For the verb meaning to grant authority or to give permission, authorize is the standard spelling in American and Canadian English. Authorise is standard in all main varieties of English outside North America.
The distinction extends to all derivative words. North Americans use authorized, authorizing, authorizes, and authorization, while English speakers from outside the U.S. and Canada use authorised, authorising, authorises, and authorisation.
Authorize is the older form, and it was standard even in British English until the second half of the 20th century. It is in the group of Classically derived words that traditionally took the -ize ending but have recently acquired -ise.
The change is not wholly engrained, though. The Oxford English Dictionary, which usually favors British or UK spellings, still lists authorize as the primary spelling,1 and some British publishing houses also favor the old spelling.
In 21st-century edited British news publications, however, authorise appears about seven times for every instance of authorize, and authorize is even less common in 21st-century Australian and New Zealand publications.
These are North American publications:
On Thursday Turkey’s parliament authorized military action against the Bashar Assad regime to its south. [Wall Street Journal]
The new framework … will provide a national strategy for authorizing and monitoring the use of orphan drugs. [Globe and Mail]
The United States has been detaining terrorism suspects indefinitely since 2001, relying on an authorization by Congress to use military force against perpetrators of the Sept. 11 attacks. [New York Times]
The committee voted Monday to ask council to authorize the aging castle’s board to undertake a request for proposals. [National Post]
And these are from outside North America (British, UK, Australian, etc.):
It authorises police to impound Pakistani newspapers in the three eastern provinces of Nangarhar, Kunar and Nuristan. [BBC News]
Mary Robinson’s life and achievements were vividly described in the authorised biography by Olivia O’Leary and Helen Burke in 1998. [Irish Times]
Virgin Australia has urged the competition watchdog to refuse a Qantas request for interim authorisation of its alliance with Emirates. [The Australian]
He said the UN Security Council had allowed itself to be “abused” last year by authorising “all necessary measures.” [Independent Online]
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