SL 4.10—Spearman’s rank correlation coefficient

$r_s=1-\frac{6\sum d_i^2}{n(n^2-1)}$ with $n=6$.

$1^2+(-2{)}^2+0^2+1^2+(-1{)}^2+1^2=1+4+0+1+1+1=8$.

$n(n^2-1)=6(36-1)=6\times 35=210$.

$\frac{6\times 8}{210}=\frac{48}{210}=0.2286$.

$r_s=1-0.2286=0.771$.

$0.886$ results from using $\sum d_i^2=5$ (miscounting or missing a square).

$0.629$ comes from using $n=5$ instead of $n=6$ in the denominator.

$0.914$ arises from forgetting to square one of the differences, using $\sum d_i=$ sum instead of $\sum d_i^2$.

Only careful squaring and correct substitution give $r_s=0.771$.

A monotonic relationship means one variable consistently rises or consistently falls as the other rises.

Unlike a linear relationship, the rate of change need not be constant, just the direction.

This matches option A exactly.

Requiring a constant rate describes a linear relationship, which is a stricter condition than monotonic.

No relationship would give $r_s$ close to 0, the opposite of what monotonic describes.

Normality is a requirement for Pearson's correlation, not a feature of monotonic relationships.

Only option A correctly captures the consistent-direction, variable-rate nature of monotonicity.