I am looking for a proof for the normal distribution.

I suppose "proof" was not a good choice of words. What I am looking for is a way to "derive" the normal distribution in simple terms so that the most average teenager can see the logic. Can you help me?

I am a secondary education teacher.
My name is Donna.
Thanks for your help!

If you are looking for a way to justify why we use the normal distribution then there are two reasons.

The first is that some "natural" phenomena have a distribution that is close to a normal distribution. Average rainfall in a particular city, heights of adult females, body temperature, IQ scores, the weight of a "five pound" bag of sugar,... In statistical language if you perform the "experiment" of randomly selecting a five pound bag of sugar and weighing it accurately then you are looking at one observation from a normal distribution. To justify this, repeat the experiment a large number of times (a few hundred), record the weight each time and construct a histogram of the weights. You should see a bell-shaped histogram that resembles a normal distribution.

The second, and more important reason, comes from a theorem in statistical theory called the Central Limit Theorem. It has to do with means or averages of samples. Suppose that you are interested in the number of TV's per household in some city. This is certainly not a normal distribution, in fact for each household the number of TV's is probably some integer between 0 and 6. In this case the experiment that a statistician would perform, if he or she were interested in the average number of TV's per household, would be to take a random sample of say 100 households, find the number of TV's in each and calculate the mean. The Central Limit Theorem says that this mean is one observation from a normal distribution. To justify this, repeat the experiment a large number of times (a few hundred), calculate the mean number of TV's in each sample and construct a histogram of these means. You should see a bell-shaped histogram that resembles a normal distribution. This is the sort of thing that statisticians and pollsters do quite often. They take a random sample calculate the mean and then make some probability statement based on the normal distribution.

I hope that this makes some sense to you.

Harley