Estimating Sample Size - Powertail

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Probabilistic statement

Let X (integrable) be a random variable with finite expected value μ and finite non-zero variance σ2. Then for any real number Template:Nowrap,

Only the case is useful. When the right-hand side and the inequality is trivial as all probabilities are ≤ 1.

As an example, using shows that the probability that values lie outside the interval does not exceed .

Because it can be applied to completely arbitrary distributions provided they have a known finite mean and variance, the inequality generally gives a poor bound compared to what might be deduced if more aspects are known about the distribution involved.

k Min. % within k standardTemplate:Ns
deviations of mean
Max. % beyond k standard
deviations from mean
1 0% 100%
Template:Sqrt 50% 50%
1.5 55.56% 44.44%
2 75% 25%
2Template:Sqrt 87.5% 12.5%
3 88.8889% 11.1111%
4 93.75% 6.25%
5 96% 4%
6 97.2222% 2.7778%
7 97.9592% 2.0408%
8 98.4375% 1.5625%
9 98.7654% 1.2346%
10 99% 1%

Central limit theorem on variance

combine

Use Feller reference for kappa