Which of the following statements relating to Correlation and Regression are true?

(a) The Coefficient of Correlation is independent of change of origin and scale.

(b) The Coefficient of Correlation between the two variables is the arithmetic average of the two Regression Coefficients.

(c) The probable error of the Coefficient Correlation is 0.6745 times its standard error.

(d) Coefficient of Correlation multiplied by the ratio between the standard deviations of the two variables denotes the slope of the regression line.

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The correct answer is (a), (c) and (d). 

Key PointsCorrelation Coefficient: 

• The correlation Coefficient is a statistical concept, which helps in establishing a relation between predicted and actual values obtained in a statistical experiment.
• The calculated value of the correlation coefficient explains the exactness between the predicted and actual values.
• The correlation coefficient value always lies between -1 to +1.
• If the correlation coefficient value is positive, then there is a similar and identical relation between the two variables.
• Else it indicates the dissimilarity between the two variables.

Important PointsMain properties of Coefficient of correlation:

•  In case the correlation is present, then the coefficient of correlation would lie between ±1. If the correlation is absent, then it is denoted by zero or -1 ≤ r ≤ +1.
• The coefficient of correlation is based on a suitable measure of variation as it takes into account all items of the variable.
• Coefficient of correlation measures both the direction as well as the degree of change
• It is independent of changes in scale and origin of the variables X and Y.
• The coefficient of correlation is a geometric mean of two regression coefficients. 
• The coefficient of correlation is independent of the unit of measurement
• The coefficient of correlation works both ways i.e., rxy = ryx
• The probable error of the Coefficient Correlation is 0.6745 times its standard error. 
• The coefficient of Correlation multiplied by the ratio between the standard deviations of the two variables denotes the slope of the regression line. 

Hence, options a, c, and d are true statements relating to Correlation and Regression.