Measurement, significant figures and uncertainty
Real lab work involves random and systematic errors. The exam tests your ability to report values to appropriate precision and estimate the uncertainty in your measurements.
Significant figures (s.f.)
Significant figures are the digits in a number that carry meaning. Rules:
- All non-zero digits are significant. (32.4 → 3 s.f.)
- Zeros between non-zero digits are significant. (102 → 3 s.f.)
- Leading zeros are not significant. (0.0034 → 2 s.f.)
- Trailing zeros after a decimal point are significant. (4.50 → 3 s.f.)
- Trailing zeros in whole numbers are ambiguous unless made explicit. (1500 could be 2, 3 or 4 s.f.; use scientific notation if needed: 1.50 × 10³ → 3 s.f.)
When to round
The final answer should match the least precise measurement used. If you measured a mass to 3 s.f. and a volume to 2 s.f., quote your answer to 2 s.f. Don't keep extra digits — they imply a precision you don't have.
But: carry extra digits through your working to avoid rounding errors, and only round at the end.
Repeating measurements — mean and range
For greater reliability, scientists usually repeat measurements and find the mean:
mean = sum of values ÷ number of values
The range = maximum − minimum value.
The uncertainty in a mean is often estimated as half the range:
uncertainty ≈ ± (max − min) / 2
So if reaction times are 12.3, 12.5, 12.4, 12.6 s:
- Mean = (12.3 + 12.5 + 12.4 + 12.6)/4 = 12.45 s
- Range = 12.6 − 12.3 = 0.3 s
- Uncertainty ≈ ±0.15 s
- Report as (12.5 ± 0.15) s (or 12.4 ± 0.2 to fewer s.f.).
Anomalous results
A value clearly outside the pattern of the others is anomalous. Discard it from the mean (note that you have done so), and explain it (e.g. apparatus issue, temperature change).
Apparatus uncertainty
Each piece of apparatus has its own uncertainty:
- 50 cm³ measuring cylinder: ±0.5 cm³
- Burette (50 cm³): ±0.05 cm³
- Pipette (25 cm³): ±0.04 cm³
- Top-pan balance (2 d.p.): ±0.005 g
- Stopwatch: ±0.1 s (reaction time)
Use the most precise instrument suitable for the volume / mass.
Random vs systematic errors
- Random errors — vary unpredictably (e.g. timing the moment a colour change occurs). Reduced by repeating measurements.
- Systematic errors — consistent bias (e.g. an unzeroed balance). Repeating doesn't help; check apparatus and zero before measuring.
✦Worked example
Five trials of a titration give burette readings (cm³): 24.10, 24.15, 24.05, 24.10, 24.45.
- The fifth reading (24.45) is anomalous (>0.3 cm³ outside the cluster).
- Discard it; mean of remaining four = (24.10 + 24.15 + 24.05 + 24.10)/4 = 24.10 cm³ (4 s.f.).
- Range of accepted values = 0.10; uncertainty ≈ ±0.05 cm³.
- Report: 24.10 ± 0.05 cm³.
⚠Common mistakes
- Quoting too many s.f. "12.5673" from a stopwatch is misleading — you didn't measure to 6 s.f.
- Rounding mid-calculation. Carry extra digits and round at the end.
- Including anomalies in the mean without commenting on them.
- Confusing precision with accuracy. Precision = repeatability (close together); accuracy = closeness to true value.
Links
Useful in every required practical, especially C8.2 (chromatography Rf), C4.7 (titrations), C6.1 (rates).
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