Spread (Variance & Standard Deviation)

핵심 개념

Spread tells us how much the values in a dataset vary around the mean. Two datasets can share the same mean yet look completely different — spread captures that difference. Lecture notes.md

Key measures at a glance

Measure	Formula / Idea	Units	When to use
Variance	Average of squared deviations: `Σ(xᵢ − x̄)² / n`	Squared (e.g. $²)	Intermediate step; useful in math proofs
Standard Deviation (σ)	`√variance`	Same as original data	Everyday interpretation of spread

Why do we square the deviations? Plain deviations (xᵢ − x̄) always sum to zero — the negatives and positives cancel out perfectly. Squaring makes every deviation positive before averaging. Lecture notes.md

Interpreting σ

σ = 0 → every value is identical; there is no spread at all. Lecture notes.md
Larger σ → values are more scattered from the mean.
Smaller σ → values cluster tightly around the mean.
σ links directly to the Normal Distribution: the 68-95-99.7 rule uses σ as its unit of distance from the mean. Lecture notes.md

Visual: from raw data to σ

큐 질문 (능동적 회상)

Use these questions for active recall — try to answer before revealing notes. Lecture notes.md

1.Why do we square the deviations when computing variance, instead of just averaging them?
2.What does a standard deviation of 0 tell you about a dataset?
3.Variance and standard deviation both measure spread — what is the key practical difference between them?
4.If one dataset has σ = 2 and another has σ = 10, what does that tell you about how the values are distributed in each?
5.How does σ relate to the 68-95-99.7 rule in a normal distribution?

📌 All of these questions are also pooled in the Review & Spaced Repetition page for scheduled practice.

요약

Variance squares the gaps from the mean to avoid cancellation; standard deviation square-roots it back to the original units — together they tell you how tightly or loosely data clusters around the mean. Lecture notes.md