A measurement written as 20.0 ± 0.2 cm contains two pieces of information: a measured value and an estimated absolute uncertainty. The uncertainty is not a correction to subtract from every reading. It describes the stated range around that value.
Percentage uncertainty expresses the size of that uncertainty relative to the measurement. That lets you compare measurements with different sizes or units, then decide which one is worth improving.
This guide uses supplied uncertainties in original practice questions. In a practical task, justify estimates from the apparatus, measurement method and repeated readings as appropriate. Do not assume every instrument has the same uncertainty. See the Cambridge Physics 9702 syllabus for your examination-year requirements.
Start with a fraction of the measured value
For L = 20.0 ± 0.2 cm, the fractional uncertainty is 0.2/20.0 = 0.01. Multiply by 100 to obtain a percentage uncertainty of 1.0%.
Use the same units in the numerator and denominator. Dividing 0.2 cm by 0.200 m without converting units gives a meaningless comparison. Either work entirely in centimetres or convert both quantities to metres first.
Apply the rule that matches the calculation
For the usual first-order maximum-uncertainty estimates used in these exercises, add absolute uncertainties for sums or differences. For products or quotients, add fractional or percentage uncertainties. For a quantity raised to a power, multiply its percentage uncertainty by the magnitude of that power.
These are estimates, not an assertion that every real error always takes its worst-case value. Statistical treatments of independent random uncertainties can use different combination rules. Follow the method and assumptions requested in your course or question.
Worked example: area of a rectangle
A rectangle has length L = 20.0 ± 0.2 cm and width W = 5.00 ± 0.05 cm. Its calculated area is 20.0 × 5.00 = 100 cm².
Length has percentage uncertainty 0.2/20.0 × 100 = 1.0%. Width has percentage uncertainty 0.05/5.00 × 100 = 1.0%. The estimated percentage uncertainty in area is 2.0%.
Two per cent of 100 cm² is 2 cm², so an appropriate reported result for this estimate is A = (100 ± 2) cm². The absolute uncertainty has square-centimetre units because it belongs to the area.
You can check the approximation with the extreme products. The upper product is 20.2 × 5.05 = 102.01 cm² and the lower is 19.8 × 4.95 = 98.01 cm². The small difference from a symmetric ±2 estimate comes from a higher-order term.
Why subtraction can produce a large percentage
Suppose you calculate a small difference from two fairly large readings. Their absolute uncertainties add in this maximum-estimate method, while the difference itself may be small. Dividing by that small result can produce a large percentage uncertainty.
This is a reason to examine the measurement method, not to hide the uncertainty by reporting more decimal places. Extra digits do not create extra information.
Try it: speed from distance and time
A distance is 1.50 ± 0.01 m and a time is 0.75 ± 0.02 s. Calculate speed and its estimated percentage uncertainty using v = s/t.
Speed is 1.50/0.75 = 2.00 m/s. Distance contributes about 0.67%; time contributes about 2.67%. Their sum is about 3.33%. The corresponding absolute uncertainty is about 0.067 m/s, so v ≈ (2.00 ± 0.07) m/s is a reasonable report here.
Timing contributes more to the uncertainty than distance. Improving timing would therefore be the first measurement change to investigate in this example.
Explore StudyForA's Physics 9702 course. For each calculation, label your uncertainty as absolute, fractional or percentage before combining it.