Say we have a dataset that contains 1 million values and Every single value is equal to $ 5$.
Say we take take a large number of samples from this set, and work out the mean of the sample each time. According to the central limit theorem, as the sample size approaches $\infty$, the distribution of sample means should tend to a normal distribution.
However, thinking about this logically, the sample mean is going to be $5$ every single time, and therefore the distribution of sample means will not be normally distributed.
How does this work?