An extra-ordinary claim might be considered something that does not fit in with the every day observance of whomever you are talking to.
So, Lets assume a person who has only ever lived on a small island occupied by only small indigenous birds.
Telling this individual that there is such a thing as a blue whale would be an incredible claim, because according to his knowledge there are no animals even close to that size or scope. An animal larger than several of my lean-to shelter, lined up end to end? That's rediculous!
But extraordinary evidence for this creature is plentiful and available. Evidence which fully confirms what might otherwise be considered an unbelievable claim. For instance, you could take this guy diving and put him right next to a blue whale.
Extraordinary claim, meet extraordinary evidence.
Before the early 1900's it was roundly asserted that heavier than air flight was impossible. Many claimed that it was, however, and that was greated as an extraordinary claim. How could a vehicle which was many tons ever fly through the air?
Extra-ordinary proof came by way of the aeronautic industry where early pioneers achieved lift off of objects which were much heavier than air. Now, indeed, C-747 and other very heavy craft regularly traverse oceans on nothing but the wind.
Or synesthesia.
Some people claim that they see colors and patterns which nobody else sees. They say that they can actually see sounds. This is an extra ordinary claim. Or that text clearly printed in black ink is in fact colored differently.
This is an extraordinary claim which would usually be dismissed as delusion out of hand, but there is concrete evidence in support of it in the form of neuroimagery and through repeated testing which demonstrates the reality of this mis-handling of labels that takes place in the brain.
http://en.wikipedia.org/wiki/Synesthesia
Synesthesia is hard to fake, and easy to prove as a genuine perception. The simplest approach is test-retest reliability over long periods of time, where synesthetes consistently score much higher—around 90% after years, compared to 30–40% after just a month in non-synesthetes even when they are warned they will be retested—using stimuli of color names, color chips, or a computer-screen color picker providing 16.7 million choices.[1][43]
The automaticity of synesthetic experience. The panel on the left is how a non-synesthete perceives the matrix, while a given synesthete might perceive it like the panel on the right.[28]Modified versions of the Stroop effect are popular. In the standard paradigm, it is harder to name the ink color of the word "red," for example, when it is printed in blue ink than when the ink is red. Similarly, if a grapheme → color synesthete is shown the digit 4 (which he sees as red, say) in blue ink, he is slower to name the ink color than when it is printed in red. He sees the blue ink, but the same sort of conflict responsible for the standard Stroop effect occurs between the ink color and the automatic synesthetic color of the grapheme. The conflict is strongest when the ink color is the opponent color to the synesthetic one (e.g., red vs. green), indicating that synesthetic color perception uses the same mechanism as the perception of real colors.[49]
Cross-sensory Stroop tests are possible: for example, a music → color synesthete must name a red swatch while listening to a sound that produces a blue sensation,[50] or a musical key → taste synesthete must identify a bitter taste while hearing a musical interval that tastes sweet .[51] Likewise, Stroop tests work even in those for whom merely thinking about a numeral elicits color. Take a person who sees 7 as yellow and 9 as blue, and make the task one of having to say a math solution out loud followed by naming a color square. In the illustration, having to answer “7” and then “yellow” is congruent with the subject’s synesthesia, which unconsciously primes him to respond faster than controls. The automatic blueness of 9, however, interferes with naming the green square, slowing him down compared to controls.
Synesthetic colors can also improve performance for some synesthetes. Inspired by tests for color blindness, Ramachandran and Hubbard presented synesthetes and non-synesthetes with a matrix of 5s in which embedded 2s formed a hidden pattern such as a square, diamond, rectangle or triangle.[28] For someone who sees 2s as red and 5s as green, for example, synesthetic colors help zero in on the embedded figure. Subsequent careful studies have found substantial variability among synesthetes in their ability to do this.[15][24] It certainly does not happen instantaneously; while synesthesia is evoked very early in perceptual processing, it does not occur prior to attention.[52][53]
In short, extra-ordinary evidence would be evidence which is irrefutable and in total acordance with the extraordinary claim. The more extraordinary the claim, the more definite, convincing and irrefutable must be the proof.
Of which you have presented not one example.
