Finite Samples
Saw et al extended Chebyshev's inequality to cases where the population mean and variance are not known but are instead replaced by their sample estimates.
where N is the sample size, m is the sample mean, k is a constant and s is the sample standard deviation. g(x) is defined as follows:
Let x ≥ 1, Q = N + 1, and R be the greatest integer less than Q / x. Let
Now
This inequality holds when the population moments do not exist and when the sample is weakly exchangeably distributed.
Kabán gives a somewhat less complex version of this inequality.
If the standard deviation is a multiple of the mean then a further inequality can be derived,
A table of values for the Saw–Yang–Mo inequality for finite sample sizes (n < 100) has been determined by Konijn.
For fixed N and large m the Saw–Yang–Mo inequality is approximately
Beasley et al have suggested a modification of this inequality
In empirical testing this modification is conservative but appears to have low statistical power. Its theoretical basis currently remains unexplored.
Read more about this topic: Chebyshev's Inequality
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