The experience of seeing meaningful patterns or connections in random or meaningless data.
In statistics, apophenia is a type I error (false positive, false alarm, caused by an excess in sensitivity). It is also used as an explanation of paranormal and religious claims, and a belief in pseudo-science.
The term was coined in 1958 by Klaus Conrad who originally described this phenomenon in relation to the distortion of reality present in psychosis, but it has become more widely used to describe this tendency in healthy individuals without necessarily implying the presence of mental illness.
(Thanks to Prof. Ian Young for the word!)