B \(z\)-score tables
This Appendix contains tables that may be useful:
- \(z\)-tables: Find the area associated with a normal distribution given the \(z\)-score (Appendix B.1).
- \(z\)-tables: Find the \(z\)-score when given an area under a normal distribution (Appendix B.2).
B.1 When the \(z\)-score is known, and the area is sought
The table gives the probability (area) that a \(z\)-score is less than the \(z\)-score looked up. For example: Look up \(z = -1.38\); the area less than \(z = -1.87\) is about 0.0838, or about 8.4%.
To use this table, enter the \(z\)-score in the search box under the z.score column.
The area will be shown.
The table includes \(z\)-values between -4 and 4.
(Alternatively, you can search through the table manually.)
The online Tables work with two decimal places.
As an example, then, consider finding the area to the left of \(z=-2.00\).
In the tables,
the value -2 is entered in the search region, just under the column labelled z.score
(see the animation below).
After pressing Enter,
the answer is shown in the column headed Area.to.left:
the probability of finding a \(z\)-score less than \(-2\) is 0.0228, or about 2.28%.
The hard-copy tables work differently. On the tables, look for \(-2.0\) in the left margin of the table, and for the second decimal place (in this case, 0) in the top margin of the table (see the animation below): where these intersect is the area (or probability) less than this \(z\)-score. So the probability of finding a \(z\)-score less than \(-2\) is 0.0228, or about 2.28%.
B.2 When the area is known, and the \(z\)-score is sought
The table gives the \(z\)-score such that a given probability (area) is to the left of the \(z\)-score. For example: Look up an area of \(0.10\) (i.e., \(10\)%), and the corresponding \(z\)-score is \(z = -1.282\). That is, the area to the left of \(z = 1.282\) is about \(10\)%.
To use this table, enter that area to the left in the search box under the Area.to.left column.
The corresponding \(z\)-score will be shown.
(Alternatively, you can search through the table manually.)
When the \(z\) score was known, the tables in Appendix B.1 were used.
However, when working backwards, the tables in Appendix B.2 are used: enter the area to the left in search box under Area.to.left, and the corresponding \(z\)-scores appears under the z.score column (see the animation below).
The hard copy tables work differently. When the \(z\) scores (which appear in the margins of the tables; see Sect. 22.8) is known, the areas appear in the body of the table. However if the area (or probability), which is in the body of the table, is known, the corresponding \(z\)-score is found in the margins of the table, and hence the observation \(x\); see the animation below.