What is Central Limit Theorm?

The Central Limit Theorem (CLT) is a fundamental concept in statistics that describes the distribution of sample means. Here's a structured explanation:

Key Principles of the Central Limit Theorem

1. Core Idea:
   Regardless of the population's original distribution, the distribution of the sample means ($\bar{X}$) will approximate a normal distribution as the sample size ($n$) increases, provided the population has a finite mean ($\mu$) and variance (${\sigma }^2$).

2. Formal Statement:
   If $X_1,X_2,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ are independent, identically distributed (iid) random variables with mean $\mu$ and variance ${\sigma }^2$, then as $n\to \mathrm{\infty }$:

   $$\frac{\bar{X}-\mu }{\sigma \mathrm{/}\sqrt{n}}\stackrel{d}{\to }N(0,1)$$

   This means the standardized sample mean converges to a standard normal distribution.

3. Parameters of the Sampling Distribution:

   • Mean: The mean of the sample means (${\mu }_{\bar{X}}$) equals the population mean ($\mu$).

   • Variance: The variance of the sample means (${\sigma }_{\bar{X}}^2$) is ${\sigma }^2\mathrm{/}n$.

   • Standard Error: The standard deviation of the sample means is $\sigma \mathrm{/}\sqrt{n}$.

Key Implications

• Normality for Large $n$: Even if the population is skewed, binomial, or otherwise non-normal, the sample means will form a normal distribution for sufficiently large $n$ (typically $n\ge 30$ is a rule of thumb).

• Inferential Statistics: The CLT justifies using normal-distribution-based methods (e.g., z-tests, confidence intervals) for inference about the population mean.

• Practical Applications:

  • Calculating confidence intervals: $\bar{X}\pm z^{\ast }(\sigma \mathrm{/}\sqrt{n})$.

  • Hypothesis testing (e.g., t-tuses when $\sigma$ is unknown).

  • Quality control (e.g., control charts).

Assumptions

1. Independence: Observations must be independent (e.g., random sampling).

2. Finite Variance: The population variance (${\sigma }^2$) must be finite.

3. Sample Size: Larger $n$ improves the approximation, especially for non-normal populations.

Examples

• Skewed Population: For a highly skewed population (e.g., income data), taking samples of $n=50$ and calculating their means will result in a roughly normal distribution of those means.

• Dice Rolls: The average of 30 dice rolls (a uniform distribution) will follow a normal distribution.

Common Misconceptions

• Applies to Small $n$: The CLT requires a sufficiently large $n$; small samples from non-normal populations may not approximate normality.

• Data Becomes Normal: The CLT applies to sample means, not the original data.

Why It Matters

The CLT enables statistical inference (e.g., hypothesis testing, confidence intervals) without requiring knowledge of the population distribution, making it foundational for data analysis, experimentation, and decision-making.

Limitations

• Fails for populations with infinite variance (e.g., Cauchy distribution).

• Dependent data (e.g., time series) may violate the independence assumption.

In summary, the CLT bridges probability theory and practical statistics, allowing us to use the normal distribution as a powerful tool for analyzing sample means.