Spearman's rank correlation of consultants' productivity and client satisfaction, with tied ranks

Ranking. Rank 1 is given to the highest score in each series. Equal scores share the average of the ranks they would occupy: in productivity, F and G (50) share ranks 7 and 8, so each gets 7.5; in client satisfaction, E and J (58) share ranks 3 and 4 (3.5 each), and A and I (54) share ranks 6 and 7 (6.5 each). For ties, Spearman's formula adds (m³ − m) / 12 to ΣD² for every group of m equal values in either series:

ρ = 1 − 6[ΣD² + Σ(m³ − m)/12] / [n(n² − 1)], D being the difference between the two ranks of a consultant.

Working table

(i) Rank correlation coefficient

Three groups of two tied values (one in productivity, two in satisfaction), each with m = 2: correction = 3 × (2³ − 2)/12 = 3 × 0.5 = 1.5.

ρ = 1 − 6(69.5 + 1.5) / [10(10² − 1)] = 1 − 426 / 990 = 1 − 0.4303 = 0.5697.

Interpretation. Yes, there is correlation: it is positive and moderate (about 0.57). Consultants with higher productivity scores tend, on the whole, to earn higher client-satisfaction scores, but the agreement between the two rankings is far from perfect (F, for example, is seventh in productivity yet second in satisfaction). As a rough check of reliability, the probable error is 0.6745 (1 − ρ²) / √n = 0.6745 × (1 − 0.3246) / √10 = 0.1441; ρ is well above the probable error but below 6 × P.E. = 0.8644, so with only ten consultants the result suggests a real link without proving it. The firm may treat productivity as one factor in client satisfaction, while correlation alone does not show which one drives the other.

(ii) Productivity score of I changed from 48 to 49

Rank correlation depends only on the order of the values, not on their size. The new score 49 still lies below 50 (F and G) and above 44 (H), so I remains ninth in productivity. No rank changes, every D and D² stays the same, the tie correction is unchanged, and so the coefficient is unchanged at 0.5697. (The ranks would change only if the new score moved I past a neighbouring value, for example to 50 or more.)