Clustering Illusions
How genuine randomness produces clumps, gaps, and streaks, and why we so often read them as meaningful or as “not random.”
This Couldn't Just Be Coincidence!
"That looks too patterned to be a coincidence." Each grid below stands in for a stretch of ordinary days, with one small event per cell. On most days, nothing lines up into anything notable. Every so often, purely by chance, a few nearby days form something that looks like a pattern.
One of the two grids below was filled in completely at random. The other was filled in by a process that really does favor placing events near other events. Uncover them below and see if you can tell which is which.
Part 4: This Couldn't Just Be Coincidence!
Rewind and watch it happen
Drag the slider below to rewind both grids to blank, then play them forward. Every cell appears in the exact order it was actually placed. This isn't a new random draw. It's a replay of the same two grids above.
Choosing a new value generates a brand-new pair of grids at that density, so you can see whether matches become more or less common.
How these grids are made
Both grids are 20×20 with exactly 80 squares filled in by default (you can adjust that above), so the only thing that can differ between them is where those squares end up, not how many there are. A qualifying cluster is a completely filled block, 2×2 or larger. We used square blocks rather than other shapes because a square just looks more "deliberate" to the eye than an odd shape of the same size.
The random grid is a plain lottery draw: 80 of the 400 cells are chosen with no regard for where any other cell landed. A random draw like that doesn't always happen to contain a qualifying cluster, so several random grids get drawn behind the scenes and thrown out until one does. The count shown above is the real number of tries that took. The non-random grid is built one cell at a time by a process called local attraction, which makes each new cell more likely to land next to cells already placed. Its clusters come from a genuine pull toward existing cells, not luck.
The rewind control replays each grid's actual history. Scrub through the random grid and cells pop up with no relationship to their neighbors. Scrub through the non-random grid and you can watch later cells crowd in next to earlier ones.
Research behind this
The math here is exact: how the grids are generated, how clusters are detected, and the odds quoted above all follow directly from the fixed-count random model and the local-attraction process described above. The bigger point ties into the site's broader theme: a tiny chance of landing on one exact, specific outcome doesn't mean that outcome wasn't just luck. Some exact outcome always has to happen.
- Diaconis, P., & Mosteller, F. (1989). Methods for studying coincidences. Journal of the American Statistical Association, 84(408), 853–861. doi:10.1080/01621459.1989.10478847
- Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102(4), 684–704. doi:10.1037/0033-295X.102.4.684
A citation on why symmetric shapes (like solid squares) read more strongly as "deliberate" than asymmetric shapes of the same size is still being sourced and verified before it's added here.
Further reading: The Clustering Illusion on Wikipedia.