Reporting rules

A number without its uncertainty and without a statement of what that uncertainty means cannot be checked, compared, or reused. This file covers what has to accompany a reported result.

Round the uncertainty first

JCGM 100:2008 7.2.6. The uncertainty is rounded to one or two significant digits, and the value is then rounded to that same decimal place — never the other way round, and never independently.

12.34567 +/- 0.02345    ->    12.346 +/- 0.023
12.34567 +/- 0.1        ->    12.35  +/- 0.10        (two digits, since 1 leads)
1234     +/- 250        ->    1230   +/- 250

Two digits are the safe default. One digit is acceptable when the leading digit is 3 or more; rounding 0.14 to 0.1 changes it by 29%, which is why a leading 1 or 2 should keep two digits. scripts/format_result.py applies the rule and warns on that case.

Trailing zeros in the uncertainty are significant and must be kept: 0.10, not 0.1.

Notations

In concise notation the digits in parentheses apply to the last digits of the quoted value, so 12.346(23) is 12.346 ± 0.023 and 1234(25) is 1234 ± 25. When the uncertainty's last significant digit falls left of the decimal point the notation is ambiguous and the scientific form must be used instead.

What has to be stated alongside

A bare ± is ambiguous. Readers cannot tell a standard uncertainty from an expanded one, a standard deviation from a standard error, or a 95% interval from a 68% one. State:

1. Which quantity the number is — combined standard uncertainty u_c, expanded uncertainty U, standard deviation of a sample, standard error of a mean, or a confidence interval.
2. The coverage factor k, when U is reported.
3. The coverage probability and how it was arrived at — from a normal assumption or from effective degrees of freedom.
4. The effective degrees of freedom, when they are small enough to matter (below about 30).
5. The method — GUM framework, Monte Carlo, or both with the validation outcome.

The two standard forms:

m = 100.02147 g with a combined standard uncertainty of u_c = 0.35 mg.

m = (100.02147 ± 0.00079) g, where the number following the symbol ± is the numerical value of an expanded uncertainty U = k·u_c with U determined from a combined standard uncertainty u_c = 0.35 mg and a coverage factor k = 2.26 based on the t-distribution for ν_eff = 9 degrees of freedom, and defines an interval estimated to have a level of confidence of 95%.

The second is verbose because it has to be. scripts/format_result.py --coverage-factor generates the sentence.

SD, SEM, and CI in figures

An error bar is uninterpretable unless the caption says what it is, and the three common choices differ by more than a factor of two for typical n:

Choosing SEM because it looks tighter is a misrepresentation when the question is about spread. Every caption needs the bar's identity, n, and whether n counts biological or technical replicates. Two SEM bars that do not overlap do not establish a significant difference, and two 95% CIs that overlap slightly do not establish the absence of one.

Relative and absolute

State which. A relative standard uncertainty is dimensionless and is written u_r(y) = 0.0035 or 0.35%; multiplying it by the value gives the absolute one. Mixing them in a single budget without conversion is a common arithmetic error — a "1%" component and a "0.2 mg" component cannot be combined until they are in the same form.

For a product or quotient of independent quantities, relative uncertainties combine in quadrature; for a sum or difference, absolute ones do. Using the wrong one is the single most common propagation mistake, and it is why deriving the budget from sensitivity coefficients rather than from remembered rules is worth the extra step.

Results near zero or below a detection limit

• Do not report a value with an uncertainty larger than itself as though it were a measurement. Report the estimate and its uncertainty, and state that it is consistent with zero.
• Do not substitute zero, LOD, or LOD/2 for a non-detect without saying so; each choice biases downstream statistics differently.
• A negative estimate of a non-negative quantity is a legitimate measurement outcome and should be reported as measured, not truncated. Truncating biases any subsequent average.
• LOD and LOQ are defined by a stated procedure (typically 3σ and 10σ of the blank). Quote the procedure with the number.

Significant figures in intermediate work

Round only at the point of reporting. Carrying rounded intermediates through a calculation accumulates error that the uncertainty budget does not account for, and it can shift the final digit. Keep full precision internally; apply scripts/format_result.py at the end.

The same applies to constants: use scipy.constants, not a value typed from memory to four digits.

Conformity statements

Deciding whether a result passes a specification is a separate step from measuring it, because a result within tolerance but with an uncertainty straddling the limit has not demonstrated conformity. ISO/IEC 17025 requires a documented decision rule; ILAC-G8 describes guard-banded acceptance, where the acceptance limit is pulled inside the specification limit by a multiple of u_c chosen for the false-accept risk you will tolerate. Report the rule alongside the verdict.

Sources

• JCGM 100:2008 (GUM), clause 7 — reporting uncertainty.
• JCGM 101:2008 (GUM Supplement 1), clause 8 — numerical tolerance and validation.
• NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results — the source of the two standard sentence forms above.
• ISO/IEC 17025:2017 clause 7.8.6 and ILAC-G8:09/2019 — decision rules and guard bands.