Error Intervals Quizzes
Visual overview of Error Intervals.
Introduction
Error intervals (or bounds) describe the range of true values that a rounded or measured number could take. They are vital for GCSE questions on measurement, estimation, and real data: they let you report answers accurately, track uncertainty through calculations, and judge whether results are plausible in science, engineering, or everyday contexts.
Example: “12 cm to the nearest cm” means the true length is at least 11.5 cm but less than 12.5 cm. Using bounds makes that uncertainty explicit.
Core Concepts
Lower & Upper Bounds
If a value is rounded to a unit of size \(u\) (e.g., 1, 0.1, 10), then:
- Lower bound \(= \text{rounded value} - \tfrac{u}{2}\)
- Upper bound \(= \text{rounded value} + \tfrac{u}{2}\) (upper bound is usually not included)
Example
12 cm to the nearest cm → \(11.5 \le L < 12.5\).
Common Rounding Units
- Nearest 10 → half unit = 5
- Nearest 1 → half unit = 0.5
- 1 d.p. → half unit = 0.05
- 2 d.p. → half unit = 0.005
Example
7.24 to 1 d.p. → reported as 7.2. Bounds: \(7.15 \le x < 7.25\). (If 7.24 is already exact and then rounded to 1 d.p. as 7.2 in a question, use \(7.15 \le x < 7.25\).)
Interval Notation
We can write error intervals using inequalities or brackets:
- Inequalities: \(a \le x < b\)
- Bracket form: \([a, b)\) means include \(a\) but exclude \(b\).
Error Intervals in Calculations (Propagation)
Use the extreme values to find bounds for results. Unless stated otherwise, GCSE problems typically involve positive quantities; if negatives are possible, consider sign carefully.
Addition / Subtraction
- Lower bound = sum/difference of lower bounds
- Upper bound = sum/difference of upper bounds
Multiplication / Division (positives)
- Lower bound = product/quotient of lower bounds
- Upper bound = product/quotient of upper bounds
Midpoint (“Best Estimate”)
The midpoint is the average of the bounds and is often used as a reasonable single estimate:
\(\text{Midpoint} = \dfrac{\text{Lower} + \text{Upper}}{2}\)
Percentage Error (Maximum)
When a measurement \(M\) is rounded to unit \(u\), the maximum absolute error is \(\tfrac{u}{2}\). A common GCSE measure is the maximum percentage error:
\(\displaystyle \text{Max % error}=\frac{\tfrac{u}{2}}{M}\times 100\%\)
Example
50 cm to the nearest cm → \(u=1\). Max % error \(=\dfrac{0.5}{50}\times100\%=1\%\).
Worked Examples
Example 1 (Foundation): Single rounded value
Measured 15 cm to nearest cm → \(14.5 \le L < 15.5\).
Example 2 (Foundation): Sum of bounds
12 cm and 7 cm, both to nearest cm.
- 12 → \([11.5, 12.5)\)
- 7 → \([6.5, 7.5)\)
- Total bounds: \(11.5+6.5=18\) and \(12.5+7.5=20\) → \(18 \le T \le 20\).
Example 3 (Higher): Area with bounds (multiplication)
Length \(4.2\) cm, Width \(3.5\) cm, both to 1 d.p.
- Length: \(4.15 \le L < 4.25\)
- Width: \(3.45 \le W < 3.55\)
- Area min ≈ \(4.15×3.45=14.32\) cm²; Area max ≈ \(4.25×3.55=15.09\) cm²
- \(14.32 \lesssim A \lesssim 15.09\) cm².
Example 4 (Higher): Division with bounds
Value \(=10.0\pm0.1\) (to 1 d.p.), divisor \(=2.0\pm0.05\) (to 1 d.p.).
- Numerator: \(9.9 \le N < 10.1\)
- Denominator: \(1.95 \le D < 2.05\)
- Min quotient ≈ \(9.9 ÷ 2.05 \approx 4.83\)
- Max quotient ≈ \(10.1 ÷ 1.95 \approx 5.18\)
- \(4.83 \lesssim Q \lesssim 5.18\).
Example 5 (Higher): Midpoint
Bounds \(49 \le x \le 51\) → midpoint \(= \dfrac{49+51}{2}=50\).
Example 6 (Higher): Max % error from rounding
60 (to nearest 2) → \(u=2\). Max % error \(=\dfrac{1}{60}\times100\% \approx 1.67\%\).
Example 7 (Real life): Board area
Length \(120\) cm ± 0.5 cm, Width \(60\) cm ± 0.5 cm.
- Length: \(119.5 \le L < 120.5\)
- Width: \(59.5 \le W < 60.5\)
- Min area ≈ \(119.5×59.5=7102.25\) cm²
- Max area ≈ \(120.5×60.5=7290.25\) cm²
- \(7102 \lesssim A \lesssim 7290\) cm² (to nearest cm²).
Common Mistakes
- Using a closed upper bound (writing \( \le \) instead of \( < \)).
- Forgetting the half unit (e.g., 1 d.p. → ±0.05).
- Mixing exact rounded values with bounds in the same calculation.
- Assuming multiplication/division bounds without checking signs.
- Confusing midpoint with the actual value.
Applications
- Science: reporting measurement uncertainty.
- Engineering/Construction: tolerances so parts fit safely.
- Finance/Data: rounding effects in totals and averages.
- Exams: bounds questions in geometry and mensuration.
Strategies & Tips
- Underline the rounding instruction and write \(u\) and \(\tfrac{u}{2}\) first.
- Use inequality form \(L \le x < U\) for clarity and method marks.
- When combining values, compute with the extreme ends to get result bounds.
- State units at every step.
- Quote a midpoint only if the question asks for a single “best estimate.”
Summary / Call-to-Action
Error intervals turn rounded measurements into precise ranges. By constructing bounds correctly and propagating them through calculations, you’ll report answers accurately and reason about uncertainty like a scientist or engineer.
- Practise converting rounded statements to intervals.
- Find bounds of sums, differences, products, and quotients.
- Use maximum percentage error to compare accuracy.