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What are some real-world applications where I can leverage Chebychev's inqueality? All the examples I find are either related to coin tosses or some school scores related problems. Are there any slightly-more complex use cases where this inequality "saves" the day?

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I once convinced a very tall woman to have dinner with me by using Chebychev's inequality to argue that her high partner height threshold would lead to many lonely nights spent with her cat. Alcohol was involved, so arithmetic mistakes were made that exaggerated the main thrust of the conclusion.

It didn't last.

In this case, the three-sigma rule would have served me better since heights are actually normal and the rule gives a tighter bound. Now, if she had an income threshold, this would have been a better anecdote.

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To steal a pun from xkcd, she did not become your statistically significant other? – Dilip Sarwate Dec 1 '12 at 2:47
What is your evidence that heights are normal? I usually give height as an example of a statistic which clearly isn't normal. – Douglas Zare Dec 1 '12 at 12:57
Since height is likely caused by many small genetic and environmental factors, treating it as normal is a reasonable approximation, especially near the center of the distribution, and if we restrict the sample to men of a certain age. In the tails, this approximation obviously breaks down. My naive CLT argument is probably wrong since things are not iid. – Dimitriy V. Masterov Dec 1 '12 at 18:43

While it's most often used for establishing bounds in various things, here's an example of it being used for constructing intervals for a real problem:

(It's not actually necessary for this problem; tighter bounds can be obtained.)

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It's not possible to tell what that paper does unless one has full text access to it. Could you at least describe its method? (I am sceptical of tests developed from Chebyshev's Inequality because I have seen at least one that is conceptually--and practically--wrong. It is difficult to use it for inference and when used correctly can be expected to give poor statistical procedures, because it is such a crude bound.) – whuber Dec 10 '12 at 22:28
Thanks, Glen. A quick look at this paper suggests it is making the same mistake of assuming Chebeyshev's Inequality applies when the true mean and variance are replaced by estimates. This is not true (as some thought would suggest and a lot of simulations have demonstrated). If I am misunderstanding it, I would be delighted to be corrected. – whuber Dec 11 '12 at 0:08
It explicitly incorporates the estimation uncertainty in the procedure. Its main problem is the intervals are much too wide; given the assumptions more precise intervals are easily generated. – Glen_b Dec 11 '12 at 2:02

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