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MrHen comments on A Request for Open Problems - Less Wrong
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If exactly half of all men have a height less than the central value c, than randomly picking sample will have a 50% chance of being below c. Picking two samples (A and B) results in four possible scenarios:
The interval created by (A, B) contains c in scenarios (1) and (4) and does not contain c in scenarios (2) and (3). Since each scenario has an equal chance of occurring, c is in (A, B) 50% of the time.
That is as far as I got just thinking about it. If I am on the right path I can keep plugging away.
In the Gaussian case, you can do better than (A, B) but the demonstration of that fact won't smack you in the face they way it does in the case of the uniform distribution.
One thing you can do in the uniform case is shorten the interval to at most length 1/2. Not sure if that's face-smacking enough.
You can do better than that. If the distance between the two data points is 7/4, you can shrink the 100% confidence interval to 1/4, etc. (The extreme case is as the distance between the two data points approaches 2, your 100% confidence interval approaches size zero.)
EDIT: whoops, I was stupid. Corrected 3/4 to 7/4 and 1 to 2. There, now it should be right
Do we know the heights of the men A and B? If so, we can get a better estimate of whether c lies between their heights by taking into account the difference between A and B...
That's the basic idea. Now apply it in the case of the uniform distribution.
If all men are (say) within 10 cm of each other, and the heights are uniformly distributed...
... if we have two men who are 8 cm apart, then c lies between their heights with 80% probability?
Getting there... 80% is too low.
Wait, what? It must be 100%...
That's it. The so-called 50% confidence interval sometimes contains c with certainty. Also, when x_max - x_min is much smaller than 0.5, 50% is a lousy summary of the confidence (ETA: common usage confidence, not frequentist confidence) that c lies between them.
If it's less than 0.5, is the confidence simply that value times 2?
"Confidence" in the statistics sense doesn't always have much to do with how confident you are in the conclusion. Something that's the real line in half of all cases and the empty set in the other half of all cases is a 50% confidence interval, but that doesn't mean you're ever 50% confident (in the colloquial sense) that the parameter is on the real line or that the parameter is in the empty set.
Whoops -- "confidence" is frequentist jargon. I'll just say that any better method ought to take x_max - x_min into account.