Bounds & Accuracy
Year 9 🔢 Number Upper and lower bounds, error intervals.
📏 Upper and Lower Bounds
Every rounded measurement has a range of possible true values. The boundaries are called upper and lower bounds.
⚡ Bounds Formula
$$\text{Lower bound} = \text{value} - \frac{1}{2} \times \text{unit of rounding}$$$$\text{Upper bound} = \text{value} + \frac{1}{2} \times \text{unit of rounding}$$Example: Length = 8.3 cm (to 1 d.p.). Unit = 0.1, so half-unit = 0.05. LB = 8.25 cm, UB = 8.35 cm. Written: $8.25 \leq l < 8.35$
🔢 Calculations with Bounds
When combining measurements, choose bounds carefully to get the maximum or minimum possible result.
| Operation | Maximum result uses | Minimum result uses |
|---|---|---|
| $A + B$ | UB($A$) + UB($B$) | LB($A$) + LB($B$) |
| $A - B$ | UB($A$) - LB($B$) | LB($A$) - UB($B$) |
| $A \times B$ | UB($A$) × UB($B$) | LB($A$) × LB($B$) |
| $A \div B$ | UB($A$) ÷ LB($B$) | LB($A$) ÷ UB($B$) |
💡 Truncation vs Rounding
Truncation removes digits without rounding up, shifting the bounds.
Truncation example: 7.89 truncated to 1 d.p. = 7.8 (not 7.9). The true value $x$ satisfies $7.8 \leq x < 7.9$.
For a truncated value, the lower bound equals the truncated value and the upper bound is truncated value + unit of truncation.
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Interactive Demonstration — Bounds & Accuracy
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Interactive demonstration available in the Calculator tab!
📏 Bounds Calculator