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Margaret Hilda Roberts dilahirkan pada 13 Oktober 1925, di Grantham, Lincolnshire. Her parents were Alfred Roberts (1892–1970), from Northamptonshire, and Beatrice Ethel Stephenson (1888–1960), from Lincolnshire. Her father's maternal grandmother, Catherine Sullivan, was born in County Kerry, Ireland.

Roberts spent her childhood in Grantham, where her father owned a tobacconist's and a grocery shop. In 1938, before the Second World War, the Roberts family briefly gave sanctuary to a teenage Jewish girl who had escaped Nazi Germany. With her  elder sister Muriel, Margaret saved pocket money to help pay for the teenager's journey.

Alfred was an alderman and a Methodist local preacher. He brought up his daughter as a strict Wesleyan Methodist, attending the Finkin Street Methodist Church, but Margaret was more sceptical; the future scientist told a friend that she could not believe in angels, having calculated that they needed a breastbone  long to support wings. Alfred came from a Liberal family but stood (as was then customary in local government) as an Independent. He served as Mayor of Grantham in 1945–46 and lost his position as alderman in 1952 after the Labour Party won its first majority on Grantham Council in 1950.

Roberts attended Huntingtower Road Primary School and won a scholarship to Kesteven and Grantham Girls' School, a grammar school. Her school reports showed hard work and continual improvement; her extracurricular activities included the piano, field hockey, poetry recitals, swimming and walking. She was head girl in 1942–43, and outside school, while the Second World War was ongoing, she voluntarily worked as a fire watcher in the local ARP service. Other students thought of Roberts as the "star scientist", although mistaken advice regarding cleaning ink from parquetry almost caused chlorine gas poisoning. In her upper sixth year Roberts was accepted for a scholarship to study chemistry at Somerville College, Oxford, a women's college, starting in 1944. After another candidate withdrew, Roberts entered Oxford in October 1943.