Every measured value has limits. A ruler, stopwatch, scale or rounded report can only describe a quantity to some degree of precision. Error intervals make that uncertainty explicit instead of pretending the recorded number is exact.
This guide connects error intervals to rounding, bounds, absolute error and percentage error. It also shows how uncertainty propagates through later calculations.
What an error interval means
If a measurement is recorded as 8.4 cm to the nearest 0.1 cm, the true value lies between 8.35 cm and 8.45 cm.
8.35 ≤ x < 8.45The interval communicates what the measurement could actually represent.
Absolute error
Absolute error is the size of the difference between an approximation and the exact or accepted value.
absolute error = |approximate value − exact value|If a true length is 12.38 cm and it is recorded as 12.4 cm, the absolute error is 0.02 cm.
Maximum possible error from rounding
If a value is rounded to the nearest 0.1 unit, the maximum possible rounding error is 0.05 unit. This is half the rounding interval.
If a value is rounded to the nearest 10 units, the maximum possible rounding error is 5 units.
Percentage error
percentage error = absolute error / exact value × 100%Suppose 49 cm is used to approximate a true length of 50 cm.
absolute error = 1 cm
percentage error = 1/50 × 100% = 2%Percentage error tells us the error relative to the size of the quantity.
The same absolute error can matter differently
An error of 1 cm is small compared with a 100 m length but large compared with a 2 cm component. Relative or percentage error captures this difference.
Measurement resolution and uncertainty
An instrument does not measure infinitely finely. A ruler marked in millimetres gives more resolution than one marked only in centimetres. A digital display may show many digits, but those digits are only meaningful if the instrument is calibrated and capable of resolving them.
Precision of display and accuracy of measurement are related but not identical ideas.
Uncertainty propagates through calculations
If both length and width are measured approximately, a calculated area inherits uncertainty from both measurements.
For positive quantities, minimum area comes from lower-bound length × lower-bound width, while maximum area comes from upper-bound length × upper-bound width.
For quotients, the pattern is different. A maximum positive quotient uses the largest possible numerator and smallest possible denominator.
Worked example: density
A sample has mass 50 g to the nearest gram and volume 20 cm³ to the nearest cubic centimetre.
49.5 ≤ m < 50.5
19.5 ≤ V < 20.5For density = mass ÷ volume, the maximum possible density uses upper mass and lower volume. The minimum uses lower mass and upper volume.
This is a direct application of bounds to measurement uncertainty.
Systematic and random uncertainty
Repeated measurements can vary because of random effects. A miscalibrated instrument can shift every measurement in the same direction, producing systematic error.
Secondary Mathematics problems usually focus on numerical bounds rather than a full experimental treatment, but the distinction helps explain why repeated measurement and calibration matter in science.
Common errors
Treating rounded values as exact. Recover the interval first.
Confusing absolute and percentage error. One is a difference in the original unit; the other is a relative percentage.
Assuming more displayed decimals always mean better accuracy. Instrument capability and calibration matter.
Using the same bound pairing for every operation. Think about what makes the expression largest or smallest.
Diagnostic table
| Error | Likely issue | Repair |
|---|---|---|
| States 8.4 cm exactly | Rounding interpreted as exact | Write the error interval |
| Percentage error has cm units | Relative error concept weak | Divide by reference value and multiply by 100% |
| Maximum quotient uses upper denominator | Bounds propagation | Ask what makes the quotient largest |
Practice
- A length is 6.2 m to the nearest 0.1 m. Write its error interval.
- A value 79 is used instead of exact value 80. Find the absolute error and percentage error.
- What is the maximum rounding error when a mass is stated to the nearest 10 g?
- Explain why a calculated result from rounded inputs should not automatically be treated as exact.
Answers
1. 6.15 ≤ x < 6.25. 2. Absolute error 1; percentage error 1.25%. 3. 5 g. 4. The inputs each represent intervals, so the calculation inherits their uncertainty.
Connected routes
Use Upper and Lower Bounds for interval construction and Estimation as a Mathematical Checking System for practical checking. Return to the Mathematics Learning Hub.