Uncertainty in sums and differences

By the end of this lesson

By the end you can write a measurement with its absolute and percentage uncertainty, estimate the uncertainty from repeated readings, and find the uncertainty in a sum or difference.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$\theta$	temperature (in °C)

Value ± uncertainty

No measurement is exact, so we give a range:

$l = (25.0 \pm 0.1)\ \text{cm}$

This says the true length very probably lies between 24.9 cm and 25.1 cm.

Give the uncertainty to 1 significant figure (2 at most).
Round the value to the same decimal place as the uncertainty: $(25.0 \pm 0.1)$ cm, not $(25.03 \pm 0.1)$ cm.

Definition

Absolute uncertainty

The size of the range either side of the measured value within which the true value is expected to lie, $\Delta x$ . It has the same unit as the quantity.

Definition

Percentage uncertainty

The absolute uncertainty as a percentage of the measured value: $\text{percentage uncertainty} = \frac{\Delta x}{x} \times 100\%$ Without the $\times 100\%$ , $\dfrac{\Delta x}{x}$ is the fractional uncertainty. Neither has a unit.

Where does Δx come from?

A single reading: usually the smallest scale division of the instrument (sometimes half of it, if you can read between the marks reliably).
Repeated readings: use the mean as the value and half the range as the uncertainty: $\Delta x = \frac{x_{\max} - x_{\min}}{2}$

Worked example: Uncertainty from repeated readings

Four timings of a trolley's run are 2.34 s, 2.38 s, 2.30 s and 2.38 s. Write the time with its absolute and percentage uncertainty.

Have a go first?

Quick check

CalculateQuestion 1 · Recall

CalculateQuestion 2 · Apply

Answer the 2 questions above to continue.