Error intervals and bounds
Error intervals
A length of 6.4 cm, rounded to 1 decimal place, could have been anything from 6.35 cm up to (but not including) 6.45 cm.
The range of values it could have been is its error interval. Go half a unit of rounding either way.
6.35 ≤ L < 6.45
The lower bound is included (6.35 rounds up to 6.4). The upper bound is not (6.45 rounds up to 6.5), but it is still called the upper bound.
The same works for any unit of rounding:
- 730 g to the nearest 10 g: half of 10 is 5, so the mass is from 725 g up to 735 g.
- 2.46 m to 2 decimal places: half of 0.01 is 0.005, so the length is from 2.455 m up to 2.465 m.
The lower bound is the least value; the upper bound is the greatest.
If 3.7 came from truncating to 1 decimal place, the number was 3.7 or more and less than 3.8: 3.7 ≤ x < 3.8.
Counting whole things
48,000 people, to 2 significant figures, means between 47,500 and 48,500.
People are whole, so the least is 47,500 and the greatest is 48,499.
Bounds in calculations (Higher)
To make an answer as big as possible, choose each bound to push it up.
- Sum or product: biggest = upper + upper, or upper × upper. Least = lower + lower, or lower × lower.
- Difference: biggest = upper − lower. Least = lower − upper.
- Quotient: biggest = upper ÷ lower. Least = lower ÷ upper.
- speed = distance ÷ time, distance 340 m (nearest 10 m), time 42.6 s (1 d.p.)
- greatest distance 345 m, least time 42.55 s
- upper bound of speed = 345 ÷ 42.55 = 8.108... m/s
For an area, multiply the bounds: a rectangle 4.2 cm by 7.5 cm (each to 1 d.p.) has least area 4.15 × 7.45 = 30.9175 cm².
The perimeter of a shape is the distance all the way round it. For a perimeter, add the bounds of every side.
For a difference such as full mass − empty mass, the least value is least full − greatest empty.
A suitable degree of accuracy: round the upper and lower bounds. Use the most figures for which they agree.
- bounds 8.108 and 7.855
- to 2 significant figures: 8.1 and 7.9, different
- to 1 significant figure: 8 and 8, the same
- so the speed is 8 m/s to a suitable degree of accuracy
Bounds on the calculator
A drone flies 340 m (to the nearest 10 m) in 42.6 s (to 1 decimal place). Work out 345 ÷ 42.55 and 335 ÷ 42.65. Round both to 3 significant figures, then to 2. At which accuracy do they agree?