He was born in Carrickfergus, Ireland, but left Ireland after the failure of the uprising of the United Irishmen in 1798 and moved to Princeton, New Jersey. He taught mathematics at various schools in the United States.
He is chiefly remembered for his formulation of the method of least squares, published in 1808. Adrain certainly did not know of the work of C.F. Gauss on least squares (published 1809), although it is possible that he had read A.M. Legendre's article on the topic (published 1804).
Adrain, Gauss, and Legendre all motivated the method of least squares by the problem of reconciling disparate physical measurements; in the case of Gauss and Legendre, the measurements in question were astronomical, and in Adrain's case they were survey measurements.
- Robert Adrain. "Research concerning the probabilities of the errors which happen in making observations, &c". The Analyst, or Mathematical Museum. Vol. I, Article XIV, pp 93-109. Philadelphia: William P. Farrand and Co., 1808.
- Brian Hayes. "Science on the Farther Shore". American Scientist, 90(6):499, 2002. (Article may be viewed at: http://www.americanscientist.org/.)
- Stephen M. Stigler. "Mathematical statistics in the early States". Annals of Statistics, 6:239–265, 1978.