Two million points drawn on one pair of axes form a solid silhouette — which remedies are honest, and what does each distort?

## What the silhouette is A **mark** is the thing drawn for each row — here, one point per row. When two million of them land in a region a few hundred pixels across, the marks stop being separable. Every pixel inside the cloud is painted by many rows, so it looks exactly like a pixel painted by one. The only information the eye can still extract is the **outline**: roughly where the extremes are. Everything the chart was drawn to show — where the mass sits, whether there are two clumps or one, whether a relationship bends — is behind the ink. This matters more than it sounds, because a saturated cloud looks confident. It is dark, it is dense, it fills the frame, and nothing about it signals that its interior carries no information. There is a cheap first move before any of the remedies below: draw smaller marks. Shrinking the mark buys back some separability at no cost to honesty, and at high density it merely postpones the problem. ## The remedies, and what each one costs | remedy | what it preserves | what it distorts | where it stops working | |---|---|---|---| | **mark transparency** — each mark partly see-through, so overlaps accumulate | every row is still drawn; shade begins to stand for density | shade is not a calibrated count, and an isolated row becomes nearly invisible | past a few dozen coincident marks the plane saturates and the silhouette returns | | **binned two-dimensional counts** — one mark per cell, shaded by how many rows fell in it | density, honestly, with a legend a reader can convert back to counts | individual rows disappear; the cell size is a choice that changes the shape of the result | when the finding is a rare point rather than the mass, since a lone row shades one cell almost imperceptibly | | **scattering marks slightly** — a small random offset per mark | the visible number of marks, so rows sharing a position stop hiding each other | position: every mark is now wrong by up to the offset | as soon as the reader needs to read a value off the axis, or the offset approaches the resolution that matters | | **a summary mark** — a contour or a fitted shape describing where the mass is | the structure of the bulk, legibly | every individual row, and it imposes a shape in sparse regions where little supports one | when the point of the chart was the rows that do not fit the summary | | **drawing a sample** — plot a random subset | the visual character of a single row, and readability | the count: density can no longer be read off the page at all | when rare structure is the finding, because the sample may not contain it | The habit to build is not "reach for the transparent version". It is to say, out loud and in the caption, **which rows the picture is about and what the remedy did to them.** ## Why transparency is the one people over-trust It is the smallest edit and it looks principled: every row is still drawn, nothing is thrown away, and the accumulation of shade genuinely does encode density — while the density is low. The trap is that its useful range is narrow. Once a few dozen marks share a pixel, that pixel is as dark as it can get, and so is its neighbour, and the accumulated-shade argument has quietly stopped applying in exactly the region the reader most wants to understand. A chart can be honest in its sparse corners and saturated in its centre at the same time, which is worse than being uniformly bad, because the sparse corners lend it credibility. ## The category case, where the silhouette also lies about composition When colour is bound to a category and the marks overlap heavily, the visible colour in a dense region is mostly **the order the marks were painted in**: the last category drawn covers the others. A reader will interpret that region as dominated by whichever category that happens to be, and re-running the drawing with the categories in a different order produces a different conclusion from identical data. Transparency softens this and does not fix it, because saturation reasserts itself. The honest moves are the same as above — count per cell, or summarise — plus one more that costs nothing: check whether the conclusion survives reversing the drawing order. ## A short procedure 1. **Estimate the coincidence**, at least roughly: how many rows land in one pixel in the densest area? That number, not taste, picks the remedy. 2. **Under a handful per pixel**, smaller marks and transparency are enough. 3. **Well past that**, move to counts per cell, and put the count legend on the chart so the density is readable rather than impressionistic. 4. **If the finding is a rare row**, do not use anything that dissolves rows into density; use a summary for the mass and draw the rare rows on top of it as themselves. 5. **If you sampled, say so on the chart**, with the sample size. A sampled cloud that does not admit it is the one remedy that misleads without leaving any trace at all.