Central Limit Theorem, Return Normality Tests, and Confidence Intervals

Summary

The article introduces the Central Limit Theorem (CLT) through simulated samples from exponential and binomial populations. It explains that, under suitable conditions, sample means approach a normal sampling distribution as sample size grows; the distribution’s mean matches the population mean, while its standard error falls with the square root of sample size. The examples illustrate this approximation, while also showing that a small sample from a skewed population may remain skewed.

It then applies statistical inference to stock returns, examining daily returns for an Indian equity, using a Shapiro-Wilk test to reject normality, and describing confidence intervals for average monthly returns. The article reports an interval estimate and uses it to discuss an investor’s threshold, but the conclusion depends on modeling assumptions and a finite historical sample. The CLT concerns distributions of sample means; it does not make individual returns normal or ensure future returns will match a historical estimate. Confidence intervals are uncertain estimates, not guarantees.

Key ideas

• The CLT describes how the sampling distribution of a mean approaches normality as sample size increases, given suitable conditions.
• The standard error of the sample mean decreases as sample size grows.
• Simulations from exponential and binomial populations illustrate the behavior of sample means.
• A Shapiro-Wilk test is used to assess whether observed stock returns are consistent with normality.
• Confidence intervals estimate a range for an average return, but depend on assumptions and do not guarantee future outcomes.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.