12.2 Central limit theorem

The Central Limit Theorem is a powerful tool in probability theory. It states that the sum or average of many random variables tends to follow a normal distribution, regardless of their original distribution. This concept is crucial for statistical inference and modeling real-world phenomena.

The theorem has wide-ranging applications in various fields. It allows us to make predictions and draw conclusions about large populations based on smaller sample sizes, making it invaluable in areas like polling, quality control, and scientific research.

Central Limit Theorem

Central limit theorem fundamentals

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• States sum or average of large number of independent and identically distributed (i.i.d.) random variables approximately normally distributed regardless of underlying distribution of individual random variables
  • Holds true under conditions such as sufficiently large sample size and finite variance of random variables (n ≥ 30)
• Provides foundation for statistical inference and hypothesis testing allows use of normal distribution approximations for various probability calculations
• Justifies widespread use of normal distribution in many real-world applications even when underlying population distribution not normal (heights, weights, test scores)

Distribution of sample means

• Distribution of sample means from population with mean $\mu$ and finite variance $\sigma^2$ approaches normal distribution as sample size increases regardless of shape of population distribution
  • Mean of sample means equal to population mean $\mu$
  • Standard deviation of sample means, known as standard error, equal to $\frac{\sigma}{\sqrt{n}}$ where $n$ is sample size
• As sample size increases distribution of sample means becomes narrower and more concentrated around population mean demonstrates consistency and efficiency of sample mean as estimator (polling, quality control)

Approximation of random variable sums

• To approximate distribution of sum of $n$ i.i.d. random variables $X_1, X_2, \ldots, X_n$ each with mean $\mu$ and variance $\sigma^2$ use CLT
  • Sum $S_n = X_1 + X_2 + \ldots + X_n$ approximately normally distributed with mean $n\mu$ and variance $n\sigma^2$
  • Standardize sum using z-score: $Z = \frac{S_n - n\mu}{\sqrt{n\sigma^2}}$
• To approximate distribution of average (or mean) of $n$ i.i.d. random variables, $\bar{X} = \frac{X_1 + X_2 + \ldots + X_n}{n}$
  • Average $\bar{X}$ approximately normally distributed with mean $\mu$ and variance $\frac{\sigma^2}{n}$
  • Standardize average using z-score: $Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}}$

Conditions for real-world application

• Random variables must be independent and identically distributed (i.i.d.)
  • Independence ensures outcome of one variable does not influence others (coin flips, dice rolls)
  • Identical distribution means all variables follow same probability distribution (uniform, binomial)
• Sample size should be sufficiently large (typically n ≥ 30) for CLT approximation to be accurate
  • Larger sample sizes lead to better approximations of normal distribution (surveys, experiments)
• Population from which samples drawn must have finite variance
  • If population variance infinite CLT may not hold
• If population already normally distributed CLT applies exactly for any sample size