Uncertainty, reproducible data and responsible reporting

A result should say what the number represents

Report the intended quantity, entity, matrix, target, basis, unit and procedure together with the value and relevant uncertainty. The result should be linked to retained indications, reference values, preparation factors, corrections and calculations. A number without those connections can be difficult to interpret or reproduce.

Measurement uncertainty describes dispersion of values attributed to the measurand using available information. It is not a measurement of the actual error or proof of absence of bias. Statistical and other evaluations can contribute; the important question is whether significant inputs and effects are represented in the model.

An uncertainty value needs its type and coverage factor

In an invented model, two independent standard uncertainty components are 0.100 and 0.200 mg L⁻¹. Combined standard uncertainty is u_c=√(0.100²+0.200²)≈0.223607 mg L⁻¹. With chosen coverage factor k=2, expanded uncertainty is U=k u_c≈0.447214 mg L⁻¹.

A selected reporting format for estimate 5.00 mg L⁻¹ is 5.00±0.45 mg L⁻¹, explicitly saying expanded uncertainty U and k=2. Approximately 95% coverage requires justified distribution and uncertainty-estimation conditions; multiplying by two does not universally establish that probability. Standard uncertainty, expanded uncertainty and a replicate SD are different reported quantities.

A shared uncertainty component does not shrink like independent noise

For an optional deeper original model, write readings Y_i=μ+B+ε_i. B is one common uncertain contribution to all readings, while ε_i are independent repeatability effects. Assume B independent of the ε_i. With n=16, repeatability SD 0.400 mg L⁻¹ and standard uncertainty of B equal to 0.300 mg L⁻¹, the repeatability contribution to u(mean) is 0.400/√16=0.100 mg L⁻¹.

The common term remains 0.300. Variance of the mean is 0.400²/16+0.300²=0.100 (mg L⁻¹)², giving combined standard uncertainty approximately 0.316228 mg L⁻¹. Incorrectly dividing both terms by √16 would give 0.125 mg L⁻¹. The repeated readings do not provide sixteen independent copies of the same shared contribution.

As n grows, the model variance approaches the common floor 0.0900 (mg L⁻¹)². Actual common effects and corrections need evidence; this calculation does not identify the cause of a real bias or make every repeated observation independent.

Correlation can raise or lower propagated uncertainty

Suppose two invented input results have standard uncertainties 0.200 mg L⁻¹ each and known correlation coefficient ρ=0.750. Their covariance is ρu₁u₂=0.0300 (mg L⁻¹)². For a sum, variance is u₁²+u₂²+2cov=0.140, giving standard uncertainty approximately 0.374166 mg L⁻¹.

For a difference, variance is u₁²+u₂²−2cov=0.0200, giving approximately 0.141421 mg L⁻¹. Treating them as independent would instead give approximately 0.282843 for either function. Positive correlation raises uncertainty of this sum but lowers uncertainty of this difference.

The covariance matrix is valid here, with eigenvalues 0.0700 and 0.0100 (mg L⁻¹)². Correlation is a stipulated input, not an inference from two measured values. Real covariance and the appropriate measurement function must be established before using such a simplification.

Readable rounding should preserve the working evidence

Suppose an invented working estimate is 12.345678 mg L⁻¹ with expanded uncertainty 0.23456 mg L⁻¹ and k=2. Choose two significant digits for the uncertainty, 0.23 mg L⁻¹, and round the estimate to the matching hundredth place: 12.35 mg L⁻¹. Preserve the unrounded working values and inputs for later checking.

A complete illustrative report identifies invented target 25-A, symbolic analyte A, mass concentration in a hypothetical aqueous mixture on final-volume basis, the educational calibration model, 12.35±0.23 mg L⁻¹ with expanded U and k=2, and the fact that the data are invented and sampling/calibration performance is unvalidated. Formatting does not itself establish traceability, coverage or a good measurement.

A reporting limit is not a substitute for a chemical identity

If a result is reported below a stated limit, name the limit, unit, basis and relevant decision procedure. Do not replace it with numerical zero. A detection limit, quantification limit and reporting threshold can express different definitions; a terse inequality without that context can conceal what was actually concluded.

Retain sample identifiers, raw indications, calibration inputs, software or calculation version, corrections, rejected-data rationale, controls and uncertainty evaluation. Reproducible calculations require that evidence trail. Reproducibility of measurements under changed locations, operators or systems is a separate performance question with specified conditions.

Report limitations alongside conclusions. A complete record allows another reader to assess the result’s scope, rather than silently converting an invented model, incomplete recovery or below-limit indication into an unsupported claim about the target.

Worked examples

Try it yourself

1. Keep the same repeatability SD and common uncertainty, but use four readings. Find u(mean) for the combined model.

   Show the answer and reasoning

   Approximately 0.360555 mg L⁻¹.

   √(0.400²/4+0.300²)=√0.130; only the independent term is reduced.

2. Keep both input standard uncertainties 0.200 mg L⁻¹ but set correlation to zero. Find uncertainty of their sum and difference.

   Show the answer and reasoning

   Both approximately 0.282843 mg L⁻¹.

   With zero covariance, either variance is 0.0400+0.0400=0.0800.

3. Should a below-detection-limit report automatically be converted to numerical zero?

   Show the answer and reasoning

   No.

   It is a scoped decision or limit statement, rather than proof of exact absence.

4. Does writing k=2 by itself justify an exact 95% coverage claim?

   Show the answer and reasoning

   No.

   The relationship depends on distribution and uncertainty-estimation conditions.

What to carry forward

• Report the defined quantity with its evidence trail.
• Label uncertainty type and coverage factor.
• Independent replication does not remove common terms.
• Use covariance with the correct measurement function.
• Retain working data and explain limits rather than substituting zero.