interval (and may be more precise if the interval is large and it is reasonable to subdivide the interval).
Experimental error
For all experiments, it is also necessary to consider whether any other factors may introduce error.
Some examples:
Uneven surfaces
Difficulty in aligning measuring tool
Difficulty in taking measurement without disrupting the system
The variable may change more rapidly than measurements can be reasonably made
Meniscus or other unevenness resulting in difficulty aligning reading
Sometimes, the reverse might be true.
E.g. human reaction time can be up to 0.3s, but for pendulums or other predictable motion, you can attain higher precision by pre-empting the motion.
Overall error
We usually sum up instrumental error and experimental error (assuming they are independent).
Because we only consider error to 1s.f., either instrumental or experimental error may be negligible.
e.g. a stopwatch has an instrumental precision of 0.01s, but human reaction time is an order of magnitude larger, so we do not need to consider the instrumental precision.
In some cases, the experimental error may be very much larger than instrumental precision - for example, the time taken for a reaction to occur or to observe temperature changes may have a precision in seconds rather than the full precision of a stopwatch or human reaction time. In this case, even if it is possible to feasibly measure higher degrees of precision when only considering equipment and technique, the nature of the experiment itself renders it meaningless to record such high degrees of precision.
Estimating error or uncertainty in calculated data
For addition of two independent variables:
If C = A+B or C = A-B, then ΔC = ΔA + ΔB
For multiplication or division of two independent variables:
If C = AB or C = A/B, then ΔC/C = ΔA/A + ΔB/B
We can also consider that for y = f(x), Δy = (largest value of y - smallest value of y)/2. Be careful to check whether y is monotonic in the range of x ± Δx.
See below for common heuristics for d.p./s.f.
If the error is known, then
Error should be reported to 1sf
Data should be measured to the same number of dp as the error.
A common heuristic for calculated data is to:
Keep the same number of d.p. as the least precise term when adding or subtracting
(e.g. 34.3 - 2.21 = 32.09, rounded to 1dp following "34.3" and expressed as 32.1)
Keep the same number of s.f. as the least precise term when multiplying or dividing or taking averages
(e.g. 15.2 x 1.5 = 22.8, rounded to 2sf following "1.5" and expressed as 23)
Note that some international scientific practices may recommend higher degrees of precision in reporting error (e.g. to 2sf), but this practice is not common in schools.