"Extraordinary claims demand extraordinary evidence."

Often attributed to Carl Sagan, but he was paraphrasing sceptic Marcello Truzzi. Doubtless the concept is even more ancient.

David Hume said, "A wise man, therefore, proportions his belief to the evidence".

One could argue this is not a quote about statistics. However, applied statistics is ultimately in the business of evaluating the quality of evidence for or against some proposition.