# How to Find Probabilities for *Z* with the *Z*-Table

You can use the *Z*-table to find a full set of "less-than" probabilities for a wide range of *z*-values. To use the *Z-*table to find probabilities for a statistical sample with a standard normal (*Z-*) distribution, do the following:

Go to the row that represents the ones digit and the first digit after the decimal point (the tenths digit) of your

*z*-value.Go to the column that represents the second digit after the decimal point (the hundredths digit) of your

*z*-value.Intersect the row and column from Steps 1 and 2.

This result represents

*p*(*Z*<*z*), the probability that the random variable*Z*is less than the value*z*(also known as the percentage of*z*-values that are less than the given*z*value ).

For example, suppose you want to find *p*(*Z* < 2.13). Using the *Z*-table below, find the row for 2.1 and the column for 0.03. Intersect that row and column to find the probability: 0.9834. Therefore *p*(*Z* < 2.13) = 0.9834.

Noting that the total area under any normal curve (including the standardized normal curve) is 1, it follows that *p*(*Z* < 2.13) + *p*(*Z* > 2.13) =1. Therefore, *p*(*Z* > 2.13) = 1 – *p*(*Z* < 2.13) which equals 1 – 0.9834 which equals 0.0166.

Suppose you want to look for *p*(*Z* < –2.13). You find the row for –2.1 and the column for 0.03. Intersect the row and column and you find 0.0166; that means *p*(*Z* < –2.13)=0.0166. Observe that this happens to equal *p*(*Z*>+2.13).The reason for this is ' because the normal distribution is symmetric. So the tail of the curve below –2.13 representing *p*(*Z* < –2.13) looks exactly like the tail above 2.13 representing *p*(*Z* > +2.13).