We are going to begin finding proportions above or below a particular
z-score. Many students find this a bit tricky, so I'd like to explain how to
think through these types of problems. The first thing you need to know, is that
because half of the scores are below the mean, and half of the scores are above
the mean, the probability of a score being below or above the mean is P =
.5000, or .5. The second thing to remember is that our Z table
tells us the proportion of scores between the mean and a particular
z-score. With these two bits of information, we can use addition or
subtraction to find the proportion of any part of the bell curve. Let's begin
by tackling the problem of what proportion of scores are below z =
+1.00. Notice that I put a dotted line at z = +1.00. We want
to find the proportion of scores at or below this point. This whole section of
the bell curve actually comprises two things: First, it includes the scores from
the mean to Z equals positive 1.00. This we could find from our Z table. The
proportion is .3413. Second, this section of the bell curve includes all
of the scores below the mean. We know that this proportion is .5, or half
of the scores. By adding these two portions together, we can determine the
whole. Adding .3413 to .5 equals .8413. But what if
our task involved finding a small portion of the bell curve? Say we want to
know what proportion of scores are below Z equals negative 1.25? This is only a
small part of the normal curve, but we can use the same techniques to figure it
out. First, we can go to the Z table and find the proportion of scores from the
mean to -1.25. That value is .3944. We know that half of the scores are
below the mean, a proportion of .5, so we need the remainder. We
subtract our Z value from .5 to get the answer. .5000 minus .3944
equals .1056. Using what we know, let's find the proportion of scores
above or below these Z scores. For each of these examples, it might help to draw
a normal curve to see what part you're trying to find. In this case, I will show
you the proportion on the normal curve below. As in the previous examples, I will
ask the question, and then provide the answer. You should pause the video while
you work on the example on your own. So what proportion of the scores are below
Z equals positive 1.00? It's a .8413. This is the example that we did
just a moment ago. So go back and review that if you still have any questions. Now
it's your turn to try. What proportion of the scores are above a Z of negative 0.50?
0.6915.
What proportion of the scores are above a negative 1.50? .9332. What proportion
of scores are below a positive 0.67? .7486. How about above 0.80? .2119. Below
-2.00? .0228. Below a - 1.01? .1562. And above positive 2.00? .0228.
Notice that the proportion of scores above a positive 2.00 is exactly the
same size that the proportion of scores below a negative 2.00. We can even apply
these skills to our example about the three guys who took the attractiveness
test, and of course, a proportion can be converted to a percentage by moving the
decimal two places to the right. So "Shifty"with his Z of negative 2.50, is
more attractive than 0.62% of all men. Not 62%,
.62%, less than one percent. Now "Mickey", with his Z of a
positive .50 is more attractive than 69.15%
of all men. And "Antonio" with his z of positive 3.00 is more attractive than 99.87%
of all men. Great. Now we're going to use these skills in a new
way. So far, I've just been giving you the z-scores, next we're going to compute Z
scores ourselves from raw data, and then figure out the proportion below the
normal curve.