Error Bounds

When we measure or round a number, we introduce uncertainty. Error bounds give us a precise way to describe that uncertainty by stating the smallest and largest values a measurement or rounded number could actually represent. Rather than pretending our answer is exact, we acknowledge a range — called the error interval — within which the true value must lie. The concept of limits of accuracy is central to this idea. If a length is measured as 8 cm to the nearest centimetre, the true length could be anywhere from 7.5 cm up to (but not including) 8.5 cm. The lower bound is 7.5 cm and the upper bound is 8.5 cm. This half-unit on either side of the rounded value is called the maximum absolute error, and it determines how confident we can be in any calculation using that measurement. Error bounds become especially important when measurements are combined — for example, when calculating area from a length and a width, each with its own uncertainty. Errors can compound, so understanding upper and lower bounds for each quantity allows us to find the most extreme possible values for any derived result. This skill is tested heavily in GCSE and A-level mathematics and is foundational to scientific measurement and engineering tolerances.

Key takeaways
  • Identify the degree of rounding
  • Calculate the maximum absolute error
  • Find the lower and upper bounds
  • Write the error interval
  • Combining bounds in calculations

Identify the degree of rounding

Before finding bounds, determine how the value has been rounded. For example, '15.3 rounded to 1 decimal place' means the rounding unit is 0.1. '200 rounded to the nearest 10' means the rounding unit is 10. The maximum absolute error is always half the rounding unit.

Calculate the maximum absolute error

Maximum absolute error = (rounding unit) ÷ 2. If a value is given to the nearest whole number (rounding unit = 1), the maximum absolute error = 0.5. If rounded to the nearest 10 (rounding unit = 10), maximum absolute error = 5.

Find the lower and upper bounds

Lower bound = stated value − maximum absolute error. Upper bound = stated value + maximum absolute error. The error interval is then written as: Lower bound ≤ x < Upper bound. Note the strict inequality at the upper bound — if the true value were exactly at the upper bound, it would have rounded up to the next value.

Write the error interval

Express the error interval using inequality notation. For example, if x = 6.4 (to 1 d.p.), the error interval is 6.35 ≤ x < 6.45. This notation makes clear that x could equal 6.35 but cannot equal 6.45, since 6.45 would round to 6.5.

Combining bounds in calculations

When adding two quantities, use: max sum = upper bound A + upper bound B; min sum = lower bound A + lower bound B. When multiplying: max product = upper A × upper B; min product = lower A × lower B. When dividing A ÷ B: maximum = upper A ÷ lower B; minimum = lower A ÷ upper B. Be careful: dividing by a smaller number gives a larger result, which is why the bounds 'swap' in division.

The Key Equation

Lower Bound≤x<Upper Bound,where LB=a−u2,UB=a+u2\text{Lower Bound} \leq x < \text{Upper Bound}, \quad \text{where } \text{LB} = a - \frac{u}{2}, \quad \text{UB} = a + \frac{u}{2}
  • x — True value (varies): The actual, unknown value of the quantity being measured or rounded
  • a — Stated (rounded) value (varies): The value as reported after rounding or measuring to a given degree of accuracy
  • u — Rounding unit (varies): The precision of the rounding, e.g. 1 for nearest whole number, 0.1 for 1 decimal place, 10 for nearest ten
  • u/2 — Maximum absolute error (varies): The greatest possible difference between the true value and the stated value; half of the rounding unit

Worked Example

Problem. A rectangle has length L = 12 cm (measured to the nearest cm) and width W = 7 cm (measured to the nearest cm). Find the upper and lower bounds for the area of the rectangle, and state the error interval for the area.

74.75≤A<93.75 cm274.75 \leq A < 93.75 \text{ cm}^2

Answer. Lower bound of area = 74.75 cm²; Upper bound of area = 93.75 cm²

Figure
Diagram
Error Interval Number Line: x = 8 (nearest whole number)678910LB = 7.5UB = 8.5x = 8 (stated)● included○ excludedError Interval: 7.5 ≤ x < 8.5Maximum absolute error = 0.5 (half of rounding unit 1)Solid line = error interval range ● = lower bound included ○ = upper bound excluded
Origins · 1 minCarl Friedrich Gauss and how this was discovered
CF
Carl Friedrich Gauss
1777–1855 · German

Gauss developed the method of least squares and made foundational contributions to the theory of measurement error. His work on the Gaussian (normal) distribution provided the mathematical framework for understanding how measurement errors are distributed. His rigorous treatment of observational error in astronomy and geodesy showed how to find the best estimate of a true value from multiple imprecise measurements, directly informing the study of error bounds and uncertainty quantification. At age 12, Gauss allegedly found the formula for the sum of an arithmetic series in seconds when his teacher asked the class to add all integers from 1 to 100. He noticed the 50 pairs each summing to 101, giving 5050 instantly — a legendary demonstration of mathematical insight.

  • Significant Figures — Rounding to significant figures determines the rounding unit, which directly sets the maximum absolute error and the width of the error interval.
  • Estimation Techniques — Estimation uses approximated values; understanding error bounds ensures we know how far our estimates might deviate from the true answer.
  • Inequalities — Error intervals are expressed using inequality notation (≤ and <), so fluency with inequalities is essential for reading and writing bounds correctly.
  • Measurement and Accuracy — Error bounds are a direct consequence of measurement precision; understanding accuracy in data collection underpins the whole concept of limits of accuracy.