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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.