Clustering Illusions
How genuine randomness produces clumps, gaps, and streaks, and why we so often read them as meaningful or as “not random.”
The Clustering Illusion
Random events don't have to spread themselves out. When unusual things happen close together, in time, place, or circumstance, they can feel connected. But chance itself can produce clusters.
First, see what a random pattern looks like to you.
Part 1: What Does Random Look Like?
Which pattern was generated by pure chance?
Correct: 0 / 0
Neighbor alternation is how often two touching squares differ, one filled and one empty. A low number means neighbors usually match, which creates clumps. A high number means they usually differ, which looks more like a checkerboard.
Random doesn't mean evenly spaced. A truly random process naturally produces clumps and gaps, while a pattern that consistently avoids clumps is actually more orderly, not less. So when unusual events happen close together, the cluster may feel meaningful. Closeness alone doesn't show that the events share a cause.
Random patterns naturally clump. A suspiciously even spread can actually be more orderly than a genuinely random pattern.
How these patterns are made
Every grid has 100 squares, exactly 50 filled and 50 empty, so the only thing that differs between patterns is how the squares are arranged.
To measure that arrangement, count the touching pairs of squares (side by side, or one above the other). A 10×10 grid has 180 of them. Neighbor alternation is the share of those pairs where one square is filled and the other is empty.
The pure chance pattern is a plain random shuffle of 50 filled and 50 empty squares. Nothing is adjusted afterward, which is why its neighbor alternation lands near 50% but varies from sample to sample. The more clumped and more spread out patterns start from a random shuffle, then swap squares one pair at a time until neighbor alternation reaches the target setting.
Research behind this
The math is exact: the grids, counts, and neighbor-alternation measure are defined above. The experimental findings are about how people judge random patterns: people tend to expect and prefer more alternation than pure chance actually produces, and to rate overly mixed patterns as especially random. The wider interpretation is more tentative. Events that cluster can feel connected by cause, but this exhibit doesn't show that misjudging chance explains any particular belief.
- Falk, R., & Konold, C. (1997). Making sense of randomness: Implicit encoding as a basis for judgment. Psychological Review, 104(2), 301–318. PDF
- Wilke, A., Minich, S., Panis, M., & Langen, T. A. (2015). A game of hide and seek: Expectations of clumpy resources influence hiding and searching patterns. PLOS ONE, 10(7), e0130976. doi:10.1371/journal.pone.0130976
- Bertamini, M., Zito, M., Scott-Samuel, N. E., & Hulleman, J. (2016). Spatial clustering and its effect on perceived clustering, numerosity, and dispersion. Attention, Perception, & Psychophysics, 78, 1460–1471. doi:10.3758/s13414-016-1100-0
- van der Wal, R. C., Sutton, R. M., Lange, J., & Braga, J. P. N. (2018). Suspicious binds: Conspiracy thinking and tenuous perceptions of causal connections between co-occurring and spuriously correlated events. European Journal of Social Psychology, 48(7), 970–989. doi:10.1002/ejsp.2507
Further reading: The Clustering Illusion on Wikipedia.