This is a very vague question, it's quite hard to know what is required!

But the main reason that the standard normal distribution is so important and widely used, is that it arises when many small numbers are added together. This makes it really fundamental for describing, for example, random noise or errors, since these are often the sum of small unpredictable events.

To be more concrete, and see this in action, try generating some very simple random numbers, in a spreadsheet for example. If you:

- Generate ten simple numbers, each with a 33% chance of being -1, 0 or 1 (e.g. -1 -1 0 1 0 0 1 -1 1 -1)

- Save the sum of these numbers (e.g. -1 in this case)

- Do this multiple times, and plot a histogram of the sums...

You'll get something that looks like a normal distribution! In fact, for almost any system where random numbers are being added together, the sum will be an approximately normally distributed variable. It doesn't matter if the original randomness is your -1,0,1 choice like above, or a binomial, or poisson variable, or something totally crazy - if you add together enough of them, the sum will be a normally distributed variable (so many of these sums will form a bell curve).

This is why the Normal is so widely used, and so important - many, many complex real-world systems have lots of small sources of randomness adding together. Fortunately we don't need to know exactly what each random element is or how its distributed - we can deal, statistically speaking, with all of it by modelling it as a Normal distribution.

Feb 27th, 2015

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