Carry measurement uncertainty through a net environmental budget

Frame the environmental question

A small net balance can be the difference of two much larger fluxes. The reported means alone do not show whether the sign of the net is well resolved. Keep uncertainty attached to the actual measured terms and distinguish standard uncertainty, a distributional interval, a hard bound and an unknown structural error.

Build the measurement or system model

For net storage rate S=I−O, independent errors give variance uI²+uO². More generally the variance is uI²+uO²−2 cov(I, O). A shared positively correlated calibration error can partly cancel in the difference. The same correlation would enter a sum with the opposite sign; covariance belongs to the measurement model, not a universal rule that errors always add.

Work at the right scale

The controls supply standard uncertainties for input 100 and output 90 kg/day. They are not estimated from the two displayed means. The model contrasts independent errors with perfect positive correlation as two explicit scenarios. A real study needs evidence about shared instruments, standards and processing before choosing a covariance structure.

Challenge the inference

For the independent case only, a normal approximation gives a displayed interval S±1.96 u. This relies on stipulated known standard uncertainties and jointly normal errors. It is not a guaranteed range, a confidence interval estimated from two observations, or a treatment-effect test. An interval crossing zero means the sign is not resolved under this particular uncertainty model.

Design a defensible comparison

Missing export, unmeasured storage and biased boundaries are structural problems. Making an instrument more precise cannot compensate for omitting an entire pathway. Report which uncertainties are quantified and which are not, and choose additional observations based on their influence on the decision.

Worked original packet

At uI=4 and uO=3 kg/day, the net is 10 and independent standard uncertainty 5. The stipulated normal interval is 0.2 to 19.8 kg/day. Under perfect positive correlation, the standard uncertainty is|4−3|=1 instead; the means have not changed.

Investigation

Compare 4/3 with 8/6 while retaining the flux means. Reconstruct variance and intervals; state exactly which assumptions permit a sign claim and which omitted flux would invalidate it.

Changed-case transfer

For uncertainties 6 and 8, calculate independent standard uncertainty, the displayed normal interval and the perfectly correlated alternative.

Feedback

Independent u=10, interval−9.6 to 29.6 kg/day; perfect-positive-correlation u=2. The independent interval crosses zero; neither scenario establishes the actual covariance.

Sources and evidence boundary

Original instructional prose, values and exercises. Factual context checked 30 October 2026: primary teaching or agency source. These are synthetic computational comparisons, not measured field data, verified mastery or authorization for physical intervention. Retain your prediction, assumptions and source of each observation.