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,
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