Upper & Lower Bounds

To find the bounds of a rounded number, first identify the unit of accuracy (e.g., if rounded to the nearest 10, the unit is 10).

Calculate the half-unit by dividing the unit of accuracy by 2 (e.g., $10 \div 2 = 5$).

The Lower Bound is found by subtracting this half-unit from the rounded value: $LB = \text{Value} - \frac{\text{Unit}}{2}$.

The Upper Bound is found by adding this half-unit to the rounded value: $UB = \text{Value} + \frac{\text{Unit}}{2}$.

Addition: To find the maximum sum, add the upper bounds ($UB_1 + UB_2$); for the minimum sum, add the lower bounds ($LB_1 + LB_2$).

Multiplication: The maximum product is found using $UB_1 \times UB_2$, while the minimum product uses $LB_1 \times LB_2$.

Subtraction: To find the maximum difference, use the largest possible starting value minus the smallest possible subtracted value ($UB_1 - LB_2$).

Division: To find the maximum quotient, divide the largest possible numerator by the smallest possible denominator ($\frac{UB_1}{LB_2}$).

Rounding vs. Truncation: Rounding moves a number to the nearest unit, whereas truncation simply cuts off digits. For a truncated number, the $LB$ is the number itself, and the $UB$ is the number plus one full unit of accuracy.

Inclusive vs. Exclusive: In an error interval $LB \le x < UB$, the $LB$ is inclusive because a value exactly at the $LB$ would round up to the given value. The $UB$ is exclusive because a value exactly at the $UB$ would round up to the next unit.

Discrete vs. Continuous: For continuous data (like height), bounds are calculated using the half-unit rule. For discrete data (like number of people), bounds must represent realistic whole-number limits.