The Islamophobic right has long repeated an extremely dubious statistic that "80 percent" of American mosques are radicalized. Media Matters found the original source for the quote, which appears to originate from a 1999 statement made by Shaykh Muhammad Hisham Kabbani. While the remark is absurd on its face, Kabbani made no effort to substantiate it.
Republicans have nevertheless attempted to give the statistic the appearance of fact through ceaseless repetition, and to this day it has been cited by Rep. Peter King as a justification for his Muslim HUAC hearings. Alongside a single document from the Holy Land Foundation case, it is the primary piece of "evidence" for the assertion that the American Muslim community is on board with radical groups secretly waging a "stealth jihad" to impose Taliban-style Sharia law through the use of decoy Muslims.
But the claim is not based on any academic research. It's not even based on evidence. It's literally just a matter of something someone said once, a claim literally pulled out of thin air. Because conservatives believed it was true, they've asserted it as fact, without the slightest effort at verification, for more than a decade.
Frank Gaffney put forth the statistic in his "Sharia: The Threat To America" report given to Congress, which in turn contained a single footnote. The footnote reads:
It's like something out of a Lewis Carroll poem. They said it thrice, it must be true.