Decide what a coarse density measurement can establish

A displayed number has limited resolution

This model uses a programmed sample density of 2.7 g/mL and an initial liquid volume of 50 mL. The reference is a teaching assumption, not a measurement or an identification of a particular material. The sample’s true volume sets its synthetic mass. Cylinder readings are rounded to the selected increment, with exact half-increment ties rounded upward. No random variation is generated.

If the increment is r mL, each displayed reading stands for values within r/2 mL of that reading under the model. The mass display has a separate ±0.05 g bound. These are stated worst-case limits, not standard deviations or confidence intervals. A real instrument may have calibration error, meniscus-reading error and other effects beyond its display increment.

Subtraction uses two readings

Let D be the displayed final reading minus the displayed initial reading. One reading could be high by r/2 while the other is low by r/2. Their difference can therefore be wrong by as much as r. A conservative displacement interval is D − r to D + r, restricted to physically positive volume for a density calculation.

A truly shared additive zero offset can cancel in a difference, but that does not make rounding errors cancel automatically. We use bounds that allow the unfavorable endpoint combination, rather than assuming cancellation or inventing an independence-based probability distribution.

For a true 5 mL sample with increment 1 mL, readings are 50 and 55 mL. D = 5 mL, the mass display is 13.5 g, and the permitted volume is 4–6 mL. The point estimate 13.5/5 = 2.7 g/mL exactly matches the programmed reference, but the permitted range is wider.

Propagate bounds through the ratio

For positive mass and volume, the smallest possible density combines the smallest permitted mass with the largest permitted volume. The largest combines the largest mass with the smallest positive volume. Here the lower bound is 13.45/6 ≈ 2.242 g/mL, and the upper is 13.55/4 = 3.3875 g/mL.

These conservative combinations need not all be simultaneously realizable in a more detailed correlated-instrument model. They are safe endpoint bounds under the stated intervals, not a probability that the density lies near the middle. Do not attach a confidence percentage to them.

Predict before changing the control: retain the 5 mL sample but select increment 3 mL. Which rounded readings will appear, and can the lower displacement limit remain strictly positive?

Check the coarse-reading case

The initial reading rounds from 50 to 51 mL, and the final reading from 55 to 54 mL. Displayed displacement is only 3 mL. Its conservative interval reaches from 0 to 6 mL. The point ratio is 13.5/3 = 4.5 g/mL. Although the true programmed density is still 2.7, the displayed ratio changes because of rounding.

When there is no finite upper bound

If the permitted positive volume can approach zero while mass stays positive, mass divided by volume has no finite upper bound. The app says so and omits an upper-bound point from the chart. It does not replace an unbounded result with zero or clip it silently to the plot ceiling. The lower bound still uses the largest permitted volume.

Larger samples can make a fixed absolute volume error less important relative to displacement, provided the same measurement assumptions remain appropriate. Test volume 25 mL at increment 3 mL. Compare the reference, point estimate and bounds. More repeated clicks on this deterministic model supply no new evidence and cannot estimate empirical variability.

Decide which claims the evidence supports

Consider two hypothetical candidate densities, 2.7 and 3.2 g/mL, under matched conditions. Both lie within the 1 mL-increment interval for the 5 mL sample. That interval alone does not distinguish them. A narrower interval might exclude one candidate under the model, but a surviving candidate is not uniquely identified if other materials or unmodeled errors remain possible.

Save coarse and finer comparisons. Propose a discriminating measurement, explain which error it addresses, and state one error it would leave unresolved. A reviewed investigation should support its identification from several relevant observations, not select the closest tabulated density and ignore uncertainty.