How do you convert a normal distribution into a standard normal distribution and what will its density be?

Question

Can anyone explain to me how to use the table of standard normal distributions?

Answer 1

Standard Normal deviation 65% 2nd SND 95% 3rd nomal deviation 99%

I guess you could say that a normal distribution, like sand falling is even and equal on both sides of the distribution, - it is not skewed.

Answer 2

First, assume that X follows a normal distribution with mean m and standarddeviation s, and assume that Z follows a standard normal distribution. Then X and Z are related as:

X = s * Z + m

or conversely

Z = (X - m) / s.

Use the second formula to transform the normal distribution X into a standard normal one (Z).

Second, the density of a standard normal distribution. Recall that for X (defined above), the density function is:

f_X(x) = (2 * pi * s^2)^(-1/2) * exp(- (x- m)^2 / (2 * s^2)).

Since a standard normal distribution has mean m = 0 and standarddeviation s = 1, it is easily seen that its density function is

f_Z(z) = (2 * pi)^(-1/2) * exp(- x^2 / 2).

The last question about using the table of standard normal distributions I'll have to pass for. I find it a bit too tricky to explain over this medium (too cumbersome). You can probably find the answer you're looking for in any good textbook on basic statistics.