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"It has long been an axiom of mine that the little things are infinitely the most important" (Sherlock Holmes)

The standard normal density function

The probability density formula simplifies quite easily.

If we make the mean equal to zero, and the standard deviation 1, this is what happens:

 Z = 1 e -(x-μ)2 / (2σ2) σ√(2π)

 = 1 e -(x-0)2 / (2 * 12) 1 * √(2π)

 = 1 e -x2/2 √(2π)

A normal distribution with a mean of zero, and a standard deviation of 1, is known as a standard normal distribution. This is the normal distribution that is used for statistical tables.

If we substitute for pi, it becomes more straightforward. π ≈ 3.142 So:

 Z = 1 e -x2/2 √(2 * 3.142)

 = 1 e -x2/2 2.507

Now we can readily calculate some values of Z.

In a standard normal distribution, Z is highest where x is zero (that is, it is the same as the mean). Since 0*0 is 0, and 0/2 is also 0, we only need to work out the antilog (to the base e) of 0. This is very easy, as the antilog (to any base) of 0 is 1. Therefore Zmax = 1/2.507 = 0.399

If, however, x were 1, then 1 * 1 = 1 and 1/2 = 0.5, so e-0.5 = 0.6065 Therefore, at one standard deviation from the mean, Z would be 0.6065/2.507 = 0.242

Of course, for x = -1, -12 also equals 1, which is why this curve comes out symmetrically.

For large values of x, things are a little different.

If x = 10 then 102 = 100
100/2 = 50, and e-50 = 0.0000000000000000000001929
So Z = .0000000000000000000000769
Which is an exceedingly tiny number, albeit not zero.