Descriptive summaries of nonempty observation lists

Mathematical statement

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, including all values when every frequency is one.

Representative formula (exact TeX)

M(D)

Assumptions

• Retain repeated observations instead of replacing the list with a deduplicated set. Sorting changes only their order.
• Exercises use original synthetic integer data. Mean and median are reduced fractions with positive denominators, and n>0.
• The maximal-frequency convention is explicit and does not negate other textbooks using the phrase no mode. Descriptive summaries alone imply no population, probability law, or causal claim.

Symbol scope

• M — specified descriptive-summary function
• D — nonempty observation list retaining multiplicity

Intuition, definition, and misconceptions

The mean need not be an observed value, and an even-length median may lie between the middle observations. For a singleton, mean and median equal its value and the range is zero.

Several values may tie for maximum frequency. Sort and deduplicate the output set, but retain every input repetition when computing frequencies; an all-distinct input yields all its values under the stated convention.

These are descriptive summaries, distinct from moments of a specified finite probability model. No independent-identically-distributed or population assumptions are introduced, and sample variance with an n−1 denominator and other quantile conventions are left for later work.

Common errors include deduplicating the observations before averaging, assuming a unique mode, or interpreting descriptive summaries as population or probability-model inference.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Arithmetic operations — Use sums, differences, and averages.
• Rational number — Represent rational summaries exactly.
• Usual order on the real numbers — Sort the observations and identify their extrema.

Verification boundary

Finite observation summaries can be checked exactly; no inference result or probability-law Evidence is attached.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind finite List sums, sorted-list median, and the maximal-frequency set while distinguishing the statistical object types.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.