SCIENCE · INVESTIGATE AND EXPLAIN

Propagate a difference without double-counting a shared offset

A gross-minus-tare measurement can cancel a common error while retaining other uncertainty.

Learning outcome: Hold each reading bound .04 g and compare shared fractions 0, .5, 1. Derive the interval before capture. Then explain why an offset drifting between readings breaks cancellation.

Starting recap

Use algebra and dimensional consistency; distinguish a measured reading, a model estimate and the evidence needed to accept the model.

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Gross 25 g minus tare 15 g gives 10 g; each ±.04 g bound half shared leaves separate ±.02 g terms, so net bound is±.04 g.

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Predict

If all error is one unchanged zero offset, does gross-minus-tare retain that offset?

Your prediction and reason

Experiment

Hold each reading bound .04 g and compare shared fractions 0, .5, 1. Derive the interval before capture. Then explain why an offset drifting between readings breaks cancellation.

Bound a mass difference with shared error

Worked teaching model · synthetic values · Indicated gross 25 g and tare 15 g share the same additive offset bounded by fraction×reading bound. Each also has its own independently bounded remainder. The common offset cancels exactly; independence here means separately variable bounded terms, not a statistical distribution.

Each reading total absolute bound: 0.04 g Shared-offset fraction of each bound: 0.5 fraction

Reset keeps your prediction, captures and practice history.

Conditional calculations; not physical measurements
Quantity	Value	Unit
Net indicated mass	10	g
Shared offset bound per reading	0.02	g
Separate remainder bound per reading	0.02	g
Net lower feasible mass	9.96	g
Net upper feasible mass	10.04	g
Naive bound if all errors were separate	0.08	g

Subtracting two readings cancels the identical additive error, not every error.

Write gross=true gross+c+eG and tare=true tare+c+eT. Their difference is true net+eG−eT. Bounds add for the two remaining independently variable terms. A drifting offset would not cancel.

Predict before changing controls, retain contrasting captures, and explain a changed case. Displayed digits are computational precision, not an uncertainty claim. Flags check stipulated numerical compatibility, not identity, reasoning quality or practical proficiency.

Subtracting two readings cancels the identical additive error, not every error.

Write gross=true gross+c+eG and tare=true tare+c+eT. Their difference is true net+eG−eT. Bounds add for the two remaining independently variable terms. A drifting offset would not cancel.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Why the measurement equation matters

Suppose a sample mass is estimated as gross reading minus tare reading. An uncertainty statement about each reading is not enough unless their relationship is specified. Two readings from the same balance may share a zero offset. Treating that one offset as two freely varying errors invents a combination that the measurement model does not allow.

Write shared and separate terms

Let G = Gtrue + c + eG and T = Ttrue + c + eT. The identical additive c appears in both. Subtracting gives G − T = Gtrue − Ttrue + eG − eT. The shared term cancels exactly, regardless of its unknown sign. The two residual terms need not cancel. This algebra is a stronger explanation than saying “uncertainties always add” or “all balance error disappears.”

Calculate a conservative interval

The synthetic indicated readings are 25 and 15 g. Each reading has total absolute bound a. A fraction b of that bound is assigned to the shared term, leaving separate terms bounded by a(1 − b). The net indicated mass is 10 g, and its worst-case absolute bound is 2a(1 − b). With a = 0.04 g and b = 0.50, each remainder is 0.02 g and the net interval is [9.96, 10.04] g.

Use limiting cases to test the formula

At b = 0, none of the bound is shared and the interval is [9.92, 10.08] g. At b = 1, this deliberately restricted model has only one common offset and the difference is exactly 10 g. That limiting result is not a claim that a real balance difference is exact: resolution, repeatability, buoyancy and changing offsets are absent from the model. Capture all three cases and name what is being assumed away.

Distinguish bounds from statistical uncertainty

These controls specify feasible absolute bounds, not standard deviations. Root-sum-of-squares propagation would require a probabilistic interpretation and appropriate assumptions about correlations. In a statistical treatment, covariance terms encode shared variation; they cannot simply be dropped because readings were written on separate lines. Here cancellation is established directly from the measurement equation.

Transfer: offset drift

If the gross reading contains cG but the tare contains cT, the difference includes cG − cT. A before-and-after reference comparison can test whether an unchanged offset is plausible. Propose what observations would distinguish a common offset from drift, and explain why repeating only the gross measurement would not test the entire difference procedure.

Source and evidence boundary

These explanations, calculations and diagrams are original. Conceptual references checked 30 October 2026: NIST uncertainty framework, OpenStax measurement uncertainty. The linked textbook is a reference, not copied lesson text or imagery. All displayed datasets are synthetic teaching packets. Saved numerical agreement or prose is not a performed practical, independent reasoning review or laboratory certification. No handling, heating or chemical preparation procedure is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Compare absolute feasible bounds with standard uncertainties; do not attach a confidence level without a distribution.

Your explanation and limitations

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Previous: Infer mass through an explicit calibration model Next: Derive exact density bounds and conditional identification

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