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According to Wikipedia, P value is defined as

the probability that data at least as surprising as the observed sample results would be generated under a model of random chance

Why is it stated this way, and what's the difference between the previous statement and the following one?

the probability that data more surprising than the observed sample results would be generated under a model of random chance

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  • Let X be "the observed sample results would be generated under a model of random chance". Then your later formulation is defining P>X but the former, original, formulation is defining P>=X. Another way of stating the original is P is no less surprising than X, or restated in logical terms similar to the preceding, P!<X.
    – Dan Bron
    Commented Apr 13, 2015 at 15:44
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    Is surprisingness accurately quantifiable? Commented Apr 13, 2015 at 15:47

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I think the answer must be that mathematically it is greater than or equal to rather than greater than. If the mathematical test were greater than, then greater than could be used. In the case 'at least as' implies that it could be equal to or greater than.

This seems a way in english of saying something indirectly a bit like a double negative, but then again I may well be wrong about that.

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  • just to be clear, is "greater than or equal to" interchangeable with "at least as extreme as"?
    – Alby
    Commented Apr 13, 2015 at 19:33
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    @Alby Yes, or "no less extreme than".
    – Dan Bron
    Commented Apr 13, 2015 at 21:36
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They are very nearly the same. However, if the probability from chance is exactly as "surprising" as the observed results (that original statement is really contorted) then the first statement would be true and the second one false. (But, of course, in any real-life example the numbers are not exact integers and the condition of exact equality would never occur.)

But the contorted phrasing of the statements nearly renders things unintelligible anyway.

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