An error interval gives the range of possible true values of a rounded number, using ≤ for the lower bound and < for the upper bound. If x is 8 to the nearest whole number, then 7.5 ≤ x < 8.5. For a truncated value, the interval starts at the value itself and goes up by one whole unit, because truncation never rounds up.
Error intervals
- Rounded: (value − ½ unit) ≤ x < (value + ½ unit)
- Truncated: value ≤ x < (value + 1 unit)
- Unit to 1 decimal place: 0.1; to 2 decimal places: 0.01
- Use ≤ at the lower end and < at the upper end
Error intervals for rounded values
When a value has been rounded, the true value lies within half a unit either side. If x is 8 to the nearest whole number, it could be anything from 7.5 up to, but not including, 8.5: 7.5 ≤ x < 8.5. The upper bound uses < because 8.5 itself would round up to 9.
Decimal places
Find the rounding unit first. To 1 decimal place the unit is 0.1, so half a unit is 0.05. A value of 5.6 to 1 decimal place gives 5.55 ≤ y < 5.65. To 2 decimal places the unit is 0.01, so half a unit is 0.005.
Significant figures
The unit depends on the place value of the last significant figure. 2400 to 2 significant figures is rounded to the nearest 100, so the interval is 2350 ≤ x < 2450. 0.36 to 2 significant figures is rounded to the nearest 0.01, giving 0.355 ≤ x < 0.365.
Error intervals for truncated values
Truncation cuts off digits and never rounds up, so the true value is at least the truncated value and less than one unit above it. If x is 3.9 truncated to 1 decimal place, the interval is 3.9 ≤ x < 4.0. This is different from rounding, where the interval is centred on the value.
Writing the interval correctly
Write the interval with x in the middle, the lower bound on the left with ≤, and the upper bound on the right with <. Include units if the question involves a measurement.
Worked examples
Example 1. A number x is 8, to the nearest whole number. Write down the error interval for x.
- The unit is 1, so half a unit is 0.5.
- Lower bound 7.5; upper bound 8.5.
Answer: 7.5 ≤ x < 8.5.
Example 2. A value y is 5.6, to 1 decimal place. Write down the error interval for y.
- The unit is 0.1, so half a unit is 0.05.
Answer: 5.55 ≤ y < 5.65.
Example 3. A crowd is 2400, to 2 significant figures. Write down the error interval for the number of people, n.
- 2 significant figures here means to the nearest 100, so half a unit is 50.
Answer: 2350 ≤ n < 2450.
Example 4. x = 3.9, truncated to 1 decimal place. Write down the error interval for x.
- Truncation never rounds up, so x is at least 3.9.
- x is less than one unit (0.1) above it.
Answer: 3.9 ≤ x < 4.0.
Example 5. A length L is 0.36 m, to 2 significant figures. Write down the error interval for L.
- Here 2 significant figures means to the nearest 0.01 m, so half a unit is 0.005.
Answer: 0.355 ≤ L < 0.365.
Common mistakes
- Using ≤ at both ends. The upper bound is not included, so use <.
- Using a whole unit for rounding. Rounded values go half a unit either side.
- Treating truncation like rounding. A truncated value is the lower bound itself.
- Getting the unit wrong for significant figures. Find the place value of the last significant figure.
- Writing the interval backwards. Lower bound on the left, upper bound on the right.
How error intervals are examined
Error intervals are tested on Higher papers for both the 1MA1 GCSE and the 4MA1 International GCSE, and simple versions appear at Foundation tier too. Questions usually:
- ask for the error interval of a rounded value
- use decimal places or significant figures
- ask for the error interval of a truncated value
- ask you to compare rounding and truncation
Practise with the error intervals and bounds worksheet, or use the GCSE number revision engine.
Error intervals lead into calculations with bounds, and they rely on significant figures and truncation.
Frequently asked questions
What is an error interval?
The range of values a number could have taken before it was rounded or truncated, written like 7.5 ≤ x < 8.5.
Why is the upper bound written with <?
Because the upper bound itself would round up to the next value, so it is not included.
How is the error interval different for truncation?
For truncation, the interval starts at the given value and goes up by one whole unit.