Math 2.0

Statistics Definition

Skewness

Builds on Central momentStandard deviation

Definition

Let XX be a real random variable with E[∣X∣3]<∞E[|X|^3] < \infty, mean μ\mu and standard deviation σ>0\sigma > 0. The skewness of XX is γ1=E[(X−μσ)3]=μ3σ3,\gamma_1 = E\left[\left(\frac{X - \mu}{\sigma}\right)^3\right] = \frac{\mu_3}{\sigma^3}, where μ3\mu_3 is the third central moment.

Remarks

A distribution that is symmetric about its mean has skewness 00. The converse is false. Positive skewness usually means a longer tail on the right. Negative skewness usually means a longer tail on the left. The skewness does not change under X↦aX+bX \mapsto aX + b with a>0a > 0. For a<0a < 0, it changes sign. The exponential distribution has skewness 22.

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. Injective function definition
  37. Preimage of a set definition
  38. Sigma-algebra definition
  39. Countable set definition
  40. Event definition
  41. Borel sigma-algebra definition
  42. Probability axioms concept
  43. Probability measure definition
  44. Probability distribution definition
  45. Probability space definition
  46. Random variable definition
  47. Cumulative distribution function definition
  48. Discrete random variable definition
  49. Law of a random variable definition
  50. Random variable with a density definition
  51. Probability mass function definition
  52. Expected value definition
  53. Probability density function definition
  54. Variance definition
  55. Law of the unconscious statistician statement
  56. Moment definition
  57. Standard deviation definition
  58. Central moment definition
  59. Skewness definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Skewness, Fisher's moment coefficient of skewness
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Kinddefinition
WritingW1 stated
MSC62E10
DefinesSkewness
UsesCentral moment, Standard deviation

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