No measurement is perfect
Every measurement has some measurement error. Instruments have limits, people make small reading mistakes, and conditions change. Understanding error helps you decide whether a result is good enough.
Greatest possible error
If a length is measured to the nearest centimetre as 24 cm, the true length could be anywhere from 23.5 cm to 24.5 cm.
Greatest possible error = ½ × smallest unit
| Measured to the nearest | Greatest possible error |
|---|---|
| 1 cm | 0.5 cm |
| 0.1 cm | 0.05 cm |
| 1 g | 0.5 g |
| 0.01 s | 0.005 s |
Upper and lower bounds
The true value lies between the upper and lower bounds.
24 cm (nearest cm) → lower bound 23.5 cm, upper bound 24.5 cm.
Bounds in calculations: A rectangle measures 8 cm × 5 cm, each to the nearest cm.
- Smallest possible area: 7.5 × 4.5 = 33.75 cm²
- Largest possible area: 8.5 × 5.5 = 46.75 cm²
Small errors in lengths can create big uncertainty in areas and volumes!
Absolute and relative error
- Absolute error = |measured − true|
- Relative error = absolute error ÷ true value (× 100% for percent error)
Example: A 2.00 m board is measured as 2.03 m.
- Absolute error: 0.03 m
- Relative error: 0.03 ÷ 2.00 = 1.5%
A 3 mm error matters more on a 10 mm part than on a 2 m board, which is why relative error is often more useful.
Tolerance
In manufacturing, a part is specified with a tolerance: the allowed variation.
Example: A bolt length of 50 mm ± 0.2 mm is acceptable from 49.8 mm to 50.2 mm.
| Measured | Acceptable? |
|---|---|
| 49.9 mm | ✓ |
| 50.2 mm | ✓ |
| 50.25 mm | ✗ |
Tighter tolerances (± 0.01 mm) cost more to produce but are needed in engines, medical devices and aircraft.
Combining tolerances
If two parts of 30 ± 0.1 mm and 20 ± 0.1 mm are joined, the total is 50 mm ± 0.2 mm. Tolerances add up, which is why engineers plan "tolerance stacks" carefully.
Reducing error
- Use a more precise instrument.
- Take several measurements and average them.
- Calibrate instruments.
- Read scales at eye level.
Worked examples
Example 1: A mass is 3.7 kg to the nearest 0.1 kg. Bounds? 3.65 kg to 3.75 kg.
Example 2: A pipe should be 120 cm ± 0.5 cm. Is 119.4 cm acceptable? Lower limit 119.5 → no.
Example 3: A 500 mL measurement is off by 8 mL. Relative error? 8/500 = 1.6%.
Common mistakes
- Using the whole unit instead of half as the greatest possible error.
- Using measured instead of true values in relative error.
- Forgetting that errors compound in area and volume.
Try it yourself
- Greatest possible error for 6.4 m (nearest 0.1 m)?
- Bounds for 85 kg (nearest kg)?
- A part is 12 mm ± 0.05 mm. Range?
- A 25.0 cm length is measured as 24.6 cm. Relative error?
Answers: 1) 0.05 m 2) 84.5 to 85.5 kg 3) 11.95 to 12.05 mm 4) 1.6%
Quick recap
- Every measurement has measurement error; the greatest possible error is half the unit.
- Absolute error vs relative error.
- Tolerance sets acceptable upper and lower bounds.