The Karl Pearson’s correlation coefficient (r) between two random variables, X and Y, is computed by:

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SSC CGL Tier-II (JSO) 2022 Official Paper (Held On: 4 March 2023)

Answer 1

The correct answer is \(\rm r(X,Y)=\frac{Cov(X,Y)}{\sigma_X \sigma _Y}\). Key Points

The Karl Pearson's correlation coefficient (r) between two random variables, X and Y, is computed using the formula: \(\rm r(X,Y)=\frac{Cov(X,Y)}{\sigma_X \sigma _Y}\)
Cov(X, Y) represents the covariance between variables X and Y. It measures how the variables vary together.
- σ_X is the standard deviation of variable X, which measures the variability or dispersion of X.
- σ_Y is the standard deviation of variable Y, which measures the variability or dispersion of Y.
The correlation coefficient (r) measures the strength and direction of the linear relationship between X and Y. It ranges between -1 and 1, where:
- r = 1 indicates a perfect positive linear relationship, where X and Y increase or decrease together.
- r = -1 indicates a perfect negative linear relationship, where X and Y have an inverse relationship.
- r = 0 indicates no linear relationship between X and Y.
The correlation coefficient not only provides information about the strength and direction of the relationship but also helps in assessing the degree to which X and Y vary together when compared to their individual variabilities (given by the standard deviations).