GNDU B.Com (Bachelor of Commerce)

Chapter 4: Correlation and Regression Analysis

Summary

Correlation analysis studies the degree and direction of the relationship between two variables, such as price and demand or advertising and sales. Correlation may be positive (variables move in the same direction), negative (they move in opposite directions) or zero (no relationship). Its importance lies in measuring how closely variables are associated and in aiding prediction. Karl Pearson's coefficient of correlation measures the linear relationship and is given by \(r=\dfrac{\sum (x-\bar{x})(y-\bar{y})}{n\,\sigma_x\,\sigma_y}\); it always lies between \(-1\) and \(+1\), where \(+1\) is perfect positive and \(-1\) perfect negative correlation. For ranked or qualitative data, Spearman's rank correlation is used, \(\rho = 1 - \dfrac{6\sum d^2}{n(n^2-1)}\). The probable error helps judge the significance of the computed coefficient. Regression analysis goes a step further by establishing the average functional relationship between the variables so that the value of one can be estimated from the other. The difference between correlation and regression is that correlation only measures the degree of association, while regression provides an equation for prediction and shows cause-and-effect direction. There are two lines of regression: the regression of \(y\) on \(x\) (used to estimate \(y\)) and of \(x\) on \(y\); their slopes are the regression coefficients \(b_{yx}\) and \(b_{xy}\), and the correlation coefficient is the geometric mean of the two, \(r=\pm\sqrt{b_{yx}\,b_{xy}}\).

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