Normal Probabilities Using the ZTable
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Any normal distribution, which is completely described by the values of its mean, , and its standard deviation, , can be transformed into the socalled standard normal via the transformation
The "Ztable" gives probabilities that values of are found between 0 and given values. These probabilities are represented by areas under the density curve of the standard normal (which is like a continuous histogram). The total area under this curve is 1. Hence, to the right or to the left of 0 is half the data, and so corresponds to a probability or area of 1/2.
Now here's how we generally use the "Ztable":
This one is straight out of the Ztable: 

Because of the beautiful symmetry of the normal density curve, we only need to give values to the right of 0; to get answers for values of to the left of zero, we simply reflect the problem from the left to the right:


Split the problem into two problems, corresponding to the two cases above:


You can imagine removing the area in the first problem above from the entire amount of area on the right hand side (which amounts to .5, or half the total area): 

Once again, imagine removing the area between 0 and .5 from the area between 0 and 1.7: 