Limits of Accuracy
Limits of Accuracy
Every measurement carries a degree of uncertainty determined by the precision of the measuring instrument or the rounding applied. The limits of accuracy are the smallest and largest values that a measurement could plausibly take — its lower bound and upper bound. This benchmark applies error intervals to practical calculations involving combined measurements, where the limits of individual quantities must be combined correctly to find the limits of the result.
Finding Limits for a Single Measurement
This follows directly from error intervals (N15). For a value rounded to a given precision \( u \):
\[ \text{Lower bound} = \text{value} - \frac{u}{2}, \qquad \text{Upper bound} = \text{value} + \frac{u}{2} \]The lower bound is included (\( \leq \)); the upper bound is excluded (\( < \)).
Example: a length of 8 cm measured to the nearest cm has limits \( 7.5 \leq L < 8.5 \text{ cm} \).
Combining Limits
When a calculation involves two or more measured values, you must choose the correct bounds for each to achieve the maximum or minimum result. The rules follow from logical reasoning:
| Calculation | Maximum result | Minimum result | Key idea |
|---|---|---|---|
| \( a + b \) | UB(\(a\)) + UB(\(b\)) | LB(\(a\)) + LB(\(b\)) | Adding more gives more |
| \( a - b \) | UB(\(a\)) \( - \) LB(\(b\)) | LB(\(a\)) \( - \) UB(\(b\)) | Subtracting less gives more |
| \( a \times b \) | UB(\(a\)) \( \times \) UB(\(b\)) | LB(\(a\)) \( \times \) LB(\(b\)) | Multiplying bigger values gives more |
| \( a \div b \) | UB(\(a\)) \( \div \) LB(\(b\)) | LB(\(a\)) \( \div \) UB(\(b\)) | Dividing by less gives more |
The subtraction and division rules are the most error-prone. For \( a - b \): to get the largest possible result, make \( a \) as large as possible and subtract as little as possible (use the smallest \( b \)). For \( a \div b \): divide by as small a denominator as possible to maximise the quotient.
Worked Examples
State the limits of accuracy for: 4.8 m (1 dp), 350 g (nearest 10 g), and 2 hours 15 minutes (nearest 5 minutes).
4.8 m to 1 dp (precision = 0.1 m, half-unit = 0.05): \( 4.75 \leq L < 4.85 \text{ m} \)
350 g to nearest 10 g (precision = 10 g, half-unit = 5): \( 345 \leq m < 355 \text{ g} \)
2h 15 min to nearest 5 min (precision = 5 min, half-unit = 2.5 min): \( 2\text{h}\,12.5\text{min} \leq t < 2\text{h}\,17.5\text{min} \)
Two lengths: \( p = 12 \) cm and \( q = 7 \) cm, each to the nearest cm. Find the maximum and minimum possible values of \( p - q \).
Limits: \( 11.5 \leq p < 12.5 \) and \( 6.5 \leq q < 7.5 \).
Maximum of \( p - q \): use UB(\(p\)) \( - \) LB(\(q\)) = \( 12.5 - 6.5 = 6 ext{ cm} \)
Minimum of \( p - q \): use LB(\(p\)) \( - \) UB(\(q\)) = \( 11.5 - 7.5 = 4 ext{ cm} \)
A rectangle has length \( l = 9 \) cm and width \( w = 5 \) cm, each to the nearest cm. Find the maximum and minimum possible area.
Limits: \( 8.5 \leq l < 9.5 \) and \( 4.5 \leq w < 5.5 \).
Maximum area: \( 9.5 \times 5.5 = 52.25 \text{ cm}^2 \)
Minimum area: \( 8.5 \times 4.5 = 38.25 \text{ cm}^2 \)
The actual area from the given measurements would be \( 9 \times 5 = 45 \text{ cm}^2 \), but the true area could lie anywhere in the range \( 38.25 \leq A < 52.25 \text{ cm}^2 \).
Key Takeaways
- The limits of accuracy for a rounded measurement are: lower bound = value \( - frac{u}{2} \), upper bound = value \( + frac{u}{2} \), where \( u \) is the unit of precision.
- To maximise \( a + b \) or \( a \times b \): use the upper bounds of both. To minimise: use the lower bounds.
- To maximise \( a - b \): use UB(\(a\)) and LB(\(b\)). To minimise: use LB(\(a\)) and UB(\(b\)).
- To maximise \( a \div b \): use UB(\(a\)) and LB(\(b\)). To minimise: use LB(\(a\)) and UB(\(b\)).
- Always clearly state which bound (upper or lower) a calculated limit represents.