Central Limit Theorem Simulation

This project presents a simulation study to illustrate the Central Limit Theorem (CLT) using height data from the Weight-Height.csv dataset.

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What is the Central Limit Theorem?

For a population with any distribution having finite mean (μ) and finite standard deviation (σ), the distribution of sample means (or standardized sums) will approximately be normal as the sample size increases.

Mathematically:

Let X₁, X₂, ..., Xₙ ~ i.i.d. X with E(X) = μ and Var(X) = σ²

• Case I: Let X̄ = ( Σᵢ₌₁ⁿ Xᵢ ) / n then: ( X̄ − μ ) / ( σ / √n ) → N(0, 1) as n → ∞

• Case II: Let Y = Σᵢ₌₁ⁿ Xᵢ then: ( Y − nμ ) / ( σ√n ) → N(0, 1) as n → ∞

Analysis

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Population Statistics:

• Population Mean: 66.367
• Population Std Dev: 3.847

Population Histogram

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Case I – CLT for Sample Means

N	Mean of Sample Means	SD of Sample Means	Theoretical SD (σ / √n)	CLT Observation
5	66.175	1.731	1.720	Poor alignment; irregular distribution
10	66.236	1.391	1.217	Slightly improved; variability high
30	66.328	0.632	0.702	Near normal; CLT emerging
50	66.336	0.543	0.544	Strong CLT evidence; symmetric
100	66.394	0.363	0.385	CLT fully realized; mean ≈ true mean

• For higher N, the Sample mean is Nearest to the Population mean
• In support of the Weak Law of Large Numbers, the SD of the sample mean keeps on decreasing as the sample size increases

Histograms

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Case II – CLT for Sum of Sample Values

N	SD of Sample Sums	Theoretical SD (σ × √n)	CLT Observation
5	8.276	8.602	Irregular distribution
10	12.777	12.165	Slightly better, still irregular
30	20.457	21.071	Near normal
50	23.250	27.202	Symmetric, nearly normal
100	35.830	38.470	Close to normal

• The SD of sample sums keeps on increasing as the sample size increases

Histograms

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Final Conclusions:

Case I – Sample Means

• N = 5: Poor alignment; irregular shape
• N = 10: Slight improvement; still irregular
• N = 30: Clearly approaching normal distribution
• N = 50: Strong CLT evidence; symmetric
• N = 100: CLT fully realized; minimal variability

Case II – Sample Sums

• N = 5: Irregular distribution
• N = 10: Slight improvement
• N = 30: Near normal
• N = 50: Symmetric, nearly normal
• N = 100: Close to normal

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Key Takeaways:

• Larger sample sizes → sampling distributions closer to normal.
• Sample means: Standard deviation decreases with larger N (Weak Law of Large Numbers).
• Sample sums: Standard deviation increases with N, but distribution shape normalizes.

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