---
slug: central_tendency
name: Central Tendency
one_liner: "The single representative value around which a collection of observations clusters — the location dual of variability along the location-versus-spread axis of a distribution."
type: prime
schema_version: 2
status: density_v2_draft
categories: [mathematical, epistemic]
origin_domain: statistics
alternate_origin_domains: [measurement_theory, signal_processing, machine_learning]
aliases: [location_parameter, average, measure_of_center, expected_value_as_center]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-01 dualization, sub-flavor 1b) — the structural dual of `variability` along the location-versus-spread axis of a distribution (where does the mass sit vs. how far it spreads). Discovered in the op-01 full sweep 2026-07-03. Verified absent via search_prime + slug check; a foundational statistical structure with no current prime, low collision risk. STATUS: RECOMMENDED ACCEPT."
- "Created 2026-07-03; STUB ONLY (frontmatter + Core Idea + Notes). Did NOT go through Phase-A triage. SI/SF/similarity grades are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT yet written."
- "Phase-C action: re-similarity check vs variability, aggregation, expectation/expected_value, and any summary-statistic or dimensionality-reduction neighbor to confirm it does not collapse into generic aggregation; draw the sibling (variability) and proposed-parent (distribution / summary-statistic) edges at incorporation. one_liner must be synced to a v1 draft per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 4
  domain_breadth: 4
  structural_abstraction: 5
  transfer_evidence: 4
  reasoning: 'Central tendency is the pure relational notion "the location a collection of values clusters around, the single point that best represents them under some loss," with no commitment to a medium — structural abstraction is a full 5. It recurs wherever a collection must be summarized by one representative: the mean/median/mode in statistics, the centroid in clustering and geometry, the fixed point or equilibrium a dynamical system settles toward, the consensus value in agreement protocols, the expected value of a random variable, and the "typical member" of a category in cognition — domain breadth 4 spanning math, ML, dynamics, and mind. Transfer is strong (4): the same object (a minimizer of aggregate deviation) is recognizably reused as centroid, expectation, and equilibrium across fields, and the location/spread decomposition is universal to any distribution-like collection. Honest note: it verges on being a fundamental structural notion, but its sharpest development is statistical, holding the composite at 4 rather than 5.'
similarity_to_nearest_existing_prime:
  slug: variability
  similarity: 0.6
structural_framed:
  label: structural
  aggregate: 0.08
  criteria: {vocab_travels: 0.1, evaluative_weight: 0.0, institutional_origin: 0.0, human_practice_bound: 0.0, import_vs_recognize: 0.1}
  rationale: 'Central tendency is the medium-neutral relation "the representative location of a collection" — a value that minimizes aggregate deviation — carrying no normative or institutional content; every diagnostic reads at or near zero. It has a formal (statistical/measure-theoretic) origin, no evaluative weight (a center is neither good nor bad), and no human-practice binding since a centroid or an equilibrium locates a collection with no human in the loop. The faint non-zeros reflect only that "mean/median/mode" are mild terms of art carried in, while the underlying clustering-point structure is recognized wherever a collection is summarized by one value.'
phase_b_passes_completed: []
---
# Central Tendency

## Core Idea

Central tendency is the structure of *the single value that best represents a collection of observations* — the location around which the collection clusters, the point that stands in for the whole. It is one pole of a single axis, *location versus spread*, whose opposite pole is `variability`. Given any collection of values (across units, conditions, or time), two complementary questions can be asked: *where does the mass sit?* and *how far does it spread?* Central tendency answers the first, variability the second. The two are duals in the precise sense that they partition the description of a distribution between them: a collection is characterized, to first order, by a center and a spread, and neither is meaningful without the other — a spread is spread *about* a center, and a center is the location *of* a spread.

The load-bearing invariant is that a central value is defined as *the point that minimizes aggregate deviation* from the collection under some choice of loss. This single idea generates the familiar measures rather than listing them arbitrarily: the value minimizing summed squared deviation is the *mean*, the value minimizing summed absolute deviation is the *median*, and the most frequent value — minimizing a 0/1 loss — is the *mode*. Choosing a measure is therefore choosing a loss function and, with it, a sensitivity profile: the squared-loss mean is efficient but pulled by outliers, the absolute-loss median is robust but discards magnitude information, the mode ignores ordering entirely. The characteristic trade-off the prime names is exactly this — *representativeness bought at the cost of information*: collapsing a whole collection to one number is the most drastic possible compression, discarding all variation, and its payoff (a single comparable, communicable, computable summary) is inseparable from its cost (everything the spread would have told you is gone, and a center reported without its spread can mislead badly).

Read as a prime, central tendency is the recognition that many "the representative one" moves across domains are the *same structure*: the arithmetic mean or median of a sample, the *centroid* of a point cloud in clustering and geometry, the *expected value* of a random variable, the *equilibrium* or fixed point a dynamical system settles toward, the *consensus value* an agreement protocol converges on, and the *prototype* or typical member of a cognitive category. Each is a location that minimizes aggregate deviation and represents a collection by one value; each pays the same compression cost; and each is only half of the location/spread pair whose other half is variability.

## Notes

**Provenance.** Produced by the op-01 dualization operator (sub-flavor 1b) as the structural dual of `variability` — flipping the pole of the *location-versus-spread* axis from spread to center. Surfaced in the op-01 full sweep 2026-07-03; the manifest rates it STRONG, noting it names the basic statistical structure (mean/median/mode) that is the complement of the already-present variability/dispersion prime, verified absent.

**For owner evaluation (keep/cut).** The case FOR: this is a foundational, medium-neutral structure with an honest structural SI profile (composite ~4), it is the exact location-pole complement of the existing `variability` prime, and its absence leaves the location/spread pair asymmetric — a clean pair-completion / lattice-gap signal. Unlike the pessimism_bias candidate, this one reads as genuinely *structural*: centroid, expected value, and equilibrium are the same object across substrates. The specific risks Kurt should weigh: (1) **Collapse into generic `aggregation` or `expectation`.** Central tendency is a *specific* aggregation — the representative-location one, defined by deviation-minimization — not aggregation in general (which also covers sums, counts, and joins). If the corpus already carries `aggregation` or an `expected_value`/`expectation` prime, confirm central_tendency is the tighter, distinct node and draw the is-a edge rather than merging. (2) **Theorem-vs-prime question.** Because "the minimizer of squared deviation is the mean" is a *theorem*, review should confirm the prime is the *structure* (representative-location-of-a-collection) and not merely a bundled statistical fact; the eager_evaluation-style read — name the invariant, not the domain instance — should carry through densification. (3) **Breadth vs. statistical framing.** Its sharpest development is statistical, but the transfer to clustering (centroid), dynamics (equilibrium), and cognition (prototype) is what earns the prime; densification should lead with the substrate-free deviation-minimizer, not the mean. Pair-completion signal: strong — accepting `variability` without its location dual leaves the distribution's description half-specified.

**At incorporation.** Re-run similarity against `variability` (expect the highest, ~0.6, as the explicit dual), `aggregation`, `expectation`/`expected_value`, and any summary-statistic or dimensionality-reduction neighbor to rule out collapse. Draw the explicit *dual* edge to `variability` and the *genus/is-a* edge to a `distribution` or summary-statistic parent (with `aggregation` as a likely super-genus). Confirm the deviation-minimization framing (mean/median/mode as loss choices) survives densification, since that is what makes it a structure rather than a list. Sync the final `one_liner` into a v1 draft per CLAUDE.md before merge, keeping it axis-naming and domain-general.
