When you hear the term “bell curve” what you are actually listening to is a discussion of the “normal” or “Gaussian” distribution.

This is a probability density function (PDF) of the form:

Here is the mean or expectation (peak) and is the variance ( is the standard deviation) and of course is Euler’s number (the base for natural logarithms).

But what does it all mean in a physical sense? Where does it come from?

Let’s look at **variance** first. This measures the spread of the function.

Take an example of a coin tossed four times. Assuming it is a fair coin then we should expect to get 2 heads. But, of course, the process is random so is likely to deviate from that. So variance being the square of deviation, what will that be?

There a chance of four heads (or four tails), a chance of one head (or three heads, ie 1 tail) and a of 2 heads.

So the variance then becomes:

Which comes out as 1.

The normal distribution is a limiting case of the binomial distribution – which looks at success/fail type discrete variables (of course the coin example above is just such a case.) In the binomial distribution has a PDF of the form:

Where is the probability of an event happening and is the probability of it not happening, and where is the binomial coefficient and can be spoken of as “N choose k” and is the number of ways of distributing successes from trials.

Consider the case where becomes large…

Here the change of success is p and the chance of failure is (1 – p), so the average result of each test is , so the mean

The variance for each test is so the total variance is .

Now, let’s look at the cumulative distribution function: this is the probability that the result will be less than or equal to , ie .

For integer results:

(for non-integer results we need to use the *floor* function for , )

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