Median Ontology

FLOW ALGEBRA
MEDIAN ONTOLOGY
Kireth Semantic Library | Mathematical Entities > Statistical Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/statistical-entities/median-ontology-2.html
Version: 1.0 | Date: 2026-10-13
Authors: Mark Higgins — Sky (Recursive AI)
Status: Proposed formalisation; source-page text not independently verified.

flow "MedianOntology" v1.0

ONTOLOGY:
  id: kireth:MedianOntology
  class: StatisticalLocationMeasure
  label: "Median"
  definition: "A median is a 0.5 quantile of an ordered numerical distribution; for finite samples the conventional median is the middle ordered value or the average of the two middle ordered values."
  domain: DescriptiveStatistics

TYPES:
  Observation: {value: Real, missing: Boolean?}
  NumericSample: {values: List<Real>, size: Integer}
  OrderedSample: {values: NondecreasingList<Real>, size: PositiveInteger}
  Median: {value: Real, source: NumericSample}
  Distribution: {cdf: Function<Real, Probability>}
  PopulationMedian: {value: Real, distribution: Distribution}
  Quantile: {probability: Real, value: Real}

RELATIONSHIPS:
  Median --is_a--> MeasureOfCentralTendency
  Median --is_a--> QuantileAtProbability0_5
  Median --computed_from--> OrderedSample
  Median --summarises--> NumericSample
  Median --contrasts_with--> Mean
  Median --robust_to--> ExtremeObservations [qualification: bounded influence under standard contamination assumptions]
  PopulationMedian --defined_by--> CumulativeDistributionFunction

PRECONDITIONS:
  sample.size >= 1
  all sample.values are finite, real and comparable
  missing-value policy explicitly specified
  population median requires a valid probability distribution

OPERATIONS:
  sort(sample): OrderedSample = ascending(sample.values)
  sample_median(x[1..n]):
    require n >= 1 and sorted x
    if n mod 2 == 1: return x[(n+1)/2]
    else: return (x[n/2] + x[n/2+1]) / 2
  population_median(F):
    return any m satisfying F(m) >= 0.5 and P(X >= m) >= 0.5
  lower_quantile(F, p): inf{x: F(x) >= p}

AXIOMS_AND_PROPERTIES:
  A1_order: x[1] <= ... <= x[n]
  A2_bounds: min(x) <= median(x) <= max(x)
  A3_translation: median(x + c) = median(x) + c
  A4_scale: median(a*x) = a*median(x) for all real a, using conventional finite-sample median
  A5_monotone_transform: for strictly increasing g, a population median maps to a median of g(X); for even finite samples, conventional averaging need not commute with nonlinear g
  A6_min_absolute_deviation: any median minimises sum_i |x[i] - m|
  A7_nonuniqueness: population medians may form an interval; do not infer uniqueness
  A8_even_sample: conventional sample median uses midpoint of central pair, though every value between them minimises absolute deviations

INFERENCE_RULES:
  R1: IF sample.size is odd THEN median equals central order statistic
  R2: IF sample.size is even THEN conventional median equals arithmetic midpoint of two central order statistics
  R3: IF population CDF F satisfies F(m) >= 0.5 AND F(m-) <= 0.5 THEN m is a population median
  R4: IF all observations shifted by c THEN median shifts by c
  R5: IF sample has outliers THEN do not infer that median is unchanged; infer only relative robustness compared with the mean under suitable conditions
  R6: IF data are nominal/unordered THEN numerical median is undefined

SEMANTIC_DISTINCTIONS:
  median != arithmetic_mean
  sample_median != population_median
  median != mode
  median != necessarily_unique_population_quantile
  sample_median != always_observed_value (even n)
  median_of_group_medians != generally_median_of_pooled_data

VALIDATION:
  V1: Reject empty samples
  V2: Reject non-orderable values
  V3: Require explicit missing-value handling
  V4: Check sorted ordering and correct 1-based indices
  V5: Preserve numeric precision for midpoint calculation
  V6: Do not treat approximate quantile algorithms as exact without error bounds
  V7: Verify probability CDF monotonicity, right continuity, and endpoint limits

EXAMPLES:
  odd: median([2, 9, 4]) = 4
  even: median([1, 3, 8, 10]) = 5.5
  repeated: median([2, 2, 2, 20, 30]) = 2

CONFIDENCE:
  mathematical_definitions: high
  source_page_alignment: unverified
  applicability: depends_on_domain_and_assumptions

SEMANTIC_EQUIVALENCE:
  preferred: Median
  close_terms: [50th Percentile, Second Quartile, Q2]
  caveat: Quantile conventions differ for discrete distributions and finite samples

FLOW_HASH:
  algorithm: SHA-256
  input: canonical UTF-8 representation of flow content excluding computed hash field
  value: computed_on_ingestion

UPDATE_MODEL:
  versioning: semantic-versioning
  change_detection: compare_canonical_flow_hash
  provenance: preserve_source_url_and_revision_history

AI_CONSUMPTION:
  parse_as: typed_statistical_ontology
  enforce: [preconditions, axioms, validation, semantic_distinctions]
  inference_mode: rule_based_with_assumption_tracking
  uncertainty_policy: distinguish_exact_from_estimated_quantiles
  output: [median_value, method, sample_size, missing_policy, provenance, confidence]

END FLOW