Estimating Sample Size - Powertail

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,

$\Pr(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}.$

Only the case $k>1$ is useful. When $k\leq 1$ the right-hand side ${\frac {1}{k^{2}}}\geq 1$ and the inequality is trivial as all probabilities are ≤ 1.

As an example, using $k={\sqrt {2}}$ shows that the probability that values lie outside the interval $(\mu -{\sqrt {2}}\sigma ,\mu +{\sqrt {2}}\sigma )$ does not exceed ${\frac {1}{2}}$.

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