Yates was born in Manchester, the eldest of five children (and only son) of seed merchant Percy Yates and his wife Edith. He attended Wadham House, a private school, before gaining a scholarship to Clifton College in 1916. In 1920 he obtained a scholarship at St John's College, Cambridge and four years later graduated with a First Class Honours degree.
He spent two years teaching mathematics to secondary school pupils at Malvern College before heading to Africa where he was mathematical advisor on the Gold Coast Survey. He returned to England due to ill health and met and married a chemist, Margaret Forsythe Marsden, the daughter of a civil servant. This marriage was dissolved in 1933 and he later married Prascovie (Pauline) Tchitchkine, previously the partner of Alexis Tchitchkine. After her death in 1976 he married Ruth Hunt, his long-time secretary.
In 1931 Yates was appointed assistant statistician at Rothamsted Experimental Station by R.A. Fisher. In 1933 he became head of statistics when Fisher went to University College London. At Rothamsted he worked on the design of experiments, including contributions to the theory of analysis of variance and originating Yates's algorithm and the balanced incomplete block design.
After the war he worked on sample survey design and analysis. He became an enthusiast of electronic computers, in 1954 obtaining an Elliott 401 for Rothamsted and contributing to the initial development of statistical computing. In 1960 he was awarded the Guy Medal in Gold of the Royal Statistical Society, and in 1966 he was awarded the Royal Medal of the Royal Society. He retired from Rothamsted to become a Senior Research Fellow at Imperial College London. He died in 1994, aged 92, in Harpenden.
- The design and analysis of factorial experiments, Technical Communication no. 35 of the Commonwealth Bureau of Soils (1937) (alternatively attributed to the Imperial Bureau of Soil Science).
- Statistical tables for biological, agricultural and medical research (1938, coauthor R.A. Fisher) sixth edition
- Sampling methods for censuses and surveys (1949)
- Computer programs GENFAC, RGSP, Fitquan.
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- O'Connor, John J.; Robertson, Edmund F., "Frank Yates", MacTutor History of Mathematics archive, University of St Andrews<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
- Neider, John (1994), "Obituary: Frank Yates: 1902–1994", The IMS Bulletin, 23 (5): 529–530<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.