An uncertainty states how far a measurement might be from the true value, written as value ± uncertainty with the same unit. In measurement and quantities it follows naturally from reading a scale and averaging repeats.
There is more than one accepted way to estimate uncertainty. The methods below are common classroom conventions, so confirm with your teacher and the Cambridge Physics 0625 syllabus which approach your course expects.
Where does an uncertainty come from?
| Situation | Common estimate |
|---|---|
| One reading on an analogue scale | half of the smallest division |
| Several repeated readings | half of the range: (largest − smallest) ÷ 2 |
| A timing of many swings, then divided | uncertainty of the total divided by the number of swings |
Percentage uncertainty = (uncertainty ÷ value) × 100%. It shows how large the uncertainty is compared with the measurement, which is helpful when comparing two different measurements.
Worked example
A student times 10 oscillations of a pendulum five times: 18.4 s, 18.7 s, 18.5 s, 18.6 s, 18.3 s. (Invented example data.)
Step 1, mean: 18.4 + 18.7 + 18.5 + 18.6 + 18.3 = 92.5. 92.5 ÷ 5 = 18.5 s.
Step 2, half the range: range = 18.7 − 18.3 = 0.4 s. Half the range = 0.2 s.
Step 3, state for 10 oscillations: 18.5 ± 0.2 s.
Step 4, percentage uncertainty: 0.2 ÷ 18.5 × 100% = 1.08%, so about 1.1%.
Step 5, period: divide both by 10. Period = 1.85 ± 0.02 s. The percentage uncertainty stays 1.1%.
Answer: period = 1.85 ± 0.02 s, which is about 1.1% uncertainty.
Timing ten swings keeps the same timing error but divides it across ten periods, so the period is known much more precisely than one swing timed alone.
The mistake to watch for
A common slip is to write an uncertainty with more digits than the measurement can support, or to give a value with unmatched decimal places.
Mistaken answer: 18.4937 ± 0.2 s, or 18.5 ± 0.183 s
The first has far more digits than the data support. The second pretends the uncertainty is known to three figures.
The correction is to round the uncertainty to one significant figure, then round the value to the same decimal place: 18.5 ± 0.2 s. Keep it consistent so the last digit of the value and the uncertainty line up.
A second slip is to use the smallest division as the uncertainty for a repeated set, ignoring the spread. When readings disagree by more than half a division, the spread is the better guide.
Check yourself
1. A ruler has 1 mm divisions. Estimate the uncertainty of a single length reading, in cm.
Show answer
Half the smallest division = 0.5 mm = 0.05 cm.
2. A length is measured as 5.1 cm, 5.4 cm, 5.2 cm and 5.3 cm. Give the mean with a half-range uncertainty.
Show answer
Sum = 5.1 + 5.4 + 5.2 + 5.3 = 21.0, so the mean = 21.0 ÷ 4 = 5.25 cm. Half range = (5.4 − 5.1) ÷ 2 = 0.15 cm. Result: 5.25 ± 0.15 cm.
3. A measuring cylinder reading is 25 mL with an uncertainty of 0.5 mL. Find the percentage uncertainty.
Show answer
0.5 ÷ 25 × 100% = 2%.
Where this leads next
With uncertainty, the measurement toolkit is complete. Try the measurement and quantities practice set to mix all five skills, and revisit calculating a repeated-measurement mean if the averaging step is unsteady. The bounds and rounding explainer shows how rounding changes the range a value could lie in.
If you understand the method but are unsure how to phrase it in a written conclusion, our teachers can practise that with you in online one-to-one Physics tuition.