He was born in Portland, Oregon, the son of attorney Lewis Cox and Elinor Cox. After Lewis Cox died, Elinor Cox married John Latané, who became a professor at Johns Hopkins University in 1913. In 1915 Richard enrolled at Johns Hopkins University to study physics, but his studies were cut short when he was drafted for World War I. He stayed in the US after being drafted and returned to Johns Hopkins University after the war, completing his BA in 1920. He earned his PhD in 1924; his dissertation was A Study of Pfund's Pressure Gauge.
He taught at New York University (NYU) from 1924 to 1943, before returning to JHU to teach. He studied probability theory, the scattering of electrons, and the discharges of electric eels. Richard Cox's most important work was Cox's theorem.
His wife, Shelby Shackleford (1899 Halifax, Virginia – 1987), whom he married in 1926, was an accomplished artist and illustrated Electric Eel Calling, a book on electric eels.