Does the central limit theorem apply to non-normally distributed variables? Why?
The central limit theorem (CLT) is a fundamental concept in statistics that states that the distribution of sample means follows an approximately normal distribution, regardless of the shape of the original population distribution. This means that even if the individual data points do not follow a normal distribution, the distribution of their means will become closer to a normal distribution as the sample size increases.
To understand why the central limit theorem applies to non-normally distributed variables, let's break down the idea behind it. The central limit theorem relies on two key assumptions:
1. Independence: The observations in the sample should be independent of each other. This means that the value of one observation should not be influenced by the values of other observations.
2. Adequate Sample Size: The sample size needs to be sufficiently large. While there is no strict cutoff for what constitutes a "sufficient" sample size, as a general rule of thumb, a sample size of at least 30 is often considered appropriate.
When these two assumptions are met, the central limit theorem holds true. It is worth noting that the larger the sample size, the better the approximation to the normal distribution.
So, in summary, the central limit theorem can be applied to non-normally distributed variables as long as the observations in the sample are independent and the sample size is sufficiently large.