Error intervals and limits of accuracy

Number • Measures and accuracy

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Explain why the upper bound is often exclusive in these intervals.

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The upper bound is exclusive to avoid overlap with the next rounded or truncated value, ensuring each reported value maps to a unique interval.

Key concepts

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Error interval - definition

An error interval gives the smallest continuous range that must contain the actual value when only a rounded or truncated value is known. The interval uses a lower bound and an upper bound. The lower bound shows the smallest possible true value; the upper bound shows the largest possible true value.

Rounding and its error interval

Rounding to a given accuracy places the true value within half of the rounding unit either side of the reported value. For example, a value reported as 3.14 to two decimal places implies a half-unit of 0.005. The true value therefore satisfies 3.135 ≤ x < 3.145, using the convention lower bound inclusive and upper bound exclusive.

Truncation and its error interval

Truncation removes digits beyond a chosen place value without adjustment. Truncation to two decimal places produces an interval that starts at the reported value and extends up to but not including the next increment. For example, 3.14 truncated to two decimals implies 3.14 ≤ x < 3.15.

Limits of accuracy (upper and lower bounds)

Limits of accuracy name the smallest lower bound and the smallest upper bound consistent with the stated accuracy. The lower bound is usually inclusive and the upper bound is usually exclusive when using standard rounding or truncation conventions. Notation uses inequalities such as a ≤ x < b to indicate the interval precisely.

Writing inequality notation

Inequality notation uses ≤ for included endpoints and < for excluded endpoints. For rounding to n decimal places: if v is the rounded value then v − 0.5×10^(−n) ≤ x < v + 0.5×10^(−n). For truncation to n decimal places: v ≤ x < v + 1×10^(−n). Examples clarify the calculation of the offset and the correct placement of inclusive/exclusive symbols.

Using bounds in calculations

When combining measured values, add corresponding lower bounds to get the smallest possible sum and add corresponding upper bounds to get the largest possible sum: L_total ≤ true_total < U_total. For multiplication and division, propagate bounds with care: sign and relative size change the effect, so evaluate extreme combinations to find the final lower and upper bounds.

Key notes

Important points to keep in mind

Lower bound is usually inclusive (≤); upper bound is usually exclusive (<).

Rounding to n decimal places uses ±0.5×10^(−n) around the reported value.

Truncation to n decimal places gives [v, v + 10^(−n)).

Write intervals in inequality form, e.g. a ≤ x < b, not only with ± notation.

Add bounds directly for sums and differences: L_total = sum of lowers, U_total = sum of uppers.

Check signs before multiplying or dividing; evaluate extreme combinations for extrema.

Use exclusive upper bounds to prevent overlap between adjacent rounded values.

Report limits of accuracy clearly when publishing calculated results.

When given significant figures, convert to place value first, then apply half-unit rule.

Remember that truncation gives a biased interval (true value at or above reported value).