Why averages are bell-shaped
Average enough random values and the bell curve appears, whatever you started from.
Take a quantity that is anything but bell-shaped: waiting times, most of them short and a few very long. Draw a handful of values and average them. Do it again and again, and plot the averages.
The averages pile up in a bell around the true mean, and the bell is narrower than the spread of the single values. The more values go into each average, the more symmetric and the narrower it gets.
This is the central limit theorem: the average of many independent values is approximately normally distributed, almost regardless of the distribution they come from.
The spread of the averages is the spread of the single values divided by √n. That is why a poll of 1,000 people is useful, and one of 4,000 is only twice as precise.
It is also why the normal distribution turns up so often: heights, measurement errors and exam totals are sums of many small independent influences.
Answer it first
Single values have a standard deviation of 12. What is the standard deviation of averages of 36 values?
Show the answer
2: that is 12 divided by √36 = 6.
Reading is not learning it
Leo teaches this at your pace, asks for your attempt before he shows his, and checks a week later what stayed.