Measures and accuracy: apply and interpret limits of accuracy

The idea

Limits of accuracy describe the range of values that round to a given stated value. To apply them, identify the degree of rounding (nearest whole number, 10, 0.1, 0.01, etc.), then add and subtract half of that unit to form the upper and lower limits. For example, a length stated as 7 cm to the nearest centimetre has a lower limit of 6.5 cm and an upper limit of 7.5 cm, because any value from 6.5 cm up to but not including 7.5 cm rounds to 7 cm. To interpret limits, read a stated rounded value and explain what the true value could be, or decide whether a claim is possible. Foundation tier applies this to simple rounded values; Higher tier extends it to upper and lower bounds in calculations. Always keep the unit and check whether the boundary is included.

What earns the marks

• Identify the unit of rounding from the stated value, such as nearest 10, nearest whole number, nearest 0.1 or nearest 0.01.
• Calculate half of that unit and add it to, and subtract it from, the stated value to obtain the upper and lower limits.
• State the lower limit as the smallest value that would round up to the stated value, and the upper limit as the boundary that would round to the next value up.
• Interpret a rounded value by describing the interval of possible true values, for example a mass of 40 kg to the nearest 10 kg lies in the interval 35 kg ≤ mass < 45 kg.
• Use limits to test a claim, such as whether a total or difference can reach a particular value, by considering the extreme cases.

Common mistakes and how to put them right

• Adding and subtracting the full unit instead of half the unit.

  Put it right: For a value rounded to the nearest 10, the limits are 5 either side, not 10 either side.

• Treating the upper limit as included when it is not.

  Put it right: For a value rounded to the nearest whole number, the upper boundary belongs to the next value up, so use strict inequality at the top.

• Forgetting to keep the original unit or converting incorrectly.

  Put it right: If a length is in metres, the limits must also be in metres unless a correct conversion is made.

• Assuming a rounded value is exact.

  Put it right: A stated value of 7 cm to the nearest centimetre represents a range of possible true lengths, not just 7 cm.

Exam tips

• Underline the degree of rounding in the question before calculating, so you know whether to halve 1, 10, 0.1 or another unit.
• Write the interval using inequality notation with the lower limit included and the upper limit excluded, for example 6.5 ≤ x < 7.5.
• Check your limits by testing a value just inside each boundary: it should round to the stated value, while a value just outside should not.
• In multiple-choice questions, eliminate options that use the full unit rather than half the unit, or that include the upper boundary incorrectly.