Descriptive statistics: mean, spread, and honest summaries
Assistance allowed only after prediction and observed evidence are recorded.
Prerequisites
Day 27 mathematics block; basic arithmetic and the prediction-review routine from Day 01.
Learning Objective
Compute and interpret mean, median, population/sample variance, and standard deviation on a small dataset, and explain why two datasets with the same mean can still differ.
Instructions
Statistics begins by compressing data into a few numbers. The mean locates the center; variance and standard deviation measure spread around it. Population variance divides the squared deviations by n; sample variance divides by n−1 (Bessel's correction) to avoid systematically underestimating spread. Predict every value by hand before checking with python3 — verification comes only after recorded predictions.
Practice & Prediction Log
For data = [2, 4, 4, 4, 5, 5, 7, 9]: compute the mean, median, sum of squared deviations, population variance, population standard deviation, and sample variance (divisor n−1). Then compute the mean and population standard deviation of data2 = [4, 5, 5, 5, 5, 5, 5, 6] and state what stays the same and what changes.
Mean = 40/8 = 5. Deviations: −3, −1, −1, −1, 0, 0, 2, 4. Squares: 9, 1, 1, 1, 0, 0, 4, 16 → sum 32. Population variance divides by 8; sample variance by 7. For data2 the mean is also 5 but the squared deviations sum to 2.
mean = 5.0, median = (4+5)/2 = 4.5, Σ(x−μ)² = 32, population variance = 32/8 = 4.0, population std = 2.0; sample variance = 32/7 ≈ 4.571 (sample std ≈ 2.138). data2 shares the mean 5.0 but has SS = 2, population variance 0.25, std 0.5 — the mean alone hides the spread.
All hand-computed values match a Python check, and the explanation states that equal means do not imply equal distributions.
Authoritative Source Resources & Provenance
NIST/SEMATECH e-Handbook of Statistical Methods
Relevant Section / Range: Section 1.3.5 (Quantitative Techniques): measures of location and scale
Verified Version & Date: Current online edition (Verified: 2026-10-30)
Primary engineering reference for descriptive statistics, exploratory data analysis, and uncertainty quantification with worked formulas.
High-school arithmetic; no calculus required for the descriptive sections.
Every formula used in the lesson (mean, variance, standard error) is reproduced as original explanation; all exercises run fully offline.
Compared Candidate Sources & Selection Analysis (2)
Undergraduate statistics text with a stronger narrative bridge from probability to inference; retained as the second linked source for inferential framing.
Strong interactive visualizations of probability and sampling, but a browser-only interactive format is a weaker fit for an offline-capable 480-minute ledger.
OpenIntro Statistics (4th edition)
Relevant Section / Range: Chapters 1-2: data basics and summary statistics
Verified Version & Date: 4th edition, 2019 (Verified: 2026-10-30)
Rigorous open textbook treatment of sampling distributions, standard error, and the difference between an inferential claim and an anecdotal observation.
Introductory algebra; comfort reading mathematical notation.
All required definitions and worked computations are given in the lesson itself; the book is a further-reading link.
Compared Candidate Sources & Selection Analysis (2)
Engineering handbook strength for formulas, tables, and measurement practice; OpenIntro supplies the probability-to-inference narrative.
Strong probability course, but lectures and the print book are under full copyright and the syllabus spans a semester rather than a single study block.