You weigh a sample in air, then weigh the same sample in water. The difference between the two numbers is the weight of the water it displaced, and the density follows. The simplest experiment in the book — until you notice that the balance is not equally accurate for every sample, and that this has nothing to do with the balance.
What makes this one different
You are given five unlabelled samples, from a tiny very heavy lump to a large light block. You measure them all on one balance, so you expect one accuracy. You do not get it — the spread is tens of times. The reason is in the formula: the answer is divided by the difference of two weights. For the small heavy sample you subtract two large numbers and a tiny one survives; the noise on both readings is still there, but it now sits on something small. This is catastrophic cancellation, and no amount of repeating the measurement fixes it. The only escape is choosing a sample that displaces more water — a decision taken before you measure, not after.
The second lesson has to be solved with your hands, not your arithmetic. When you lower the sample into the water, small bubbles cling to its surface. Bubbles lift, so the sample reads lighter in water, the difference comes out too large, and the density is reported low. In calorimetry we removed a systematic by calibration and in the momentum experiment by symmetry; neither works here. Only procedure does: brush the sample before you weigh it. The numbers will not save you — you have to do the right thing in the laboratory.
And a third, which none of the earlier experiments has shown you: every error you remove makes the next one visible. While you work on the small sample, the bubbles and the water temperature are both buried under the noise. On the large sample the bubbles suddenly become the largest error; and once you have brushed them off, the water temperature — which until that moment looked like an irrelevant detail — becomes the dominant one. Water at 25 °C is not 1.000 g/cm³, and that third digit is exactly where your answer is standing. In a real laboratory “which error matters?” has no fixed answer; it depends on how far you have already got.
The Python section does what the page cannot. The page writes ρ ± σ like every textbook, as though the distribution were symmetric. Python refuses that assumption and simulates the worst sample two hundred thousand times to show that when the denominator is itself uncertain, the tail stretches one way — so the plus-or-minus sign you wrote with such confidence is a small lie.