Error Intervals
An error interval shows the range of values that could have rounded to a given number. This is closely linked to rounding, estimation and bounds.
An error interval is usually written using inequalities.
lower bound ≤ value < upper bound
What are lower and upper bounds?
The lower bound is the smallest possible value. The upper bound is the value just above the largest possible value.
Example 1: Number rounded to nearest 10
A number is rounded to 70 to the nearest 10.
The halfway point below 70 is 65.
The halfway point above 70 is 75.
65 ≤ x < 75
The error interval is 65 ≤ x < 75.
Error intervals for decimal places
If a number is rounded to 1 decimal place, the interval goes halfway to the decimals on either side.
Example 2: Rounded to 1 decimal place
A length is 4.6 cm to 1 decimal place.
The lower bound is 4.55.
The upper bound is 4.65.
4.55 ≤ x < 4.65
Error intervals for whole numbers
If a number is rounded to the nearest whole number, the interval goes 0.5 below and 0.5 above.
Example 3: Rounded to the nearest whole number
A mass is 18 kg to the nearest kilogram.
17.5 ≤ m < 18.5
Error intervals for significant figures
For significant figures, first identify the place value of the final significant digit. Then go halfway below and halfway above.
Example 4: Rounded to 2 significant figures
A number is 340 to 2 significant figures.
The final significant digit is in the tens column.
Half of 10 is 5.
335 ≤ x < 345
Why the upper bound is not included
The upper bound would round to the next value, not the original rounded value. That is why we usually use < rather than ≤ for the upper bound.
Example 5: Why not include the upper bound?
If a number is 4.6 to 1 decimal place, the upper bound is 4.65.
But 4.65 rounds to 4.7 to 1 decimal place.
So the interval is:
4.55 ≤ x < 4.65
A common mistake is writing the upper bound with ≤. Usually the upper bound is not included, so write < for the upper bound.
Find the rounding unit first. Then halve it. For example, nearest 10 means half of 10 is 5, so the bounds are 5 below and 5 above.
Video explanation
A short Worthing Maths Tutor video explanation for error intervals can be embedded here later to improve student engagement and time on page.
Practice questions
- A number is 50 to the nearest 10. Write the error interval.
- A length is 7.3 cm to 1 decimal place. Write the error interval.
- A mass is 24 kg to the nearest kilogram. Write the error interval.
- A number is 860 to 2 significant figures. Write the error interval.
- Why is the upper bound usually written with < rather than ≤?
Answers
- 45 ≤ x < 55
- 7.25 ≤ x < 7.35
- 23.5 ≤ x < 24.5
- 855 ≤ x < 865
- Because the upper bound would round to the next value.
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