Norman Alexander was born in Mangapiko, New Zealand, and graduated from Auckland University with first class honours in physics in 1927. He won a two-year scholarship to Cambridge University’s Cavendish Laboratory in 1930 and worked as Demonstrator in Physics at King's College, London, while completing his doctorate. In 1936 he was appointed Professor of Physics at Raffles College in Singapore, where one of his research projects involved assisting the Royal Navy in establishing a network of radio direction-finding stations. Alexander was interned at Changi Gaol and Sime Road civilian internment camps following the fall of Singapore to Japan in 1942. In Changi, he contributed to the construction of a salt evaporation plant and the production of medical supplies for the camp hospital.

Alexander returned to Raffles College after the end of the Second World War, then in 1952, became Professor of Physics at Nigeria’s University College, Ibadan, an internal college of London University (later Ibadan University after Nigeria's independence in 1960). Alexander and the department conducted significant magnetic and ionospheric geophysical research. In 1958, he accepted a post at the Middle East Technical University, in Ankara, to teach engineering physics, but a year later was appointed the first Vice-Chancellor of Ahmadu Bello University, Northern Nigeria, where he remained until 1966. From this time, Alexander offered advice and support for the establishment of universities within the Commonwealth, including the University of the West lindies in 1966 and the University of the South Pacific in Fiji (1966-68). In 1970, as an Advisor to the Ministry of Overseas Development, Alexander authored a report on the development of a federal university for Botswana, Lesotho and Swaziland, combining higher education with vocational and teacher training. In 1971, he joined the Governing Body of the School of Oriental and African Studies, from which he retired in 1973.