What will be the Karl Pearson's Coefficient of Correlation between X and Y as given below ? (Table with X values 10, 20, 30, 40, 50 and Y values 50, 40, 30, 20, 10)

Question

Accepted Answer

The correct option is 4:
Karl Pearson's Coefficient of Correlation ($r$) quantifies the strength and direction of the linear association between two variables, with its value always falling between -1 and +1. In the provided data set, the X values (10, 20, 30, 40, 50) show a constant increase, while the corresponding Y values (50, 40, 30, 20, 10) show a constant and proportional decrease. This scenario, where a perfect change in one direction of X is always accompanied by a perfect change in the opposite direction of Y, is defined as a perfect negative or perfect inverse linear correlation. When this perfect inverse relationship exists, meaning all points lie exactly on a downward-sloping straight line, the coefficient of correlation is precisely equal to -1.

The incorrect option is 1: Can not be determined.
This option is incorrect because the given data set is complete, numerical, and clearly shows a discernible linear pattern. Karl Pearson’s coefficient is designed specifically to be calculated for such data. The only circumstances under which it could not be determined are if the data were nominal, or if one of the series had no variation (e.g., all X values were the same). Since the relationship is perfectly linear and the coefficient is -1, the value is very much determined and precisely calculated.

The incorrect option is 2: 0.
A correlation coefficient of $r = 0$ implies that there is no linear relationship between X and Y. Graphically, the data points would appear scattered with no clear directional pattern. This would mean that a change in X gives no useful information about the direction or magnitude of the change in Y. Since the given data clearly exhibits a strong, predictable, and perfectly inverse linear trend, the variables are highly dependent, and therefore, the correlation value cannot be zero.

The incorrect option is 3: 1.
A coefficient of correlation of $r = 1$ would represent a perfect positive linear relationship. This means that as the X values increase, the Y values would also have to increase by a perfectly proportional amount. In other words, for $r$ to be +1, the X and Y series would need to move in the exact same direction. The provided data shows X and Y moving in opposite directions, thus eliminating the possibility of a perfect positive correlation.