5.4 Using the Standard Normal Table
The standard normal table (usually found in the appendix of a Statistics textbook) can be used to solve problems related to normal distributions.
5.4.1 Find the Area (Probability) for a Given ZScore
In general, the standard normal table gives the area under the standard normal curve to the left of a specified zscore. Using the table, we can calculate the area under the curve to the left of a zscore, to the right of a zscore, between two zscores, or beyond two zscores. Figure 5.6 shows that the area to the left of 1.96 under the standard normal curve is 0.975.
If random variable [latex]Z[/latex] follows a standard normal distribution, more detailed examples of using the standard normal table can be found in Figure 5.7:
 Left panel: the area to the left of 1.96 is 0.975, i.e., [latex]P(Z<1.96)=0.975.[/latex]
 Middle panel: the area to the right of 1.96 is 0.025. There are two ways to solve this problem:
 Recall that the total area under any density curve is one, the area to the right of 1.96 equals one minus the area to the left of 1.96, i.e, [latex]\begin{eqnarray*} P(Z>1.96)&=&\mbox{area under the standard normal curve to the right of 1.96}\\&=&1\mbox{area under the curve to the left of 1.96}=10.975=0.025.\end{eqnarray*}[/latex]
 [latex]P(Z>1.96)=P(Z<1.96)=0.025.[/latex] This is because the standard normal curve is symmetric at 0. The area to the right of 1.96 equals the area to the left of 1.96.
 Right panel: the area between 1.96 and 1.96 is 0.95, i.e., [latex]\begin{eqnarray*}P(1.96 \lt Z \lt 1.96)&=&\mbox{area between 1.96 and 1.96}\\&=&(\mbox{area to the left of 1.96)  (area to the left of 1.96})\\&=&0.9750.025=0.95.\end{eqnarray*}[/latex]
[latex]P(Z \leq a)[/latex]  [latex]P(Z \geq b)[/latex]  [latex]P(a \leq Z \leq b) \\ = P(Z \leq b)  P(Z\leq a)[/latex] 
Figure 5.7: Area to the Left (left panel), Right (middle panel) of a Zscore, and Between Two Zscores (right panel). [Image Description (See Appendix D Figure 5.7)]
Example: Finding Areas Under Standard Normal Curve
Suppose that [latex]Z\sim N(0, 1)[/latex], follows a standard normal distribution.
1. Draw a graph to show and find [latex]P(Z<2)[/latex].
We can find the area to the left of 2 using the standard normal table directly. [latex]P(Z<2)=P(Z<2.00)=0.0228.[/latex] Graph showing the area can be found in Panel (1) of Figure 5.6.
2. Draw a graph to show and find [latex]P(Z>2).[/latex]
This is the area to the right of 2. Recall that the table gives the area to the left of a [latex]z[/latex]score. There are two ways to answer this question:

 Apply the symmetry property of the standard normal curve. The standard normal curve is symmetric at 0, and the area to the right of 2 equals the area to the left of 2. [latex]P(Z>2)=P(Z<2)=0.0228.[/latex]
 Use the property of a density curve: all density curves have an area of one under the curve. The area to the right of 2 equals one minus the area to the left of 2. [latex]P(Z>2)=1P(Z<2)=1P(Z<2.00)=10.9772=0.0228.[/latex] Graph showing the area can be found in Panel (2) of Figure 5.8.
3. Draw a graph to show and find [latex]P(Z<2 \mbox{ or } Z>2).[/latex]
Area beyond 2 and 2, i.e., to the left of 2 or to the right of 2. The two events {Z<2} and {Z>2} don’t overlap and, hence, are mutually exclusive; the special addition rule applies. [latex]P(Z<2 \mbox{ or } Z>2)=P(Z<2)+P(Z>2)=0.0228+0.0228=0.0456.[/latex] Graph showing the area can be found in Panel (3) of Figure 5.8.
4. Draw a graph to show and find [latex]P(4 < Z < 5).[/latex]
The area between 4 and 5 equals the area to the left of 5 minus the area to the left of 4. P(4<Z<5)=P(Z<5)P(Z<4)=10=1. Graph showing the area can be found in Panel (4) of Figure 5.8.
Note: the standard normal table gives the area (in four decimal places) to the left of the zscore between 3.90 and 3.90. Therefore, the area to the left of any zscore below 3.90 is 0, and the area to the left of any zscore above 3.90 is 1.
(1) 
(2) 
(3) 
(4) 
Figure 5.8: Graphs showing the areas under the standard normal curve corresponding to the probabilities in the example. [Image Description (See Appendix D Figure 5.8)] Click on the image to enlarge it.
5.4.2 Find the ZScore for a Given Area (Probability)
We can use the standard normal table in another way: find the [latex]z[/latex]score for a specified area or probability (percentage). The steps are as follows:
 Express the given area in terms of a lefttailed probability (or probabilities if there are 2 zscores).
 Search the main body of the standard normal table for the closest value to the lefttailed probability.
 Obtain the [latex]z[/latex]score that corresponds to the given area. If multiple values are equally close to the given lefttailed probability, take the average of their corresponding [latex]z[/latex]scores.
Example: Given the Area, find the corresponding zscore
Find the [latex]z[/latex]score corresponding to the shaded area in each graph:
Closest value to [latex]0.1[/latex] is [latex]0.1003 \Longrightarrow z= 1.28.[/latex]  Method 1: Area on the right is [latex]0.1 \Longrightarrow[/latex] area on the left is [latex]0.9[/latex], the closest value is[latex]0.8997 \Longrightarrow z= 1.28.[/latex] Method 2: due to symmetry, [latex]z=1.28.[/latex] 
Area on the right is [latex]0.05 \Longrightarrow[/latex] area on the left is [latex]0.95[/latex], two values [latex]0.9495 (z=1.64)[/latex] and [latex]0.9505 (z=1.65)[/latex] are equally close to 0.95 [latex]\Longrightarrow z = \frac{1.64 + 1.65}{2} = 1.645.[/latex] 
The notation [latex]z_{\alpha}[/latex] (read as z alpha) has a special meaning: the zscore that has an area of [latex]\alpha[/latex] to its right under the standard normal curve. The middle and right panels of the example above show that [latex]z_{0.1}=1.28[/latex] and [latex]z_{0.05}=1.645[/latex].
Exercise: Finding Zscore [latex]\color{white}Z_{\alpha}[/latex]
Use the standard normal table to find
 [latex]Z_{0.25}[/latex]: zscore with an area of 0.25 to its right
 [latex]Z_{0.6}[/latex]: zscore with an area of 0.6 to its right
 [latex]Z_{0.005}[/latex]: zscore with an area of 0.005 to its right
Show/Hide Answer
 [latex]Z_{0.25}[/latex]: zscore with an area of 0.25 to its right, so the area to the left of [latex]Z_{0.25}[/latex] is 10.25=0.75. Search the main body of the table; the closest value to 0.75 is 0.7486, which corresponds to the zscore 0.67; therefore, [latex]Z_{0.25}=0.67[/latex].
 [latex]Z_{0.6}[/latex]: zscore with an area of 0.6 to its right, so the area to the left of [latex]Z_{0.6}[/latex] is 10.6=0.4. Search the main body of the table; the closest value to 0.4 is 0.4013, which corresponds to the zscore 0.25; therefore, [latex]Z_{0.6}=0.25[/latex].
 [latex]Z_{0.005}[/latex]: zscore with an area of 0.005 to its right, so the area to the left of [latex]Z_{0.005}[/latex] is 10.005=0.995. Search the main body of the table for 0.995. Two values that are equally close to 0.995 are 0.9949 and 0.9951; the corresponding zscores are 2.57 and 2.58, respectively. Therefore, [latex]Z_{0.005}=\frac{2.57+2.58}{2}=2.575[/latex].