Why must r be between -1 and +1?

r cannot be greater than 1 because r is the proportion of covariance to the geometric mean of the two variances [try it out, starting with the standard formula for r = sum of y1y2 over s1s2(n-1)]. In the extreme case of two identical data columns, the covariance is identical to the geometric mean of the two variances, and the value for r = 1. In the other extreme case of orthogonal data, the covariance is 0 and r must be 0.

The sign for r stems from the sign of the covariance---if all negative values of the x1 column (mean deviated) are paired with all the negative values of the x2 column, and all positive values of the x1 column with all the positive values of the x2 column, then the sum of covariance products is going to be positive [e.g., (-5)(-5) + (-4)(-4) + ... + (5)(5)]. If, however, the positive values of the x1 column and the negative values of the x2 column are paired, and vice versa, the sum of products must be negative [e.g., (-5)(5) + (-4)(4) + ... + (5)(-5)].