AnimG LogoAnimG

Central Limit Theorem

Audience: UniversityCategory: Math

Description

Demonstrates the Central Limit Theorem by starting with a non-normal (uniform) population, then repeatedly drawing samples of size n=30 and computing their means. As more sample means are collected, the histogram of means converges to a bell curve regardless of the original distribution's shape.

Inspired by this animation?

Central Limit Theorem

Description

Demonstrates the Central Limit Theorem by starting with a non-normal (uniform) population, then repeatedly drawing samples of size n=30 and computing their means. As more sample means are collected, the histogram of means converges to a bell curve regardless of the original distribution's shape.

Phases

# Phase Name Duration Description
1 Introduction 4s Title, explain what CLT states
2 Population Distribution 7s Show uniform distribution, label ΞΌ and Οƒ, emphasize it's NOT normal
3 Taking One Sample 6s Animate selecting n=30 values, compute sample mean xΜ„
4 Building Histogram 12s Animate histogram growing bar by bar as more xΜ„ values added (50, 100, 500)
5 Convergence 8s Overlay Gaussian curve on histogram, show it fitting perfectly
6 CLT Formula 8s Display XΜ„ ~ N(ΞΌ, σ²/n), explain each part
7 Summary 5s Key insight: shape of population doesn't matter for the mean

Layout

+--------------------------------------------------+
|  Title: "Central Limit Theorem"                  |
+--------------------------------------------------+
|  Population dist.  |  Sampling distribution      |
|  (uniform, left)   |  (histogram of xΜ„, right)    |
|                    |                              |
|  U(0,1) shown     |  Histogram bars grow          |
|  ΞΌ = 0.5          |  β†’ converges to bell curve   |
|  σ² = 1/12        |                              |
+--------------------------------------------------+
|  CLT Formula: XΜ„ ~ N(ΞΌ, σ²/n)                   |
+--------------------------------------------------+

Area Descriptions

  • Left 40%: Population distribution (uniform) β€” static after Phase 2
  • Right 60%: Sampling distribution histogram that grows and converges
  • Bottom strip: CLT formula and explanation text

Assets & Dependencies

  • Fonts: LaTeX
  • Manim version: ManimCE 0.19.1

Notes

  • Use fixed random seed (np.random.seed(42)) for reproducibility
  • Population: Uniform(0,1); ΞΌ=0.5, σ²=1/12
  • Sample size n=30; show growing histogram for 10, 30, 100, 500 samples
  • Histogram bins: ~20 bins over [0.2, 0.8] range
  • Overlay Normal curve N(0.5, 1/(12*30)) when histogram is dense