Lower Bound Calculator
Find the lower and upper bounds when a value is rounded to a given precision.
What is a lower bound?
A lower bound is the smallest possible original value that could round to a stated number. If a measurement is rounded, the true value is not exact—it lies in a range. The lower bound is the bottom of that range.
For example, if a length is recorded as 12 cm to the nearest 1 cm, the true value could be anywhere from 11.5 cm up to (but not including) 12.5 cm. So the lower bound is 11.5 cm.
Core formula
If a value R is rounded to the nearest u, then:
- Lower bound = R − (u / 2)
- Upper bound = R + (u / 2)
- Interval = Lower bound ≤ true value < Upper bound
The lower bound is inclusive, while the upper bound is usually exclusive in standard interval notation.
How to use this calculator
Step 1: Enter the rounded value
Type the reported or rounded number (for example: 7, 12.4, 350).
Step 2: Enter what it was rounded to
Choose the nearest unit (such as 1, 0.1, 10, or a custom value).
Step 3: Calculate
Click Calculate Bounds to get the lower bound, upper bound, interval form, and maximum absolute error.
Worked examples
Example 1: Rounded to nearest 10
Reported value: 90, nearest unit: 10
Lower bound = 90 − 5 = 85
Upper bound = 90 + 5 = 95
So: 85 ≤ true value < 95
Example 2: Rounded to nearest 0.1
Reported value: 4.2, nearest unit: 0.1
Lower bound = 4.2 − 0.05 = 4.15
Upper bound = 4.2 + 0.05 = 4.25
So: 4.15 ≤ true value < 4.25
Where lower bounds are useful
- Science: understanding uncertainty in measurements.
- Finance: estimating conservative outcomes from rounded reports.
- Manufacturing: tolerance and quality control decisions.
- Education: exam problems involving bounds and error intervals.
Common mistakes to avoid
- Using the full rounding unit instead of half when computing bounds.
- Forgetting that upper bound is typically exclusive.
- Mixing decimal place rounding with significant figure rounding rules.
- Using negative or zero rounding units.
Quick takeaway
Lower bound calculations are simple but powerful: subtract half of the rounding unit from the rounded value. Use the full interval whenever precision matters.