The Central Limit Theorem (CLT) is a fundamental principle in statistics that explains how the distribution of sample means approximates a normal distribution as the sample size becomes larger, regardless of the original population's distribution. Here are the key aspects:
- Definition: The theorem states that the sum or average of a large number of independent, identically distributed random variables, each with finite variance, will be approximately normally distributed.
- Importance:
- It allows statisticians to make inferences about population parameters using the normal distribution, even when the population distribution is unknown.
- It underpins many statistical methods like hypothesis testing and confidence intervals.
- Historical Context:
- The roots of the theorem can be traced back to Abraham de Moivre in the 18th century, who derived it in the context of binomial distributions.
- The term "Central Limit Theorem" was coined by Georg Pólya in 1920.
- Significant contributions were made by Pierre-Simon Laplace in the 19th century, who generalized the theorem to other distributions.
- Conditions for Application:
- The samples must be independent.
- The samples must come from the same distribution.
- The distribution of the population should have a finite variance.
- The sample size should be large enough (usually n > 30 is considered adequate).
- Limitations:
- If the population distribution is extremely skewed or has heavy tails, larger sample sizes might be necessary for the normal approximation to hold.
- The theorem does not apply to distributions with infinite variance (like the Cauchy distribution).
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