Central Limit Theorem

The sum of many independent random variables will approximately follow a Normal distribution

Key Points of the CLT

• If a variable X has mean μ and standard deviation σ
• And if the sample size (n) is large
• Then the sampling distribution for $\overline{x}$(sample mean) is approximately Normal with:
  • Mean = $\mu$
  • Standard error = $\frac{\sigma }{\sqrt{n}}$
• Expressed as $\overline{x}\sim N(\mu ,\frac{\sigma }{\sqrt{n}})$

Importance of CLT

• Natural phenomena: Many natural phenomena result from many independent random events, leading to Normal distributions
• Statistical inference: Allows us to make inferences about population parameters using sample statistics