AREAS UNDER THE
NORMAL CURVE
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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.