Math 2.0

Statistics Definition

Correlation coefficient

Builds on CovarianceStandard deviation

Definition

Let XX and YY be real random variables on one probability space with finite and positive variances. The correlation coefficient of XX and YY is ρX,Y=Corr⁡(X,Y)=Cov⁡(X,Y)σX σY,\rho_{X, Y} = \operatorname{Corr}(X, Y) = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \, \sigma_Y}, where Cov⁡\operatorname{Cov} is the covariance and σX\sigma_X, σY\sigma_Y are the standard deviations.

Remarks

By the Cauchy–Schwarz inequality, −1≤ρX,Y≤1-1 \le \rho_{X, Y} \le 1. ∣ρX,Y∣=1|\rho_{X, Y}| = 1 if and only if Y=aX+bY = aX + b almost surely for some reals a≠0a \ne 0 and bb. The sign of ρ\rho is the sign of aa. The coefficient does not depend on units: ρaX+b, cY+d=ρX,Y\rho_{aX + b, \, cY + d} = \rho_{X, Y} if ac>0a c > 0. For a sample of pairs (xi,yi)(x_i, y_i), the sample correlation coefficient rr uses sample means and sample sums of squares instead of expectations.

Reading path

To understand this page from the ground up, read these 52 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 space definition
  45. Random variable definition
  46. Discrete random variable definition
  47. Random variable with a density definition
  48. Expected value definition
  49. Variance definition
  50. Covariance definition
  51. Standard deviation definition
  52. Correlation coefficient definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Pearson_correlation_coefficient, For a population
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