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I haven't done very much statistics but in a Finnish programming forum I found a some exercises where answers are integers. In every exercise the best result $r$ gets 100 points. Suppose you get the result $s$. If in a given problem the best score is as high as possible, you will be awarded by $100(s/r)^2$ points rounded to the nearest integer. If in a given problem the best score is as small as possible, you will be awarded by $100(r/s)^2$ points rounded to the nearest integer.

Now I have a table of the best scores where top contestants gets many scores $100$ for single problem, some gets lower scores, some haven't done a problem or send any valid solution in which case the score is empty.

What kind of methods there are to approximate how difficult a single problem have been?

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"If in a given problem the best score is as small as possible" I don't understand that. Perhaps an example? –  leonbloy Jan 19 '11 at 17:51
    
So in some problems the best score is the highest score, and in others the best score is the smallest one? –  PEV Jan 19 '11 at 19:00
    
@PEV: That's right. There are some minimization problems where the lowest integer value is the best but there are also some problems where one has to maximize something. –  Jaakko Seppälä Jan 19 '11 at 21:55
    
Good question! It deserves a high score in difficulty! :) –  user37153 Aug 3 '12 at 14:46

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