Math 2.0

Statistics Hypothesis testing Concept

Chi-squared test

Builds on Statistical hypothesis testChi-squared distributionIndependence of random variables

Summary

A chi-squared test is a hypothesis test whose test statistic has approximately a chi-squared distribution under the null hypothesis. The main example is Pearson's chi-squared test for counts in categories. For observed counts O1,…,OkO_1, \dots, O_k and expected counts E1,…,EkE_1, \dots, E_k under H0H_0, Pearson's statistic is χ2=∑i=1k(Oi−Ei)2Ei.\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}. One rejects H0H_0 when χ2\chi^2 is large.

Remarks

In the goodness-of-fit test, H0H_0 fixes the probabilities of the kk categories. For a large sample, χ2\chi^2 has approximately the law χk−12\chi^2_{k - 1} under H0H_0. In the test of independence for an r×cr \times c contingency table, H0H_0 states that the two categorical variables are independent. Then EijE_{ij} is (row total ×\times column total) / nn, and the law is approximately χ(r−1)(c−1)2\chi^2_{(r-1)(c-1)}. The approximation is poor when expected counts are small. A common rule asks for Ei≥5E_i \ge 5 in each cell. For small tables, Fisher's exact test is an alternative. Karl Pearson introduced the test in 1900.

Reading path

To understand this page from the ground up, read these 79 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. Indicator function definition
  32. Injective function definition
  33. Preimage of a set definition
  34. Sigma-algebra definition
  35. Countable set definition
  36. Event definition
  37. Borel sigma-algebra definition
  38. Probability axioms concept
  39. Probability measure definition
  40. Dirac measure definition
  41. Probability distribution definition
  42. Probability space definition
  43. Conditional probability definition
  44. Random variable definition
  45. Discrete random variable definition
  46. Cumulative distribution function definition
  47. Independence of two events definition
  48. Law of a random variable definition
  49. Random variable with a density definition
  50. Joint probability distribution definition
  51. Independence of two random variables definition
  52. Mutual independence of events definition
  53. Expected value definition
  54. Probability density function definition
  55. Identically distributed random variables definition
  56. Marginal distribution definition
  57. Variance definition
  58. Independent sigma-algebras definition
  59. Linearity of expectation statement
  60. Density of the normal distribution definition
  61. Exponential distribution definition
  62. Independence means the joint law is the product of the marginals statement
  63. Independence of random variables definition
  64. Covariance definition
  65. Normal distribution definition
  66. Gamma distribution definition
  67. Expectation of a product of independent random variables statement
  68. Variance of a sum statement
  69. Independent and identically distributed random variables definition
  70. Independent random variables are uncorrelated statement
  71. Statistic definition
  72. Variance of a sum of independent random variables statement
  73. Statistical hypothesis test concept
  74. Estimator definition
  75. Sample mean definition
  76. Bias of an estimator definition
  77. Sample variance definition
  78. Chi-squared distribution definition
  79. Chi-squared test concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Chi-squared_test, Pearson's chi-squared test
  2. Web https://en.wikipedia.org/wiki/Chi-squared_test, Example chi-squared test for categorical data
About this page
Tag0EN0V3NY20
URIm2://0EN0V3NY20
Aliaseschi-square-test, pearsons-chi-squared-test
Kindconcept
WritingW1 stated
MSC62H17, 62G10
SpecializesStatistical hypothesis test
UsesStatistical hypothesis test, Chi-squared distribution, Independence of random variables
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