Bounds and Error Intervals

A rounded measurement is not exact. When someone says a length is 80 to the nearest 10, the real value could be anything that rounds to 80 — and that is a whole range of values, not a single one.

To the nearest 10, every value from 75 up to (but not including) 85 rounds to 80. So if x = 80 to the nearest 10, its lower bound is 75 and its upper bound is 85:

75 \le x < 85

This is called the error interval. The two bounds sit half a rounding-unit either side of the rounded value: half of 10 is 5, so the bounds are 80 - 5 = 75 and 80 + 5 = 85. Notice the upper bound uses <, not \le: a value of exactly 85 would round up to 90, so it is not included.

The same recipe works for rounding to any unit — the nearest whole number, the nearest 0.1, the nearest 100. Halve the unit and step that far either side of the rounded value.

A value rounded to a unit u lies within \tfrac{u}{2} of the rounded value:

lower bound = \text{value} - \tfrac{u}{2};
upper bound = \text{value} + \tfrac{u}{2};
write the error interval as \text{lower} \le x < \text{upper} — the upper bound uses < because that value would round up.

Seeing the interval

Here is x = 80 (to the nearest 10) on a number line. The shaded band is the error interval 75 \le x < 85 — an open circle at 85 marks that the upper bound is not included.