Math 2.0

Statistics Definition

Sample mean

Builds on StatisticIndependent and identically distributed random variablesLinearity of expectationVariance of a sum of independent random variables

Definition

Let X1,…,XnX_1, \dots, X_n be real random variables, for example an i.i.d. sample. The sample mean is the statistic Xˉn=1n∑i=1nXi.\bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i. For observed values x1,…,xnx_1, \dots, x_n, the sample mean is the number xˉ=(x1+⋯+xn)/n\bar{x} = (x_1 + \dots + x_n)/n.

Remarks

If the XiX_i are i.i.d. with mean μ\mu, then E[Xˉn]=μE[\bar{X}_n] = \mu by linearity of expectation. So Xˉn\bar{X}_n is an unbiased estimator of μ\mu. If also Var⁡(Xi)=σ2<∞\operatorname{Var}(X_i) = \sigma^2 < \infty, then Var⁡(Xˉn)=σ2/n\operatorname{Var}(\bar{X}_n) = \sigma^2 / n. For i.i.d. 𝒩(μ,σ2)\mathcal{N}(\mu, \sigma^2) variables, Xˉn∼𝒩(μ,σ2/n)\bar{X}_n \sim \mathcal{N}(\mu, \sigma^2/n). By the law of large numbers, Xˉn→μ\bar{X}_n \to \mu almost surely.

Reading path

To understand this page from the ground up, read these 65 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 definition
  10. Ring of integers statement
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Ordered pair definition
  15. Union definition
  16. Sample space definition
  17. The rationals form a field statement
  18. Existential quantifier definition
  19. Universal quantifier definition
  20. Cartesian product definition
  21. Subset definition
  22. The rationals form an ordered field statement
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Order on the reals definition
  27. Power set definition
  28. Complement definition
  29. Function definition
  30. Interval definition
  31. Injective function definition
  32. Preimage of a set definition
  33. Sigma-algebra definition
  34. Countable set definition
  35. Event definition
  36. Borel sigma-algebra definition
  37. Probability axioms concept
  38. Probability measure definition
  39. Probability distribution definition
  40. Probability space definition
  41. Conditional probability definition
  42. Random variable definition
  43. Discrete random variable definition
  44. Law of a random variable definition
  45. Independence of two events definition
  46. Random variable with a density definition
  47. Joint probability distribution definition
  48. Identically distributed random variables definition
  49. Expected value definition
  50. Mutual independence of events definition
  51. Independence of two random variables definition
  52. Marginal distribution definition
  53. Variance definition
  54. Linearity of expectation statement
  55. Independent sigma-algebras definition
  56. Independence means the joint law is the product of the marginals statement
  57. Covariance definition
  58. Independence of random variables definition
  59. Variance of a sum statement
  60. Independent and identically distributed random variables definition
  61. Expectation of a product of independent random variables statement
  62. Statistic definition
  63. Independent random variables are uncorrelated statement
  64. Variance of a sum of independent random variables statement
  65. Sample mean definition

Where this leads

44 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Sample_mean_and_covariance, Definition of the sample mean
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Kinddefinition
WritingW1 stated
MSC62F10
DefinesSample mean
UsesStatistic, Independent and identically distributed random variables, Linearity of expectation, Variance of a sum of independent random variables

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