Descriptive statistics: mean, median, range and mode set with repetitions kept

Goal and scope

For the observation list D=(4,10,1,3,8,1,6,3), keeping repeated values, find the number of observations n, the arithmetic mean, the median, the range and the mode set under this entry's convention; prove the general fact 'minimum ≤ mean ≤ maximum' and use it as a check; finally show that deduplicating first would give a different (wrong) mean and median.

Treats only a nonempty finite list of integer observations and the four summaries defined by the MKO; fully proves that the mean lies between the minimum and the maximum. It does not treat sample variance, other quantiles or charts, draws no population, probability or causal conclusion from the summaries, and does not reject other textbooks' phrase 'no mode'; it does not change the canonical MKO or its English edition.

Givens and symbols

• numeric_evaluation
• finite_sum_rules
• frac_rules
• order_basics
• finite_counting_basics

• G1 — D=(4,10,1,3,8,1,6,3) is an original synthetic list of integer observations; it is a list that keeps order and repeated values, not a deduplicated set (MKO assumptions a1, a2).
• G2 — Mode-set convention: all values whose frequency is maximal, listed in increasing order, each once (MKO statement and assumption a3).
• G3 — This task is a descriptive summary only; D is not assumed to be an independent sample from a population (MKO assumption a3).

• D=(d_i)_{i=1}^{n} — the observation list, with order and repetitions
• n — the number of observations
• \bar d — the arithmetic mean
• s_{(k)} — the k-th value after sorting
• f(v) — the number of times v occurs in D
• m,\ M — the minimum and the maximum

Rules and sources

• numeric_evaluation · prerequisite — Exact evaluation of an explicit integer or rational expression; independently recomputable.
• finite_sum_rules · prerequisite — Finite sums: Σ_{i=1}^{0} a_i=0 and Σ_{i=1}^{n+1} a_i=Σ_{i=1}^{n} a_i + a_{n+1}; Σ(a_i+b_i)=Σa_i+Σb_i; Σ c·a_i=c·Σa_i.
• frac_rules · prerequisite — Fraction rules (all denominators nonzero): a/c+b/c=(a+b)/c; (a/b)(c/d)=(ac)/(bd); (a/b)/(c/d)=(ad)/(bc) (also needing c≠0); a/b=(ak)/(bk) for k≠0.
• order_basics · prerequisite — Order of the reals: exactly one of u<v, u=v, u>v holds; u<v and v<w give u<w; u<v gives u+w<v+w; u<v and w>0 give uw<vw; u<v and w<0 give uw>vw.
• finite_counting_basics · prerequisite — Finite counting: for disjoint finite sets A, B, |A∪B|=|A|+|B|; |A×B|=|A|·|B|; there is a bijection between finite sets ⇔ they have the same number of elements.
• def_unfold · prerequisite — Unfolding a definition: replacing a defined term by its definition (or folding it back) does not change whether a statement is true.
• nat_induction · prerequisite — Principle of induction: for a family P(n) starting at n₀, if P(n₀) holds and P(n)⇒P(n+1) for every n≥n₀, then P(n) holds for every n≥n₀.
• mul_inverse_nonzero · prerequisite — Inverse of a nonzero number: in ℚ, ℝ or ℂ, if u≠0 there is 1/u with u·(1/u)=1, and v/u means v·(1/u). 0 has no inverse, so v/0 is undefined.
• divides_def · prerequisite — Divisibility: for integers a, b, a∣b means there is an integer k with b=ak.
• and_intro_elim · prerequisite — Conjunction: from P and Q infer P∧Q; from P∧Q infer P and infer Q.
• mko_descstat_def · cited — For a nonempty finite real observation list D of length n, preserve repeated values. The mean is the sum divided by n. After sorting, the median is the middle value for odd n and the mean of the two middle values for even n. The range is max(D)−min(D). In this entry, the mode set contains all values with maximum frequency, listed in increasing order, including all distinct observed values when every frequency is one. · mko-descriptive-statistics
• perm_invariance · prerequisite — Rearranging the entries of a finite list changes neither its sum nor the number of times each value occurs (generalized commutativity of finite sums, obtainable from add_assoc and add_comm by induction).
• L_mean_bounds · lemma — For a nonempty finite real list (d_i)_{i=1}^{n}, if m≤d_i≤M for every i, then m≤(Σd_i)/n≤M.

Main derivation

D=(d_i)_{i=1}^{n},\ n>0

\bar d=\tfrac{1}{n}\sum_{i=1}^{n}d_i,\ \tilde d,\ R=\max D-\min D,\ \operatorname{Mo}(D)

D=(4,10,1,3,8,1,6,3)

n=8>0

\sum_{i=1}^{8}d_i

4+10+1+3+8+1+6+3=36

\bar d=\tfrac{36}{8}

\bar d=\tfrac{9}{2}

(4,10,1,3,8,1,6,3)

(s_{(k)})_{k=1}^{8}=(1,1,3,3,4,6,8,10)

\tilde d,\ n=8\ \text{even}

\tilde d=\tfrac{s_{(4)}+s_{(5)}}{2}=\tfrac{3+4}{2}=\tfrac{7}{2}

R=\max D-\min D

R=10-1=9

f(v)\ \text{for } v\in\{1,3,4,6,8,10\}

f(1)=2,\ f(3)=2,\ f(4)=f(6)=f(8)=f(10)=1

\max_v f(v)=2

\operatorname{Mo}(D)=\{1,3\}

1\le d_i\le 10\ \text{for every } i

1\le \tfrac{9}{2}\le 10

\bar d=\tfrac{9}{2},\ \tilde d=\tfrac{7}{2}

\bar d\notin D,\ \tilde d\notin D

\{1,3,4,6,8,10\}\ \text{(deduplicated)}

\tfrac{32}{6}=\tfrac{16}{3}\neq\tfrac{9}{2},\quad \tfrac{4+6}{2}=5\neq\tfrac{7}{2}

\text{tasks of the goal}

n=8,\ \bar d=\tfrac{9}{2},\ \tilde d=\tfrac{7}{2},\ R=9,\ \operatorname{Mo}(D)=\{1,3\}

Substeps and lemmas

m\le d_i\ \text{for every } i

nm=\sum_{i=1}^{n}m\le\sum_{i=1}^{n}d_i

nm\le\sum_{i=1}^{n}d_i,\ n>0

m\le\tfrac{1}{n}\sum_{i=1}^{n}d_i=\bar d

d_i\le M\ \text{for every } i

\bar d\le M

Reverse audits

• S13 · sufficient_goal_reduction · S02, S04, S06, S07, S09, S10, S12 — Goal G: the summaries and checks asked for. Sufficient condition H: the conclusions of S02 to S12; H⇒G uses only conjunction. H is derived forwards from G1 to G3; G is never used as a premise.
• S12 · premise_audit · S04, S06, numeric_evaluation — A counterexample needs only one concrete instance (N12 to N14); it shows that deduplication does not in general preserve the mean and the median.
• S11 · premise_audit · S04, S06, divides_def — It claims only that these two summaries are not in D; it does not claim that a mean or median can never be an observation.
• S10.3 · premise_audit · S10.1, S10.2, order_basics — It likewise needs termwise upper bounds and n>0.
• S10.2 · premise_audit · S10.1, order_basics, mul_inverse_nonzero — n>0 is essential: multiplying by a negative number would reverse the inequality, and for n=0 the mean is undefined.
• S10.1 · premise_audit · finite_sum_rules, order_basics, nat_induction — Every term must satisfy m≤d_i; if one term were below m the conclusion could fail. No converse is claimed.
• S10 · premise_audit · L_mean_bounds, S07 — The lemma needs n>0 and termwise bounds; no converse is claimed: a number in [1,10] need not be the mean.
• S09 · premise_audit · S08, G2, mko_descstat_def — It needs the complete frequency table; looking only at the first value with maximum frequency would miss 3.
• S08 · premise_audit · finite_counting_basics, S02 — The frequencies summing to n is a necessary check (N8); it is not claimed that the frequency table recovers the original order.
• S07 · inverse_equivalence · mko_descstat_def, numeric_evaluation — An exact computation (N7), two-way.
• S06 · inverse_equivalence · mko_descstat_def, numeric_evaluation — After unfolding the definition it is an exact computation (N6), two-way; the positions must be taken in the sorted list.
• S05 · premise_audit · perm_invariance, finite_counting_basics — What is needed is that each value occurs equally often (N5, N8); the sorted list is in general not equal to the original list, so only 'the same observations' is claimed, not equality of the lists.
• S04 · inverse_equivalence · frac_rules, numeric_evaluation — Backward: multiply numerator and denominator by 4 to return to 36/8; this needs 8≠0 and 4≠0.
• S03 · inverse_equivalence · finite_sum_rules, numeric_evaluation — An exact integer computation (N1) that can be recomputed, two-way.
• S02 · premise_audit · G1, finite_counting_basics — What is counted is list positions, not distinct values; this is exactly what MKO assumption a1 requires.
• S01 · premise_audit · G1, G2, mko_descstat_def — This step only states the definitions and derives no new fact; n>0 is what makes the mean defined.

Error controls and negative controls

Do not deduplicate first

One misconception the MKO names is deduplicating before computing. After deduplication the mean becomes 16/3 and the median 5 (S12), neither of which summarizes D. Repeated values are part of the data: two 1s mean two observations equal to 1.

Every tied value of maximal frequency belongs to the mode set

1 and 3 both occur twice, so the mode set is {1,3}, not {1}. If every value occurred once, under this entry's convention all distinct values would be in the mode set; other textbooks may call that 'no mode', which is a different convention, not a contradiction (MKO assumption a3).

Take the median after sorting

In the unsorted D the 4th and 5th entries are 3 and 8, whose mean 11/2 is not the median. One must first sort to (1,1,3,3,4,6,8,10) and then take the 4th and 5th entries 3 and 4, giving 7/2 (S06).

• The median is the mean of the 4th and 5th entries of D, (3+8)/2=11/2 → reject — The median is taken after sorting; the sorted 4th and 5th entries are 3 and 4, so the median is 7/2.
• The mean is (1+3+4+6+8+10)/6=16/3 → reject — That is the mean after deduplication; with repetitions kept it is 36/8=9/2.
• The mode set is {1} → reject — 3 also occurs twice, the maximum frequency; the mode set is {1,3}.
• The mean of D is 9/2, so the population that produced D has mean 9/2 → reject — A descriptive summary describes D only; there is no population or sampling model here (MKO assumption a3).

Result and limitations

For D=(4,10,1,3,8,1,6,3): n=8, mean 36/8=9/2, median after sorting (3+4)/2=7/2, range 10−1=9, mode set {1,3}; and 1≤9/2≤10. Deduplicating first would wrongly give mean 16/3 and median 5. These summaries describe D only and support no population inference.

• perm_invariance (generalized commutativity of finite sums and preservation of frequencies under sorting) is used as a prerequisite and is not proved here.
• Sample variance, other quantiles, charts and probabilistic moments are outside this guide; the type distinction between descriptive summaries and probabilistic expectations is left to later slices.
• The MKO's formal next step (bridging finite list sums, sorted medians and maximal-frequency sets) is not done.

Verification and source binding

17 declared checks for this guide; only the listed expressions are covered.

• source commit: b28a5edbd46279b1db2a3f3f48bb3f447e18899f
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• original source SHA-256: de10c74e217f968eb9b391752651fd9f7f2626b798bfb8e87c9af6eadff7e2e3
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