Math 2.0

Probability theory and stochastic processes Example

68–95–99.7 rule

Builds on Normal distributionStandard deviationStandard score

Example

Let X∼𝒩(μ,σ2)X \sim \mathcal{N}(\mu, \sigma^2) with σ>0\sigma > 0, and let Φ\Phi be the cumulative distribution function of 𝒩(0,1)\mathcal{N}(0, 1). Since Z=(X−μ)/σ∼𝒩(0,1)Z = (X - \mu)/\sigma \sim \mathcal{N}(0, 1), P(∣X−μ∣≤kσ)=2Φ(k)−1P(|X - \mu| \le k\sigma) = 2\Phi(k) - 1 for each k>0k > 0. Numerical values of Φ\Phi give: P(∣X−μ∣≤σ)≈0.6827,P(∣X−μ∣≤2σ)≈0.9545,P(∣X−μ∣≤3σ)≈0.9973.P(|X - \mu| \le \sigma) \approx 0.6827, \quad P(|X - \mu| \le 2\sigma) \approx 0.9545, \quad P(|X - \mu| \le 3\sigma) \approx 0.9973. So about 68%68\%, 95%95\% and 99.7%99.7\% of the mass of a normal distribution lie within one, two and three standard deviations of the mean.

Remarks

The rule holds for normal distributions only. For an arbitrary distribution with finite variance, Chebyshev's inequality gives only P(∣X−μ∣≤kσ)≥1−1/k2P(|X - \mu| \le k\sigma) \ge 1 - 1/k^2, for example 3/43/4 for k=2k = 2. The interval μ±1.96σ\mu \pm 1.96\sigma contains 95%95\% of the mass. It is the base of the usual 95%95\% confidence intervals.

Reading path

To understand this page from the ground up, read these 59 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Real numbers object
  9. Set bounded above definition
  10. Set definition
  11. Ring of integers statement
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Supremum definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. The rationals form a field statement
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Completeness axiom statement
  23. Cartesian product definition
  24. Subset definition
  25. The rationals form an ordered field statement
  26. Uniqueness quantifier definition
  27. Existence of square roots of nonnegative reals statement
  28. Binary relation definition
  29. Universal set definition
  30. Order on the reals definition
  31. Power set definition
  32. Square root definition
  33. Complement definition
  34. Function definition
  35. Interval definition
  36. Indicator function definition
  37. Injective function definition
  38. Preimage of a set definition
  39. Sigma-algebra definition
  40. Countable set definition
  41. Event definition
  42. Borel sigma-algebra definition
  43. Probability axioms concept
  44. Probability measure definition
  45. Dirac measure definition
  46. Probability distribution definition
  47. Probability space definition
  48. Random variable definition
  49. Discrete random variable definition
  50. Standard score definition
  51. Cumulative distribution function definition
  52. Random variable with a density definition
  53. Expected value definition
  54. Probability density function definition
  55. Variance definition
  56. Density of the normal distribution definition
  57. Normal distribution definition
  58. Standard deviation definition
  59. 68–95–99.7 rule example

Further reading

  1. Web https://en.wikipedia.org/wiki/68–95–99.7_rule, Table of numerical values
About this page
Tag0EYQ03XCR3
URIm2://0EYQ03XCR3
Aliasesempirical-rule, three-sigma-rule
Kindexample
WritingW0 stub
MSC60E05
Instance ofNormal distribution
UsesNormal distribution, Standard deviation, Standard score
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