Understanding Karl Pearson's Correlation Coefficient

Karl Pearson's correlation coefficient, often denoted by \(r\), is a statistical measure that quantifies the strength and direction of a linear relationship between two continuous variables, let's say X and Y. The value of \(r\) ranges from -1 to +1. A value of +1 indicates a perfect positive linear correlation, -1 indicates a perfect negative linear correlation, and 0 indicates no linear correlation.

Formula for Karl Pearson's Correlation Coefficient

The formula for computing Karl Pearson's correlation coefficient \(r(X,Y)\) between two random variables X and Y involves their covariance and their respective standard deviations.

The formula is defined as the covariance between X and Y divided by the product of their standard deviations:

$$ r(X,Y) = \frac{Cov(X,Y)}{\sigma_X \sigma_Y} $$

Where:

• \(Cov(X,Y)\) is the covariance between variables X and Y. Covariance measures the extent to which two variables change together. A positive covariance indicates that X and Y tend to increase or decrease together, while a negative covariance indicates that one increases as the other decreases.
• \(\sigma_X\) is the standard deviation of variable X. Standard deviation measures the dispersion or spread of the data points in variable X around its mean.
• \(\sigma_Y\) is the standard deviation of variable Y. Standard deviation measures the dispersion or spread of the data points in variable Y around its mean.

The covariance provides information about how X and Y vary together, but its magnitude is influenced by the scales of X and Y. Dividing by the product of the standard deviations normalizes the covariance, resulting in a dimensionless measure (the correlation coefficient \(r\)) that is not affected by the units of the variables.

Analyzing the Given Options

Let's compare the standard formula for Karl Pearson's correlation coefficient with the options provided:

1. r(X,Y) = Cov(X,Y)*σXσY
   This option suggests multiplying the covariance by the standard deviations. This is incorrect as the formula requires division by the product of standard deviations.
2. \(\rm r(X,Y)=\frac{\sigma_X \sigma _Y}{Cov(X,Y)}\)
   This option presents the reciprocal of the correct formula, with the product of standard deviations in the numerator and covariance in the denominator. This is incorrect.
3. \(\rm r(X,Y)=\frac{Cov(X,Y)}{\sigma_X \sigma _Y}\)
   This option correctly represents the covariance divided by the product of the standard deviation of X and the standard deviation of Y. This matches the standard definition of Karl Pearson's correlation coefficient.
4. \(\rm r(X,Y)=\frac{Cov(X,Y)}{(\sigma_X \sigma _Y)^2}\)
   This option suggests dividing the covariance by the square of the product of the standard deviations. This is incorrect.

Based on the analysis, the formula \( \rm r(X,Y)=\frac{Cov(X,Y)}{\sigma_X \sigma _Y} \) accurately defines Karl Pearson's correlation coefficient between two random variables X and Y.

Revision Table: Karl Pearson Correlation Coefficient

Additional Information: Properties of Correlation Coefficient

• The value of Karl Pearson's correlation coefficient \(r\) always lies between -1 and +1, inclusive (\(-1 \le r \le +1\)).
• A correlation coefficient close to +1 indicates a strong positive linear relationship.
• A correlation coefficient close to -1 indicates a strong negative linear relationship.
• A correlation coefficient close to 0 indicates a weak or no linear relationship. It's important to note that a zero correlation coefficient does not necessarily mean the variables are independent; they could have a non-linear relationship.
• Correlation does not imply causation. A high correlation between X and Y does not mean that X causes Y or that Y causes X.
• The correlation coefficient is symmetric, i.e., \(r(X,Y) = r(Y,X)\).