Correlation

Correlation, $\rho$

Description:

• Refers to any broad class of statistical relationships involving dependence
• The correlation of two random variables $X$ and $Y$, denoted by $\rho (X,Y)$
  • defined as long as $Var(X).Var(Y)$ is positive
  • Denotes the degree of linearity:
    • Near +1 or -1 denotes there is a high degree of linearity
    • Near 0 indicates such linearity is absent

Correlation coefficient:

• $\rho (X,Y)=\frac{\text{Cov}(X,Y)}{\sqrt{Var(X)Var(Y)}}=\frac{n\sum x_i{y}_i-\sum x_i\sum y_i}{\sqrt{n\sum x_i^2-(\sum x_i{)}^2}.\sqrt{n\sum y_i^2-(\sum y_i{)}^2}}$
  • $-1\le \rho (X,Y)\le 1$
  • $(x_i,y_i)$ are pairs of data for two variables $X$ and $Y$
• $\rho (X,Y)=1$ implies $m={\sigma }_y/{\sigma }_x>0$ for $y=mx+c$
• $\rho (X,Y)=-1$ implies $m=-{\sigma }_y/{\sigma }_x<0$ for $y=mx+c$

Degrees of correlation:

• Perfectly Correlated
  • Perfect positive vs perfect negative correlation
  • Has no margin of error
• Partly Correlation
• Uncorrelated

Coefficient of Determination

Spearman’s Rank Correlation Coefficient