Statistical & Psychometrics
Spearman’s Footrule Rank Distance & Normalized Rank Similarity Formula
Measure L1 rank-order disagreement D = SUM(|R_A,i - R_B,i|) and Diaconis-Graham normalized similarity 1 - D / FLOOR(n^2 / 2) across search engine SERPs and judge rankings.
Linear L1 rank displacement without squaring outlier jumps like Spearman’s rho!
When comparing two search-engine rankings, recommender lists, or clinical triage priority queues of length n, squaring rank differences (as in Spearman’s ρ) lets a single swapped item dominate the entire score! Charles Spearman’s 1906 "footrule" distance (formalized by Persi Diaconis and Ronald Graham, Journal of the Royal Statistical Society B, 1977) sums absolute L1 rank displacements D = SUM(|R_A,i - R_B,i|), whose exact theoretical maximum over permutations of 1..n is INT(n^2 / 2) (= n^2 / 2 for even n, (n^2 - 1) / 2 for odd n)!
Compare 6 ranked search results A = [1, 2, 3, 4, 5, 6] vs B = [2, 1, 5, 3, 6, 4]: absolute displacements |R_A - R_B| = [1, 1, 2, 1, 1, 2] sum to D = 8 out of maximum INT(6^2 / 2) = 18, yielding normalized footrule similarity 1 - 8/18 = 0.5556 (and dropping to 0.0000 under exact reversal [6, 5, 4, 3, 2, 1])!
| Candidate Ranking B (n = 6 vs [1..6]) | Raw Footrule Distance D | Max Possible INT(n²/2) | Normalized Similarity (1 - D/D_max) |
|---|---|---|---|
| B = [2, 1, 5, 3, 6, 4] (Moderate Reordering) | 8 | 18 | 0.5556 |
| B = [6, 5, 4, 3, 2, 1] (Exact Reversal) | 18 | 18 | 0.0000 |
| Insufficient Items (n < 2) | Undefined | Undefined | "Need >= 2 Ranked Items" |
Raw Spearman Footrule L1 Distance D = Σ |R_A,i - R_B,i|
Sums per-item absolute rank displacements D2:D7 across n >= 2 ranked items.
=IF(COUNT(B2:B7) < 2, "Need >= 2 Ranked Items", SUM(D2:D7))Normalized Footrule Rank Similarity (1 - D / INT(n² / 2))
Scales footrule distance onto [0, 1] using exact Diaconis-Graham maximum disarray INT(n^2 / 2), which works identically for both even and odd n!
=IF(COUNT(B2:B7) < 2, "Need >= 2 Ranked Items", ROUND(1 - (SUM(D2:D7) / INT((COUNT(B2:B7) ^ 2) / 2)), 4))Try the example
Use the original values or change them to test your understanding.