Determining uncertainties in your raw data

Where an uncertainty comes from

Every raw measurement in this standard needs an uncertainty attached, because the Achieved criterion asks for "appropriate uncertainties in raw data". There are only three sources you need, and knowing which applies is most of the skill.

1. Reading uncertainty (from the instrument)

• For an instrument with a scale you read against (ruler, thermometer, protractor), the uncertainty is usually half the smallest division.
  • A millimetre ruler: $\pm 0.5$ mm on a single reading.
• For a digital instrument, the uncertainty is usually the last digit displayed.
  • A balance reading $2.34$ g: $\pm 0.01$ g.
• Where a measurement requires two readings (e.g. a length measured between two marks on a ruler), the uncertainty doubles, because each end contributes.
  • Measuring a length from $2.0$ mm to $153.0$ mm on a mm ruler: $\pm 1$ mm overall.

2. Uncertainty from repeated readings

• When you have repeats, the scatter of the readings is usually a better estimate than the instrument's resolution — and if it is larger, it is the one to quote.

$$\delta x = \frac{x_{\text{max}} - x_{\text{min}}}{2}$$

• Use the larger of the reading uncertainty and half the range. Quoting $\pm 0.01$ s from a stopwatch's display when your three readings differ by $0.4$ s is not an "appropriate" uncertainty.

3. Uncertainty from the experimenter

• Some measurements are limited by you, not the instrument.
  • Reaction time in hand timing: about $\pm 0.2$ s, far larger than any stopwatch's resolution.
  • Judging a position by eye: the uncertainty is how confidently you can say where the mark was, which may be several millimetres even on a mm scale.
• State this explicitly in the write-up: "the stopwatch reads to $0.01$ s, but the timing is limited by reaction time, so I have taken $\pm 0.2$ s."

Absolute and percentage uncertainty

• Absolute uncertainty $\delta x$ carries the same unit as the measurement: $L = 0.850 \pm 0.002$ m.
• Percentage uncertainty expresses it as a fraction of the reading:

$$\%\,\delta x = \frac{\delta x}{x} \times 100$$

• Percentage uncertainty is what you need when combining measurements, and it is what tells you which measurement is limiting your result.

Presenting raw data properly

• State the uncertainty once per column in the table heading if it is the same for every reading: "Length $L$ (m) $\pm 0.002$".
• Give the uncertainty to one significant figure, and round the measurement to the same decimal place.
  • Write $2.34 \pm 0.05$ s, not $2.3417 \pm 0.0523$ s.