Absolute uncertainty calculator

The formulas

relative uncertainty = δx ÷ |x| percentage uncertainty = δx ÷ |x| × 100 δx = (percentage ÷ 100) × |x|

For a sum or difference q = a + b − c of independent measurements:

δq = √(δa² + δb² + δc²) (quadrature) δq ≤ δa + δb + δc (worst case)

Worked example: converting

The default is a length of 25.0 cm measured to ± 0.5 cm. The relative uncertainty is 0.5 ÷ 25 = 0.02, which is 2%. The result can be written as 25.0 ± 0.5 cm or as 25.0 cm ± 2%. Both say the same thing; the percentage form makes it easy to compare the precision of measurements with different sizes or units. A 2% uncertainty on a length is worse than a 0.5% uncertainty on a mass, even though you cannot compare 0.5 cm with 0.05 g directly.

In the other direction, a value of 25.0 with a 2% uncertainty has an absolute uncertainty of 0.02 × 25 = 0.5.

Worked example: combining

Choose the combine option for the second default: three lengths laid end to end, 25.0 ± 0.5, 12.4 ± 0.3 and 8.1 ± 0.2. The total is 45.5.

δ = √(0.5² + 0.3² + 0.2²) = √0.38 = 0.616 δ_max = 0.5 + 0.3 + 0.2 = 1.0

So the total is 45.50 ± 0.62 in quadrature, or 45.5 ± 1.0 using the worst-case rule. The quadrature figure is smaller because the three errors are unlikely to all be at their maximum in the same direction at once. The relative uncertainty of the total, 0.62 ÷ 45.5 = 1.4%, is smaller than that of the biggest piece: adding measurements tends to dilute relative error, while subtracting close values does the opposite.

Interpreting and rounding

Round the absolute uncertainty to one or two significant figures, then round the value to the same decimal place: 45.50 ± 0.62, not 45.5 ± 0.616441. The absolute uncertainty carries the unit; the relative and percentage uncertainties have none.

Where the absolute uncertainty comes from

For a single reading, start from the instrument: half the smallest division on an analogue scale, or one unit in the last digit of a digital display. For repeated readings, the standard uncertainty s/√n is the better estimate, and half the range is a quick alternative. Whichever you use, say so in your report, because the same measurement can honestly carry different uncertainties under different conventions.

Common mistakes

• Dividing by the uncertainty instead of the value: 25 ÷ 0.5 = 50 is not a percentage uncertainty.
• Subtracting uncertainties when values are subtracted. δ(a − b) is the same as δ(a + b).
• Adding percentage uncertainties for a sum. Percentages combine for products and quotients; for sums use absolute uncertainties.
• Quoting the value to more decimal places than the uncertainty.