Measures and accuracy: use inequality notation to specify simple error intervals due to truncation or rounding

The idea

An error interval shows all values that would round or truncate to a given result. For rounding to the nearest unit, a value reported as 8 has interval 7.5 ≤ x < 8.5: the lower bound is included because 7.5 rounds up to 8, and the upper bound is excluded because 8.5 rounds up to 9. For truncation, a value reported as 8 has interval 8 ≤ x < 9, because any value from 8 up to but not including 9 truncates to 8. The same method applies to decimal places and significant figures: find the smallest and largest values that would produce the reported result, then write the interval with correct inequality signs.

What earns the marks

• Identify whether the reported value comes from rounding or truncation.
• For rounding to the nearest unit, use half a unit below and half a unit above the reported value.
• For truncation, use the reported value as the lower bound and the next unit value as the upper bound.
• Use ≤ for the included lower bound and < for the excluded upper bound in a standard rounding interval.
• Apply the same interval method to decimal places and significant figures by finding the relevant unit size.
• Write the interval with the variable, correct inequality signs and consistent units.

Common mistakes and how to put them right

• Using the same interval for rounding and truncation.

  Put it right by remembering that truncation keeps the reported value as the lower bound, while rounding uses half a unit below.

• Including both bounds with ≤ or excluding both with <.

  Put it right by using ≤ for the lower bound and < for the upper bound in a standard rounding interval.

• Forgetting to change the unit size when the accuracy is in decimal places or significant figures.

  Put it right by identifying the place value of the last kept digit and halving that unit for rounding.

• Writing the interval in the wrong order or with the variable in the middle incorrectly.

  Put it right by writing the lower bound first, then the variable, then the upper bound, for example 7.5 ≤ x < 8.5.

Exam tips

• Decide first whether the question says rounded or truncated, because the interval bounds differ.
• Write the interval using the same variable and units as the question, and check that the lower bound is less than the upper bound.
• Test a value just inside and just outside your interval to confirm it would round or truncate to the reported result.