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R0056-28960 · q-20261005-advance-20261006-r07-descriptive-statistics-0035
The nonempty observation list is D=[12, 12, 12, -1, -1, -11]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers.
G001 — The nonempty observation list is D=[12, 12, 12, -1, -1, -11]. Retain every repetition. Find the median after sorting: one middle value for odd length, or the average of the two middle values for even length. Give a reduced fraction with positive denominator, retaining /1 for integers.Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [qf-statistics-observation-median, observation_median]
After
derived = [odd_middle_or_even_two_middle_mean]
Conditions: none
Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions.
Rule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001
Before
source roles = [[12, 12, 12, -1, -1, -11]]
After
derived = [[-11, -1, -1, 12, 12, 12], 6]
Conditions: none
Retain EVERY observation,including repeated values.Numeric sorting changes order only,not n or the sum;do not deduplicate as a set or sort lexically.These are sample summaries,not inferred population laws.
Rule: Statistics_FULL_observations_MULTIPLICITY_and_numeric_order · Uses: G001, S001
Before
Statistics EVEN ORIGINAL two middle values sum := sum(-1, 12)
After
Statistics EVEN ORIGINAL two middle values sum = 11
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Statistics EVEN ORIGINAL middle sum over two := div(11, 2)
After
Statistics EVEN ORIGINAL middle sum over two = 11/2
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
source roles = [[-11, -1, -1, 12, 12, 12], 6]
After
derived = [-1, 12, 11/2]
Conditions: none
For even n average BOTH middle values,not just one.Repetitions retain their positions,and the median need not be attained.
Rule: Statistics_EVEN_median_TWO_middle_mean_can_be_unattained · Uses: G001, S004
S005 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.S004 · premise_and_result_audit — Recompute Statistics EVEN ORIGINAL middle sum over two with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Statistics EVEN ORIGINAL two middle values sum with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.11/2
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: e32abae811de9940821103fdd4f5254920c1c5cf0a2ce37234f29352d92d3c57candidate record SHA-256: 3f56f0722bca81a3515e825caf247ba23e951267a9f06a1792a94a7629095b23semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759fLean: NOT_RUN · human review: NOT_PERFORMED