Absolute uncertainty
Measurements & Uncertainties - OCR A-Level Physics
Key Definition
Absolute uncertainty
The size of the confidence range around a measurement, expressed in the same units as the measurement itself and written as ± a value after the reading. The way you calculate it depends on whether the measurement came from one reading or from a set of repeats.
The size of the confidence range around a measurement, expressed in the same units as the measurement itself and written as ± a value after the reading. The way you calculate it depends on whether the measurement came from one reading or from a set of repeats.
Case 1 : A single reading
When you take one reading from an analogue instrument, the absolute uncertainty is ± half the smallest division on the scale. A 30 cm ruler with mm markings gives a reading of ± 0.5 mm. A protractor reading to 1° gives ± 0.5°. For a digital instrument, use the manufacturer's tolerance, or ± 1 in the last digit if no tolerance is stated.
$$\Delta x = \pm \tfrac{1}{2} \times (\text{smallest division})$$
Case 2 : A set of repeat readings
When you take two or more repeat readings of the same quantity, the measurement is reported as the mean, and the absolute uncertainty is ± half the range (half the gap between the largest and smallest reading). This captures the spread caused by random error.
$$\Delta x = \pm \frac{x_{\max} - x_{\min}}{2}$$
- Worked example for Case 2: timings of 2.34, 2.38, 2.31 s. Mean $= \frac{2.34 + 2.38 + 2.31}{3} = 2.34 \text{ s}$. Range $= 2.38 - 2.31 = 0.07 \text{ s}$. Absolute uncertainty $= \pm \frac{0.07}{2} = \pm 0.04 \text{ s}$. Final result: $t = 2.34 \pm 0.04 \text{ s}$.
- Pick the case that matches the data in the question. If the stem gives only one reading, use Case 1. If it gives a table of repeats, use Case 2.
- Round the absolute uncertainty to 1 significant figure, then round the mean to match the last decimal place of the uncertainty.
Percentage uncertainty
The absolute uncertainty expressed as a percentage of the measured value. Has no units, so you can compare uncertainties across quantities measured in different units.
$$\% \text{ uncertainty} = \frac{\Delta x}{x} \times 100\%$$
- Worked example: $\ell = 0.45 \pm 0.01 \text{ m}$. Percentage uncertainty $= \frac{0.01}{0.45} \times 100\% = 2.2\%$.
- A small percentage uncertainty means a high-quality measurement. Below 5% is generally good for an A-level practical.
Common Mistake
MEDIUM
Wrong: Using the absolute uncertainty as the percentage (e.g. writing "0.01%" when the absolute uncertainty is 0.01 m). Or using the full smallest division for a single reading instead of half it.
Right: $\% \text{ uncertainty} = \frac{\Delta x}{x} \times 100\%$. You must divide by the measured value first. 0.01 m out of 0.45 m is 2.2%, not 0.01%. And for a single reading: $\Delta x = \pm \tfrac{1}{2}$ smallest division, not the full smallest division.
Right: $\% \text{ uncertainty} = \frac{\Delta x}{x} \times 100\%$. You must divide by the measured value first. 0.01 m out of 0.45 m is 2.2%, not 0.01%. And for a single reading: $\Delta x = \pm \tfrac{1}{2}$ smallest division, not the full smallest division.