Accuracy vs precision
These words are often mixed up, but they mean different things.
| Term | Meaning |
|---|---|
| Accuracy | how close a measurement is to the true value |
| Precision | how fine the scale is, and how consistent repeated measurements are |
Dartboard picture:
- Accurate and precise: darts clustered on the bullseye.
- Precise but not accurate: tight cluster, off to one side.
- Accurate but not precise: scattered around the bullseye.
- Neither: scattered and off-centre.
Precision and instruments
A ruler marked in millimetres is more precise than one marked only in centimetres. A measurement like 12.3 cm says "to the nearest 0.1 cm"; 12.30 cm says "to the nearest 0.01 cm".
Significant figures
Significant figures (sig figs) show how precise a measurement is.
| Rule | Example | Sig figs |
|---|---|---|
| All non-zero digits are significant | 345 | 3 |
| Zeros between non-zero digits are significant | 4,005 | 4 |
| Leading zeros are not significant | 0.0072 | 2 |
| Trailing zeros after a decimal point are significant | 2.500 | 4 |
| Trailing zeros in a whole number are ambiguous | 1,300 | 2, 3 or 4 |
Use scientific notation to remove ambiguity: 1.30 × 10³ has 3 sig figs.
Calculating with measurements
Multiplying and dividing: keep the fewest significant figures in the inputs.
- 4.56 cm × 1.4 cm = 6.384 → 6.4 cm² (2 sig figs)
Adding and subtracting: keep the fewest decimal places.
- 12.11 + 0.3 = 12.41 → 12.4 (1 decimal place)
Do the whole calculation, then round once at the end.
Rounding to significant figures
| Number | 2 sig figs | 3 sig figs |
|---|---|---|
| 0.004567 | 0.0046 | 0.00457 |
| 72,849 | 73,000 | 72,800 |
| 9.995 | 10 | 10.0 |
Why it matters
Reporting 12.384 km when your trip meter only reads to 0.1 km claims precision you don’t have. In science, engineering and medicine, honest precision prevents costly errors.
Worked examples
Example 1: Sig figs in 0.05020? 4 (5, 0, 2, 0).
Example 2: 3.2 m × 1.25 m = 4.0 m² (2 sig figs).
Example 3: 105.3 g − 2.47 g = 102.83 → 102.8 g (1 decimal place).
Common mistakes
- Counting leading zeros as significant.
- Rounding too early in a multi-step calculation.
- Thinking more decimal places always means more accurate. (It means more precise.)
Try it yourself
- Sig figs in 6,070?
- Sig figs in 0.000300?
- Calculate 2.5 × 3.14159 to the right sig figs.
- Calculate 15.2 + 3.456 to the right decimal places.
- A scale reads 50.1, 50.2 and 50.1 g for a 52.0 g mass. Precise, accurate, or both?
Answers: 1) 3 (or 4 if the last zero is measured) 2) 3 3) 7.9 4) 18.7 5) precise but not accurate
Quick recap
- Accuracy = close to true; precision = fine and consistent.
- Significant figures show precision; leading zeros never count.
- Round results to match the least precise input.