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Samples getting 6 standard deviations away from the mean can happen 1/6² ≈ 2.7% of the time. You might be assuming properties of the distribution that aren't true.

https://en.wikipedia.org/wiki/Chebyshev%27s_inequality



Isn't that an upper bound? It says that it happens at most 2.7% of the time


Yea, it's an upper bound. The actual percentage depends on the distribution and can be anywhere from 0-2.7%. The Wikipedia link has more details.




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