Reporting uncertainty: rounding and significant figures

An expanded uncertainty of 0.11734 °C claims knowledge no budget can support. The GUM recommends reporting U to at most two significant figures, and the measured value rounded to the decimal place of U. Rounding of intermediate values is avoided; rounding happens once, at the end.

Examples

• Correction +0.0473 °C, U = 0.0316 °C → report +0.047 °C, U = 0.032 °C (k = 2).
• Reading 21.8367 °C, U = 0.12 °C → report 21.84 °C ± 0.12 °C (k = 2).

Form of words

The GUM suggests a form such as: 'The reported expanded uncertainty is based on a standard uncertainty multiplied by a coverage factor k = 2, providing a level of confidence of approximately 95 %.' Accredited certificates carry an equivalent statement.

The rule, and why it exists

GUM §7.2.6 asks for the expanded uncertainty to be given to at most two significant figures, with the reported value rounded to the same decimal place. The reason is not tidiness. An uncertainty is itself an estimate, built from other estimates, several of which came from a specification sheet or an engineering judgement. Quoting it to five figures claims a precision about the uncertainty that no budget in this field can support, and it is one of the things an assessor will notice first because it signals that the numbers were copied out of a spreadsheet rather than read.

Round once, at the end

Every intermediate value carries full precision until the final report. Rounding each component before combining introduces an error that compounds through the quadrature sum, and rounding u_c before expanding it is the commonest version. Carry the digits, expand, then round the pair together.

The exception worth knowing is GUM §7.2.6's own allowance: where rounding down would reduce the uncertainty by more than about 5 %, round up instead. Reporting 0.0996 as 0.099 is fine; reporting a computed 1.04 as 1.0 when the difference matters is the case the allowance covers.

The statement that goes with the number

A number without a coverage factor cannot be compared with any other number. The GUM's recommended form, which accredited certificates carry in some variation, is: the reported expanded uncertainty is based on a standard uncertainty multiplied by a coverage factor k = 2, providing a level of confidence of approximately 95 %.

• Always state k, or the coverage probability, or both. A bare ± figure is ambiguous by a factor of two or more.
• Say approximately 95 %, not 95 %. The coverage is approximate unless effective degrees of freedom were evaluated and k chosen accordingly.
• Give the unit of the uncertainty explicitly, and say whether it is absolute or relative. Relative uncertainties quoted as a bare percentage next to a temperature are a recurring source of confusion.
• Keep the result and its uncertainty at the same decimal place throughout the document, including in tables and summaries, so a reader is never asked to reconcile two versions of the same figure.