Limits of accuracy Revision Notes

When a number or measurement is rounded to a certain degree of accuracy, the exact original value is not known but lies within a range. This range is defined by the upper and lower limits (also called bounds).

The lower limit is the smallest possible value that could have been rounded to the given number, and the upper limit is the largest possible value that could have been rounded to it.

For example, if a length is recorded as 5 metres rounded to the nearest metre, the actual length could be slightly less or more than 5 metres but not enough to round to 4 or 6 metres. The limits define this range.

• Upper limit: maximum possible value before rounding up to the next number.
• Lower limit: minimum possible value before rounding down to the previous number.

The size of the limits depends on the rounding unit (the place value to which the number is rounded). The general rule is:

The upper limit is the rounded value plus half the rounding unit.

The lower limit is the rounded value minus half the rounding unit.

• If rounding to the nearest 1 (unit), half the rounding unit is 0.5.
• If rounding to the nearest 10, half the rounding unit is 5.
• If rounding to the nearest 0.1, half the rounding unit is 0.05.
• Therefore, limits = rounded value ± (½ × rounding unit).

Example 1: A length is recorded as 12 m, rounded to the nearest metre. Find the upper and lower limits.

Step 1: Identify rounding unit = 1 m. Half unit = 0.5 m.

Step 2: Calculate lower limit = 12 − 0.5 = 11.5 m (inclusive). Upper limit = 12 + 0.5 = 12.5 m (exclusive). The actual length is ≥ 11.5 m and < 12.5 m.

• Example 2: A mass is recorded as 250 g, rounded to the nearest 10 g.
• Step 1: Rounding unit = 10 g. Half unit = 5 g.
• Step 2: Lower limit = 250 − 5 = 245 g, upper limit = 250 + 5 = 255 g.
• So the actual mass is ≥ 245 g and < 255 g.

Knowing the limits helps understand the possible range of the true value behind a rounded figure. This is important when combining measurements or calculating with rounded data, as it shows the potential variation or uncertainty.

For example, if a length is 5 m rounded to the nearest metre, the actual length could be as low as 4.5 m or as high as just under 5.5 m. This range must be considered when using this measurement in further calculations or comparisons.

Note that limits are always inclusive of the lower limit and exclusive of the upper limit, meaning the actual value can be equal to the lower limit but less than the upper limit.

• Limits provide a range of possible values, not a single exact value.
• They help estimate the accuracy and precision of measurements.
• Limits are essential when working with rounded data in calculations.

Here are some common rounding units and the corresponding half units used to find limits:

• Nearest 1 (unit): half unit = 0.5
• Nearest 10: half unit = 5
• Nearest 100: half unit = 50
• Nearest 0.1: half unit = 0.05
• Nearest 0.01: half unit = 0.005
• To find limits, add and subtract the half unit from the rounded value.