Central Limit Theorem › Mathematical Statement

The Central Limit Theorem can be stated formally as follows. Let $X_1, X_2, \ldots, X_n$ be a sequence of independent and identically distributed (i.i.d.) random variables with mean $\mu$ and finite variance $\sigma^2$. Define the sample mean as:

$$\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i$$

Then, as the sample size $n$ approaches infinity, the standardized sample mean converges in distribution to a standard normal distribution:

$$\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{\text{d}} N(0, 1)$$

This can be equivalently stated as:

$$\bar{X}_n \sim N\left(\mu, \frac{\sigma^2}{n}\right) \quad \text{for large } n$$

Where:

• $\bar{X}_n$: Sample mean of $n$ observations
• $\mu$: Population mean (expected value of $X_i$)
• $\sigma^2$: Population variance
• $\sigma / \sqrt{n}$: Standard error of the mean
• $N(0, 1)$: Standard normal distribution
• $\xrightarrow{d}$: Converges in distribution

The key insight here is that the standard error of the mean decreases at a rate proportional to $1/\sqrt{n}$. This means that as we collect more data, our estimate of the population mean becomes more precise, and the distribution of sample means becomes increasingly concentrated around the true population mean.

To illustrate that the Central Limit Theorem applies regardless of the original population distribution, the following examples show convergence to normality across several population shapes: uniform and exponential distributions, plus a bimodal one.