Propagating Upper and Lower Bounds

Each measurement has two bounds

A length of 8 cm to the nearest cm could be anything from 7.5 cm up to, but not including, 8.5 cm. 7.5 is its lower bound and 8.5 its upper bound.

A calculation with rounded measurements inherits that uncertainty. Its result has bounds too, and each one comes from choosing the right bound of every measurement.

Sums and products

Adding or multiplying positive numbers, a bigger input gives a bigger result. So the upper bound uses every upper bound, and the lower bound every lower bound.

A rectangle 8 cm by 5 cm, each to the nearest cm: its area lies between 7.5 × 4.5 = 33.75 cm² and 8.5 × 5.5 = 46.75 cm², and its perimeter between 2(7.5 + 4.5) = 24 cm and 2(8.5 + 5.5) = 28 cm.

A square of side 6 cm to the nearest cm has area between $5.5^2=30.25$5.5² = 30.25 cm² and $6.5^2=42.25$6.5² = 42.25 cm².

Differences and quotients

Subtracting, the number taken away pulls the result down. So the largest difference is the upper bound minus the lower bound: 12 − 5, each to the nearest whole number, is at most 12.5 − 4.5 = 8 and at least 11.5 − 5.5 = 6.

Dividing works the same way: a bigger divisor gives a smaller answer. The largest quotient is the upper bound divided by the lower bound. 100 m to the nearest m in 12.3 s to the nearest 0.1 s is a speed of at most $100.5/12.25=8.204\dots \frac{m}{s}$100.5 / 12.25 = 8.204… m/s and at least $99.5/12.35=8.056\dots \frac{m}{s}$99.5 / 12.35 = 8.056… m/s.

How accurate the answer can be

Round both bounds to the same level of accuracy and compare. 8.204… and 8.056… are 8.2 and 8.1 to 1 decimal place, which differ, so that level claims more than the measurements know.

To 1 significant figure both are 8. The speed is $8\frac{m}{s}$8 m/s to 1 significant figure: the most accurate level at which the two bounds agree.

Writing the result as an error interval

The bounds of a result go into an error interval like any measurement: $33.75\le A<46.75$33.75 ≤ A < 46.75 for the rectangle's area.

The upper bound is never quite reached, since 8.5 would round to 9, but it is the number written as the bound.

The usual slips

Calculating with the rounded values. 8 × 5 = 40 is the area of the rounded sides, which is neither bound.

Upper minus upper. 12.5 − 5.5 = 7 is not the largest difference: the number taken away should be as small as it can be.

Dividing by the upper bound. 100.5 ÷ 12.35 gives a smaller speed than 100.5 ÷ 12.25, so it is not the upper bound.