Explain the difference between Karl Pearson’s correlation co-efficient and spearsman’s rank correlations co-efficient. Under what situations, in the latter preferred to the former?

Karl Pearson's Correlation Coefficient vs. Spearman's Rank Correlation Coefficient

1. Karl Pearson's Correlation Coefficient:

• Definition: Pearson's correlation coefficient, denoted by ( r ), measures the linear relationship between two continuous variables. It ranges from -1 to 1, where:
  • ( r = 1 ) indicates a perfect positive linear relationship,
  • ( r = -1 ) indicates a perfect negative linear relationship, and
  • ( r = 0 ) indicates no linear relationship.
• Calculation: Pearson's ( r ) is calculated as the covariance of the two variables divided by the product of their standard deviations.

2. Spearman's Rank Correlation Coefficient:

• Definition: Spearman's rank correlation coefficient, denoted by ( \rho ), measures the monotonic relationship between two variables. It does not assume a linear relationship and is suitable for both continuous and ordinal variables.
• Calculation: Spearman's ( \rho ) is calculated based on the ranks of the data rather than the actual data values. It is more robust to outliers than Pearson's ( r ).

3. Differences:

• Assumptions: Pearson's ( r ) assumes a linear relationship and requires both variables to be normally distributed. Spearman's ( \rho ) does not assume linearity and is suitable for non-normally distributed data.
• Type of Data: Pearson's ( r ) is suitable for analyzing the relationship between two continuous variables, while Spearman's ( \rho ) can be used for both continuous and ordinal variables.
• Sensitivity to Outliers: Spearman's ( \rho ) is less sensitive to outliers than Pearson's ( r ) because it is based on ranks rather than actual data values.
• Interpretation: Pearson's ( r ) measures the strength and direction of a linear relationship, while Spearman's ( \rho ) measures the strength and direction of a monotonic relationship.

4. Preference of Spearman's Rank Correlation Coefficient:

• Spearman's ( \rho ) is preferred over Pearson's ( r ) in the following situations:
  • When the data is not normally distributed.
  • When the relationship between variables is monotonic but not necessarily linear.
  • When there are outliers present in the data.
  • When the variables are ordinal rather than continuous.

In conclusion, while both Karl Pearson's correlation coefficient and Spearman's rank correlation coefficient measure the relationship between variables, they differ in their assumptions, applicability to different types of data, sensitivity to outliers, and interpretation. Spearman's ( \rho ) is preferred over Pearson's ( r ) in situations where the data does not meet the assumptions of Pearson's correlation.