Significant figures: the digits we can trust
A measured number shows how good the instrument is. Significant figures = all the sure digits + the first uncertain digit.
Rules to count them
- All non-zero digits count. 2.47 has 3.
- Zeros between non-zero digits count. 1.05 has 3.
- Zeros at the start (before the first non-zero digit) never count. 0.0052 has 2.
- Zeros at the end count only if there is a decimal point. 2.300 has 4. 2300 is unclear; write 2.3 × 10³ (2 s.f.) or 2.300 × 10³ (4 s.f.).
- In scientific notation a × 10ᵇ, count only the digits in a. The power of ten does not count.
- Changing units does not change the count: 2.30 m = 230 cm = 0.00230 km, all 3 s.f.
- Exact numbers (like 2 in 2πr, or counted objects) have unlimited significant figures.
Rounding off and arithmetic with significant figures
Rounding off
- First dropped digit more than 5: add 1 to the last kept digit. 1.746 → 1.75.
- Less than 5: keep it the same. 1.743 → 1.74.
- Exactly 5 (nothing after it): if the last kept digit is even, leave it; if odd, add 1. 2.745 → 2.74, 2.735 → 2.74.
Multiplying and dividing
Keep as many significant figures as the number with the fewest significant figures. 4.237 g ÷ 2.51 cm³ = 1.688… → 1.69 g cm⁻³ (3 s.f.).
Adding and subtracting
Keep as many decimal places as the number with the fewest decimal places. 436.32 + 227.2 + 0.301 = 663.821 → 663.8.
Round only the final answer, not every middle step.
Uncertainty: kinds of errors
Error = measured value − true value. It is not a "mistake"; it is the doubt that every reading has.
Systematic errors
They push every reading the same way (all too big or all too small). Causes: a faulty instrument (zero error in a screw gauge), a bad method (ignoring air friction), or a personal habit (always looking at the scale from one side, called parallax). Fix: find the cause and correct it.
Random errors
They go up and down without a pattern, e.g. from small changes in temperature or your reaction time. Fix: take many readings and use the mean.
Least count error
The smallest value an instrument can read is its least count: 1 mm for a metre scale, 0.01 cm for a vernier, 0.001 cm for a screw gauge. Every reading may be off by up to one least count. Fix: use a finer instrument.
Try it at home
Use a phone stopwatch to time 10 swings of a key tied on a string, five times. Write the five times, find the mean and the mean absolute error. Then time 20 swings: see how the percentage error drops.
Absolute, relative and percentage error
Suppose readings are a₁, a₂, … aₙ.
- Mean (best value): amean = (a₁ + a₂ + … + aₙ) ÷ n
- Absolute error of each reading: Δaᵢ = |amean − aᵢ|
- Mean absolute error: Δamean = (|Δa₁| + … + |Δaₙ|) ÷ n
- Result is written as a = amean ± Δamean
- Relative error = Δamean ÷ amean
- Percentage error = (Δamean ÷ amean) × 100%
Combination of errors
| Operation | Error rule (maximum error) |
|---|---|
| Z = A + B or Z = A − B | ΔZ = ΔA + ΔB (absolute errors always add, even when subtracting) |
| Z = A × B or Z = A ÷ B | ΔZ/Z = ΔA/A + ΔB/B (relative errors add) |
| Z = Aⁿ | ΔZ/Z = n (ΔA/A) |
| Z = Aᵖ Bᑫ ÷ Cʳ | ΔZ/Z = p ΔA/A + q ΔB/B + r ΔC/C |
The quantity with the highest power adds the most error, so measure it most carefully.
Key formulas and definitions
- a_mean = (a₁ + a₂ + … + aₙ)/n
- Δaᵢ = |a_mean − aᵢ|; Δa_mean = Σ|Δaᵢ|/n
- Relative error = Δa/a; % error = (Δa/a) × 100
- Z = A ± B → ΔZ = ΔA + ΔB
- Z = A × B or A/B → ΔZ/Z = ΔA/A + ΔB/B
- Z = Aᵖ Bᑫ/Cʳ → ΔZ/Z = pΔA/A + qΔB/B + rΔC/C
Worked examples
1. How many significant figures are in (a) 0.007020 (b) 6.320 × 10⁴ (c) 5000?
(a) Leading zeros do not count; 7, 0, 2, 0 count → 4. (b) 6, 3, 2, 0 → 4. (c) No decimal point, so trailing zeros are unclear; normally 1 (write 5.000 × 10³ if all are measured).
2. Round 2.5463 to 3 s.f. and 7.850 to 2 s.f.
2.5463 → first dropped digit is 6 > 5 → 2.55. 7.850 → dropped part is exactly 50, kept digit 8 is even → 7.8.
3. A block has mass 5.74 g and volume 1.2 cm³. Find its density with correct significant figures.
Density = 5.74 ÷ 1.2 = 4.783… The fewest s.f. is 2 (in 1.2), so density = 4.8 g cm⁻³.
4. Five readings of a pendulum's period are 2.63, 2.56, 2.42, 2.71 and 2.80 s. Find the mean, mean absolute error, relative and percentage error.
Mean = 13.12 ÷ 5 = 2.624 ≈ 2.62 s. Errors: 0.01, 0.06, 0.20, 0.09, 0.18 → sum 0.54 → mean absolute error = 0.108 ≈ 0.11 s. Relative error = 0.11 ÷ 2.62 = 0.04. Percentage error = 4%. Result: 2.6 ± 0.1 s.
5. Two resistors are 100 ± 3 Ω and 200 ± 4 Ω. Find the total in series.
R = 100 + 200 = 300 Ω. ΔR = 3 + 4 = 7 Ω. So R = 300 ± 7 Ω.
6. Temperatures are 20.0 ± 0.5 °C and 50.0 ± 0.5 °C. Find the rise in temperature.
Δθ = 50.0 − 20.0 = 30.0 °C. Errors add even for subtraction: 0.5 + 0.5 = 1.0 °C. Answer: 30.0 ± 1.0 °C.
7. Length = 5.0 ± 0.1 cm, breadth = 4.0 ± 0.1 cm. Find the area with its error.
A = 20.0 cm². % errors: 0.1/5 × 100 = 2%, 0.1/4 × 100 = 2.5%. Total = 4.5%. ΔA = 4.5% of 20 = 0.9 cm². A = 20.0 ± 0.9 cm².
8. Z = A²B³/√C. The percentage errors in A, B and C are 1%, 2% and 4%. Find the percentage error in Z.
% error in Z = 2(1%) + 3(2%) + ½(4%) = 2 + 6 + 2 = 10%.
Common mistakes
- Counting the zeros in 0.0045 as significant. Leading zeros never count.
- Subtracting errors when subtracting quantities. Absolute errors always add.
- Rounding at every middle step. Round only the final answer.
- Giving more digits than the data allow, like 4.783333 g cm⁻³ from 1.2 cm³.