What is an error interval?
An error interval is simply the upper and lower bounds of a rounded value, written together as a single inequality. If a length rounds to 8cm, its error interval is written:
7.5 ≤ length < 8.5
The lower bound uses "≤" (the true value could equal it exactly), while the upper bound uses "<" (the true value can get arbitrarily close but never actually reach it, since reaching it would round up to the next figure instead).
Writing an error interval
Worked Example 1
A length is measured as 12cm, correct to the nearest cm. Write down the error interval for the length, L.
1
The rounding interval is 1cm, so half of it is 0.5cm
2
Lower bound: 12 − 0.5 = 11.5. Upper bound: 12 + 0.5 = 12.5
Answer11.5 ≤ L < 12.5
The thick line shows every value the length could actually be — all of them round to 12cm.
Worked Example 2
A number is 5.8, correct to 1 decimal place. Write the error interval.
1
The rounding interval is 0.1, so half of it is 0.05
2
Lower bound: 5.8 − 0.05 = 5.75. Upper bound: 5.8 + 0.05 = 5.85
Answer5.75 ≤ n < 5.85
Worked Example 3
A number is 3000, correct to 1 significant figure. Write the error interval.
1
1 significant figure of 3000 means rounded to the nearest 1000, so half the interval is 500
2
Lower bound: 3000 − 500 = 2500. Upper bound: 3000 + 500 = 3500
Answer2500 ≤ n < 3500
Working backwards from an error interval
Worked Example 4
The error interval for a number x is 6.5 ≤ x < 7.5. What was x rounded to the nearest whole number?
1
The rounded value sits exactly halfway between the two bounds
2
Halfway between 6.5 and 7.5: (6.5 + 7.5) ÷ 2 = 7
Answerx = 7
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1A number is 24, correct to the nearest whole number. Write the error interval.Foundation
Q2A length is 8.3cm, correct to 1 decimal place. Write the error interval.Foundation
Q3A mass is 70kg, correct to the nearest 10kg. Write the error interval.Foundation
Q4A number is 500, correct to 1 significant figure. Write the error interval.Foundation
Q5The error interval for a number y is 11.5 ≤ y < 12.5. What is y rounded to the nearest whole number?Foundation
Higher (Grade 5–7)
Q6A number is 0.06, correct to 1 significant figure. Write the error interval.Higher
Q7A number is 250, correct to 2 significant figures. Write the error interval.Higher
Q8A distance is 3.45km, correct to 2 decimal places. Write the error interval.Higher
Q9The error interval for a number x is 14.5 ≤ x < 15.5. Write down x, rounded to the nearest whole number.Higher
Q10A number is 8000, correct to 1 significant figure. Write the error interval.Higher
Higher — Hard (Grade 8–9)
Q11A number n, correct to 3 significant figures, is 0.0470. Write the error interval.Grade 8–9
Q12The error interval for a number p is 3.15 ≤ p < 3.25. Find the value p was rounded to, and state the degree of accuracy used.Grade 8–9
Q13A square has an area of 36cm², correct to the nearest whole number. Find the error interval for the side length, to 3 significant figures.Grade 8–9
Q14Two numbers a and b satisfy 4.5 ≤ a < 5.5 and 2.5 ≤ b < 3.5. Find the error interval for a + b.Grade 8–9
Q15A number is truncated (not rounded) to 6.7, to 1 decimal place. Explain how the error interval for a truncated value differs from a rounded one, and write down the error interval.Grade 8–9
Answers
Foundation (Q1–Q5)
Q123.5 ≤ n < 24.5
Q28.25 ≤ L < 8.35
Q365 ≤ m < 75
Q4450 ≤ n < 550
Q5y = 12
Higher (Q6–Q10)
Q60.055 ≤ n < 0.065
Q7245 ≤ n < 255
Q83.445 ≤ d < 3.455
Q9x = 15
Q107500 ≤ n < 8500
Higher — Hard (Q11–Q15)
Q110.04695 ≤ n < 0.04705(the 3rd significant figure sits in the ten-thousandths column)
Q12p = 3.2, to 1 decimal place(midpoint of 3.15 and 3.25)
Q135.96cm ≤ side < 6.04cm(√35.5 and √36.5)
Q147 ≤ a+b < 9(4.5+2.5 and 5.5+3.5)
Q156.7 ≤ n < 6.8. Truncation always cuts digits off without rounding, so a truncated value is never adjusted upward — the true value must be at least the truncated value, but could be anywhere up to (not including) the next interval, unlike a rounded value where the true value could be slightly below the stated figure too.
Common mistakes
Common Mistake 1
Using ≤ on both sides of the inequality
Writing 7.5 ≤ L ≤ 8.5 is incorrect — the upper bound should use "<" since a value of exactly 8.5 would round up to 9, not to 8.
Common Mistake 2
Reversing the inequality direction
The smaller bound always goes on the left: 11.5 ≤ L < 12.5, not 12.5 < L ≤ 11.5. Reading the inequality "left to right, small to large" avoids this error.
Common Mistake 3
Using the wrong half-interval for significant figures
For 3000 to 1 significant figure, the half-interval is 500 (since the rounding interval is 1000), not 0.5 — always work out the actual place value being rounded to first.
Common Mistake 4
Treating truncation the same as rounding
A truncated value has already had digits removed without any rounding adjustment, so its error interval only extends upward from the stated value, not both ways like a normally rounded number.
Exam tips
💡 Exam Tip 1
Always use the letter given in the question
If the question defines a variable (e.g. "the length, L"), use that same letter in your error interval — examiners specifically check that the inequality refers to the correct quantity.
💡 Exam Tip 2
Double-check the ≤ and < symbols
The lower bound always uses ≤ and the upper bound always uses < — writing the wrong symbol is a very common way to lose a mark despite understanding the maths correctly.
💡 Exam Tip 3
Find the midpoint to work backwards
Given an error interval, the original rounded value is always exactly halfway between the two bounds — add them together and divide by 2.
💡 Exam Tip 4
State the degree of accuracy alongside the rounded value
When asked to find both the rounded value and its accuracy, give both parts clearly — e.g. "3.2, to 1 decimal place" — since the accuracy alone doesn't fully answer the question.
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