Trying to understand Central Limit Theorem via example.
My question is,
In a simple random sample of $1000$ physicists taken among all universities in a country, the number of papers published by the sampled physicists in the past year had a mean of $1.1$ and a standard deviation of $1.8$.
In this example, does the Central Limit Theorem say that the distribution of the number of papers published by the sampled faculty in the past year is roughly normal?
I know that the sample mean is close to the population mean and can be considered as the population mean. The sample size is also quite large, and I assume it's large enough to say the means are distributed roughly normal by CLT. However, the standard deviation is large. I know that on a normal the standard deviation are $+ - 1$, but i don't know how to use the standard deviation given.
If it's not normal then what shape is it?
There's no telling what distribution the number of papers per person has. That is not what the theorem is about. Here's the point. Since each value in the data set is a random variable, the sample mean $\overline{X}_n$ is itself a random variable, with an approximately normal distribution provided the sample is large. More precisely, when the sample size $n$ goes to infinity, $$\sqrt{n}\,\frac{\overline{X}_n-\mu}\sigma\to N\left( 0,\,1\right)$$