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.

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Phases

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

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Layout

+--------------------------------------------------+
|  Title: "Central Limit Theorem"                  |
+--------------------------------------------------+
|  Population dist.  |  Sampling distribution      |
|  (uniform, left)   |  (histogram of x̄, right)    |
|                    |                              |
|  U(0,1) shown     |  Histogram bars grow          |
|  μ = 0.5          |  → converges to bell curve   |
|  σ² = 1/12        |                              |
+--------------------------------------------------+
|  CLT Formula: X̄ ~ N(μ, σ²/n)                   |
+--------------------------------------------------+

Area Descriptions

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

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Assets & Dependencies

• Fonts: LaTeX
• Manim version: ManimCE 0.19.1

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Notes

• Use fixed random seed (np.random.seed(42)) for reproducibility
• Population: Uniform(0,1); μ=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