AREAS UNDER THE NORMAL CURVE


We are often interested in finding the areas under the normal curve which are associated with some given z score. Remember from your previous lessons that the normal curve can be divided into sections by each standard deviation, beginning with a z score of zero in the center.

 

 

FIGURE 1

 

 

Notice that the total area under the curve is 1.000 by definition (that would correspond to 100% in Figure 1 above). The area under the curve is depicted in yellow in Figure 2 below.

 

 

FIGURE 2

 

 

Now try to see (Figure 3 below) that the curve can theoretically be split in half at z=0, and there will be .5000 area above z=0 and .5000 area below z=0.

 

 

FIGURE 3

 

 

Frequently we ask questions in statistics which require us to know the area above or below a given z value, or the area between two z values. Referring to Figure 3 above, note that I can say that the area above z=0 is .50, and the area below z=0 is .50. Now look back at Figure 1 at the top of the page. The area between z=0 and z=+1 is .34. Notice that it can be stated that the area above z=+1 is .16. This is because the area in purple is .02 (which should be easily deducible since there is .50 above z=0, and of that .50, there is .34 + .14 between z=0 and z=+2.  That is a total of .48, so that leaves the area in purple to be .02 (since the total above z=0 is .50).

Notice from Figure 1 we can make all kinds of statements about integer z values. For instance, the area between z=-1 and z=+1 is .68 (because .34 + .34 = .68). The area between z=-2 and z=+1 is  .14  +  .34  +  .34,  or .82.

Frequently, though, our z values of interest are no perfect integers. For instance, we might need to know the area above z= +1.74. That is why we have z tables, and now is the time to learn about them.

There are basically three types of z tables.

They are illustrated below for a z value of +1 :

 

The first type lists the area between the mean (z=0) and the z of interest:

 

 

The second type lists the area above the z of interest.

 

 

The third type lists the area below the z of interest.

 

The best way to determine what type of z table you have is to look up a z of 1.00.  From Figure 1 above, if it is the first type of table (area between z=0 and z of interest), then the tabled value for z=+1 will be .34 (rounded). If the table is of the second type (area above the z of interest), then the tabled value for z=1.00 will be .16 (rounded). If it is the third type (area below the z of interest), the tabled value for z= 1.00 will be .84 (rounded). Please do not continue unless you understand why this is true.

What type of z table is the NIST table used in our course? The way to find out is to look up the value for z=+1.

The NIST table in our course is obviously the first type. It lists the area between z=0 (the mean) and the z of interest. Notice the columns in our table are for the third digit of the z value. For instance, z = 1.01 is .34375, and z= 1.02 is .34614.

Now let's look at a "real world" problem. Suppose we need to know the normal curve area which falls above z=+1.74. The problem is solved by setting up the theoretical model below. Looking at our NIST table above, we see the tabled value for z=1.74 is .4591 (that is the area below in yellow). We are interested in the blue area (that would be the area above z=+1.74).

Since .4591 + (blue area) = .5000, the blue area must be .5000 - .4591. That is .0409. Refer to figure 3 above if you don't see where the .5000 comes from. The total of the yellow and blue areas MUST BE .5000.

 

 

 

Many students ask for a rule about "...when do I subtract .5 and when do I add .5. I just can't get it right". I don't know. Neither can I. It depends on the type of table you have and the area you are looking for. I have to draw the picture above (perhaps only in my head) to know what needs to be done.

Now, let's look at the area above z = -1.14.  Let's draw it:

The two sides of the curve are mirror images of each other, so I just use my reasoning backwards this time. By looking up a z of 1.14 in the NIST table above, I see the area between that z and the mean is .3729. Then, from the picture above, I know that I need the sum of the yellow and green areas (I think that would be the area above z= -1.14, wouldn't it?).

Ask questions if you do not understand.