Central Limit Theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the context of different conditions.

Some of the important versions of this are:

Variant	Assumptions	Formula Type	Use Case
Classical / Lindeberg–Lévy	i.i.d., finite variance	$Z_n\to \mathcal{N}(0,1)$	General inference, sample mean
Lyapunov	Independent, not identically dist.	Normalized sum $\to \mathcal{N}(0,1)$	Heterogeneous data
Lindeberg–Feller	Independent, with Lindeberg condition	General normalized sum	Most general CLT
Multivariate CLT	i.i.d. vectors	$\sqrt{n}(\overline{X}-\mu )\to {\mathcal{N}}_d(0,\mathrm{\Sigma })$	Multivariate stats, finance
Functional CLT	i.i.d., viewed as stochastic process	$S_n(t)\Rightarrow W(t)$	Time series, Brownian motion modeling
Dependent Variable CLT	Weakly dependent vars	Sum $\to \mathcal{N}(0,1)$	Time series, Markov chains

Classical Central Limit Theorem

If $X_1,X_2,...,X_n$ are independent and identically distributed (i.i.d.) random variables with:

• Mean $\mu$,
• Variance ${\sigma }^2<\mathrm{\infty }$,

Then the standardized sum:

$$Z_n=\frac{\sum _{i=1}^n{X}_i-n\mu }{\sigma \sqrt{n}}\stackrel{d}{\to }\mathcal{N}(0,1)$$

as $n\to \mathrm{\infty }$.

Application 1 - Usage of Standard Deviation(σ)

CLT says:

The sampling distribution of the sample mean ( $\overline{x}$ ) will be approximately normal with:

• Mean: $\mu$ (same as the population mean)

• Standard Deviation (of the sample mean):

  $${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$$

Where:

• ${\sigma }_{\overline{x}}$ = standard deviation of the sampling distribution (also called standard error)
• $\sigma$ = population standard deviation
• $n$ = sample size

How σ Works with CLT

1. Reduces Variability with Bigger Samples

As sample size $n$ increases:

$${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\text{gets smaller}$$

This means: Your sample mean becomes more stable and reliable as you take larger samples.

2. Allows Confidence Interval Estimation

In business, we often construct a confidence interval for the mean:

$$\overline{x}\pm z\cdot \frac{\sigma }{\sqrt{n}}$$

Where:

• $\overline{x}$ = sample mean
• $z$ = z-score (e.g., 1.96 for 95% confidence)
• $\sigma /\sqrt{n}$ = standard error

You need σ to determine how “wide” or “narrow” your estimate is.

3. Enables Hypothesis Testing

When comparing two groups (A/B testing, quality control, etc.), you calculate:

$$Z=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}$$

• A smaller σ → more precise Z-scores → clearer decisions.

Example

Let’s say a factory produces nails with a known population standard deviation $\sigma =2\text{ mm}$, and you take a sample of 100 nails.

• Standard error:

  $${\sigma }_{\overline{x}}=\frac{2}{\sqrt{100}}=0.2$$

So your sampling distribution of the mean length will have:

• Mean = population mean (say, 50 mm)
• Std dev = 0.2 mm, not 2 mm!

Your estimate of the average length from the sample is much more precise than any one nail.

Summary

Concept	Role of σ (sigma)
Sampling distribution	Determines spread of sample means
Standard error	$\sigma /\sqrt{n}$: lower with bigger n
Confidence intervals	Used to calculate margin of error
Hypothesis testing (Z-tests)	Critical in test statistic formula

If the population σ is unknown, we use the sample standard deviation $s$ and apply the t-distribution — another extension of CLT.