Below is the distribution of word lengths for the whole Gettysburg Address.

There are 271 words in the Gettysburg address, and the average word is \(\mu =4.24\) letters long, with a standard deviation of \(\sigma = 2.11\) letters.

In class we each selected random samples of 10 words. The different sample means we collected were an example of a sampling distribution.

Here is the sampling distribution for the sample mean \(\bar{x}\) of 10 randomly selected words based on a simulation of 1000 trials.

Sampling distributions have 3 important features:

1. Shape. You can see that it is more bell-shaped than the original distribution.
2. Center. The center is the same, \(\mu_{\bar{x}} = 4.24\) words.
   
3. Spread. There is less variability than in the original distribution.
   \[\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{10}} = 0.67.\]