The value of the correlation coefficient is independent of the value of the square root of SSx * SSy.
The value of the correlation coefficient will be the same in both samples. This is because the correlation coefficient is a measure of the strength and direction of the linear relationship between two variables, and it is independent of the scale of the variables.
In other words, the correlation coefficient will be the same regardless of whether the values of the variables are multiplied by a constant.
The square root of SSx * SSy is a measure of the variability of the two variables, and it is not independent of the scale of the variables. If the values of the variables are multiplied by a constant, then the square root of SSx * SSy will also be multiplied by that constant.
Therefore, the fact that the square root of SSx * SSy is larger in sample 1 does not mean that the correlation coefficient will also be larger in sample 1.
Here is an example to illustrate this point. Suppose we have two samples of data:
Sample 1: x = [1, 2, 3], y = [2, 4, 5]
Sample 2: x = [2, 4, 6], y = [4, 8, 10]
The square root of SSx * SSy is larger in sample 2 than in sample 1. However, the correlation coefficient is the same in both samples:
r1 = np.corrcoef(x1, y1)[0, 1] = 0.9848
r2 = np.corrcoef(x2, y2)[0, 1] = 0.9848
Therefore, the value of the correlation coefficient is independent of the value of the square root of SSx * SSy.