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
Famous quotes containing the words finite and/or samples:
“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (1623–1662)
“Good government cannot be found on the bargain-counter. We have seen samples of bargain-counter government in the past when low tax rates were secured by increasing the bonded debt for current expenses or refusing to keep our institutions up to the standard in repairs, extensions, equipment, and accommodations. I refuse, and the Republican Party refuses, to endorse that method of sham and shoddy economy.”
—Calvin Coolidge (1872–1933)