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Major Greenwood was born in 1880 and was the third generation and only surviving son in a family of East End General Practitioners. He was expected to follow suit, but was rescued for medical research by the physiologist Sir Leonard Hill, father of Bradford Hill. Trained in the laboratories of Hill; instructed in biometry and statistics by Karl Pearson, Greenwood developed Karl Pearson's rigorous mathematical logic in a way which made medical statistics acceptable to a previously hostile and uncomprehending medical profession.
Greenwood became a medical statistician to the Lister Institute, 1910, where he published numerous studies which added to his fame, among others, with his friend Arthur Bacot, on the epidemiology of plague in India. He was then called during World War One to the medical research subsection of the Ministry of Munitions and became immersed in industrial problems. After the end of war, working for the Medical Research Council, he was appointed first senior medical statistician to the new (1919) Ministry of Health with Sir George Newman. Having already collaborated with WWC Topley on Medical Research Council sponsored studies in experimental epidemiology, their collaboration continued when, in 1927, both men were appointed to new chairs in the new London School of Hygiene and Tropical Medicine. Greenwood was appointed Professor of Epidemiology and Vital Statistics, a post which he held until his retirement in 1945. When Brig. Parkinson was recalled to service in 1943, Greenwood stood in and carried out the onerous duties of the Dean of the School until his successor could be appointed.
He was the Milroy Lecturer at the Royal College of Physicians in 1922, received the Buchanan Medal of the Royal Society in 1927 and was a Gold Medallist of the Royal Statistical Society. He died very suddenly in October 1949.