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Bigge was the son of John Frederic Bigge (1814–1885) Vicar of Stamfordham, Northumberland and the grandson of Charles William Bigge (1773–1849) of Benton House, Little Benton, Newcastle on Tyne and Linden Hall, Longhorsley, Northumberland, High Sheriff of Northumberland and a prominent merchant and banker in Newcastle on Tyne.