Spearman correlation concept

Definition	The ordinary correlation between two lists of ranks. Equation 0.14. Also Spearman.
Example	Ranks (1, 2, 3) against (1, 3, 2) give Spearman correlation 0.5.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `spearman-correlation`, kind concept.
Status	refutes or corrects [refuted]. Corrections: 1 item(s), see below.
Defining equation	Book equation 0.14.
Assumptions and scope	The ordinary correlation between two lists of ranks, one minus six times the sum of squared rank differences over n(n² − 1). It is at most one, one when the rankings agree, and minus one for three items in opposite orders, and since ranks depend on the ordering alone a strictly increasing transform of either score leaves it unchanged. A high Spearman between a proxy and a consumer’s loss is not a certificate. The variance-ratio proxy with Spearman above 0.9 is a refuted ledger row, and the rank certificate’s floor on Spearman is the strict setting’s guarantee, not a measured correlation.
Prior art	none recorded
Evidence	`geometric-observation/claims/LEDGER.md:101`, `lean/DataMiningAsObservation/Spearman.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

Pearson's correlation of the ranks.

Equation

Book equation 0.14.

\[\rho_S=1-\frac{6\sum_i d_i^{2}}{n(n^{2}-1)},\qquad d_i=\text{difference of the two ranks of item } i.\]

Conditions

The ordinary correlation between two lists of ranks, one minus six times the sum of squared rank differences over n(n² − 1). It is at most one, one when the rankings agree, and minus one for three items in opposite orders, and since ranks depend on the ordering alone a strictly increasing transform of either score leaves it unchanged.
A high Spearman between a proxy and a consumer’s loss is not a certificate. The variance-ratio proxy with Spearman above 0.9 is a refuted ledger row, and the rank certificate’s floor on Spearman is the strict setting’s guarantee, not a measured correlation.

Conditions are curated in entries.toml rather than read from a record.

Ledger

refutes or corrects. NEG-8 [refuted]. The Var-ratio tang_qproj is a ≥0.9-Spearman rank proxy for softmax-KL under every consumer. geometric-observation/claims/LEDGER.md:101.

First stated

Spearman, the proof and measurement of association between two things, 1904, as chapter 0 section 0.9 states it, and the rank statistic of the recognizer battery in the-angular-observer and of the rank certificate in readscope.

Measurements

none

Failures and corrections

NEG-8, [refuted]. The Var-ratio tang_qproj is a ≥0.9-Spearman rank proxy for softmax-KL under every consumer. geometric-observation/claims/LEDGER.md:101.

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Spearman.lean, theorems rho_le_one, rho_identical, rho_reversed_three, rank_comp, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation primer S, chapters 0, 3, 4, 7, 9, 11.

Kendall correlation; rank certificate; recognizer; Monotone Invariance Theorem.

Status

Generated 2026-09-10 by encyclopedia/generate.py; book at observation-data-mining f3914f0; the commit of every record is listed in the encyclopedia’s provenance.

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