Matemáticas
What Is an Error Interval?
A length given as 7.4 cm to 1 d.p. has the error interval 7.35 ≤ x < 7.45. Why one end is included and the other is not, and how truncation changes it.
Error interval
An error interval is the range of values a rounded or truncated measurement could have come from, written as an inequality between its lower and upper bounds.
- También llamado
- Bounds, Upper and lower bounds
- Dónde lo encuentra el alumnado
- Grade 9 or 10 mathematics, on both GCSE tiers and on IGCSE 0580, usually worded 'write down the error interval for x'.
La respuesta corta
An error interval is the range of values a rounded measurement could actually have come from, written as an inequality. A length recorded as 7.4 cm to 1 decimal place has the error interval 7.35 ≤ x < 7.45: anything from 7.35 up to, but not including, 7.45 rounds to 7.4.
Un ejemplo
x = 7.4 cm to 1 d.p. → 7.35 ≤ x < 7.45
Rounding to 1 decimal place means the true value was pushed to the nearest tenth. Half a tenth is 0.05, so the interval reaches 0.05 either side of 7.4. Its width is 0.1, the same as the rounding unit — a useful check on any error interval you write.
Why one end is ≤ and the other is <
The asymmetry comes from the rounding convention, not from anything about measurement. A value of exactly 7.35 rounds up to 7.4, so it is one of the values that could have produced the reading, and it belongs inside the interval — hence ≤. A value of exactly 7.45 rounds up to 7.5, so it could not have produced 7.4, and it is excluded — hence <.
That upper number is still written down, even though it is not itself a possible value. Students often want to write 7.4499… instead, which is understandable and wrong: there is no largest number below 7.45, which is precisely why the strict inequality exists.
Reading the interval off the stated accuracy
Find the rounding unit — the size of the place you rounded to — and go half of it either side. To the nearest 10, the unit is 10 and the half is 5. To 2 decimal places, the unit is 0.01 and the half is 0.005.
Significant figures work the same way once you identify which column the last significant digit sits in. In 8.60 to 3 significant figures the last digit is hundredths, so the half is 0.005.
- 12 to the nearest whole number: 11.5 ≤ x < 12.5
- 250 to the nearest 10: 245 ≤ x < 255
- 7.4 to 1 decimal place: 7.35 ≤ x < 7.45
- 8.60 to 3 significant figures: 8.595 ≤ x < 8.605
- 0.070 to 2 significant figures: 0.0695 ≤ x < 0.0705
Truncated numbers give a different interval
Truncating means chopping the digits off rather than rounding them. A value truncated to 7.4 could have been 7.4 itself, or 7.49, or anything up to but not including 7.5 — nothing below 7.4 survives truncation as 7.4. So the interval is 7.4 ≤ x < 7.5.
Exam questions state which has happened, and the word 'truncated' is the only signal you get. The interval is the same width either way; it sits in a different place, starting at the stated value rather than straddling it.
Preguntas frecuentes
Why is the upper bound not 7.4499999?
Because there is no largest number below 7.45 to write down. Between 7.4499 and 7.45 sits 7.44999, and so on forever. The strict inequality x < 7.45 says exactly what is meant — every value below 7.45 is allowed and 7.45 itself is not — without needing a last number.
Is the upper bound included in the error interval?
No, for a rounded value. 7.45 would round to 7.5, so it is not one of the values that could have given a reading of 7.4. It is still the number you write, with a strict < sign. The lower bound is included, because 7.35 does round up to 7.4.
What is the difference between rounding and truncating here?
A rounded 7.4 sits in the middle of its interval, 7.35 ≤ x < 7.45. A truncated 7.4 sits at the bottom of its interval, 7.4 ≤ x < 7.5, because truncation only ever removes value, never adds it. Same width, different position, and the question will tell you which applies.
How does an error interval connect to bounds calculations?
The two ends of the interval are the lower and upper bounds, and bounds questions carry them through a calculation. For the largest possible area of a rectangle, multiply the two upper bounds; for the smallest, multiply the two lower bounds. The error interval is where those numbers come from.
Fuentes
- Pearson Edexcel GCSEs Mathematics (9–1) from 2015 — Pearson Education
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