The normal curve is based on a complex mathematical equation. Its usefulness stems from the fact that many variables in nature are distributed normally or near normally. This allows us to use the equation for a normal curve to do some useful computations.

The bell-shaped curve above represents the shape of a normal
curve. In a normal curve, most of the subjects score near the middle
of the distribution. More importantly, you can compute exactly what
proportion of people score between any two points. For example, 34%
of people fall between the mean (the middle of the distribution) and
one standard deviation above the mean. Another 34% fall between the
mean and one standard deviation below the mean. Therefore, 68% score
within one standard deviation of the mean, and roughly 95% fall
within two standard deviations from the mean. These values can be
computed from the equation for the normal curve, but an easier way
is to access the standard
normal table, which is available on this CD supplement. To use
the standard normal table, you have to convert a score to a *Z*-score.
You compute a *Z*-score by subtracting the mean of the
distribution from the score and dividing the difference by the
standard deviation of the distribution as shown in this equation.

*Z* = (score - mean)/standard deviation

For example, if someone gets a score of 40 on a test with a mean
of 50 and a standard deviation of 20, their *Z*-score would be
-.50. In other words, they scored half a standard deviation below
the mean for the class.

*Z* = (40 - 50)/20 = -10/20 = -.50

What can we do with such a score? Well, if we know that the
scores on the test are distributed normally, we can use the standard
normal table to compute the percentile rank. To use the standard
normal table, we compute the *Z*-score and then look up the
proportion of people using the table. Notice that the
standard normal table
is divided into sets of three columns. The first column (labeled a)
is the *Z*-score; the second column (labeled b) is the
proportion of people between the mean and that *Z*-score; the
third column (labeled c) is the proportion of people beyond that *
Z*-score. If you look closely, you will see that columns (b) and
(c) always add up to .5000 because the two together represent the
entire area from the mean to the end of one tail. Of course, .5000
of the distribution will fall on the other side of the mean because
the normal curve is symmetric.

If someone scored right at the mean, their *Z*-score would
be 0, and their percentile rank would be 50--that is, 50% of the
class would score below them. If *Z*-scores are positive, they
are above the mean and therefore the percentile rank would be
greater than 50. Conversely, if subjects scored below the mean,
their *Z*-score is negative, as in our example, and their
percentile rank is less than 50. We can use the standard normal
table to compute the percentile rank.

It is helpful to draw a quick picture when doing the computation
for percentile rank. In our example, the *Z*-score was a -.50.
That means that the person scored below the mean. We should draw a
line in the middle of the distribution and then another line a bit
to the left of the middle to represent a *Z*-score of -.50. To
get a rough idea of proportion, the distance from the mean to near
the end of either tail of the distribution is about 2 standard
deviations. The percentile rank is the proportion of scores below
that line representing -.50. We can see precisely what proportion of
the distribution that represents by using the
standard normal table.
Notice from your drawing that we want to know how many people are
below a -.50. In the table, that would be the proportion between a
*Z* of .50 and the tail, which is what is shown in column (c).
Notice that that figure is .3085. Multiplying by 100 to convert this
proportion to a percentage gives us 30.85%, which we can round to
31%. A score of 40, which represented a *Z*-score of -.50, is
at the 31st percentile. In other words, 31% of the class scores
below that score.

What if our student had a score of 60 instead of 40. The
student's *Z*-score would be a +.50 (a half standard deviation
above the mean). It helps to draw this by putting a line
representing a +.50 on a normal curve. Clearly the percentile rank
will be greater than 50 because the area below the line at +.50
includes the 50% below the mean and some percentage between the mean
and the +.50 *Z*-score. Using the standard normal table, we
would look up the proportion between the mean and the *Z*-score
and add it to .5000 (the proportion below the mean). Notice that the
proportion between the mean and a *Z*-score of .50 is .1915.
Therefore the percentile rank is (.5000 + .1915) * 100 or 69.15%,
which we round to 69th percentile. Notice that we could have
obtained the same result by looking up in column (c) the proportion
above our *Z*-score and subtracting it from 1.000 and
multiplying by 100 to convert it to percentages. That is, the
percentage scoring below is equal to 100% minus the percentage
scoring above the score. It can be a bit confusing, but with a
little practice it will become second nature.

We have included a few exercises to make sure you understand the principles of using a standard normal table to convert a score to a percentile rank. Remember, to do this you must know the mean and that standard deviation of the distribution and the shape of the distribution should be the bell shape of a normal curve. If you have that information, you compute a percentile rank by:

- Computing a
*Z*-score by subtracting the distribution mean from the score and dividing by the standard deviation. - You draw the situation so that you can see visually what the
*Z*-score represents. Minus*Z*-scores are below the mean and plus*Z*-scores are above the mean. - You then determine the total proportion of the curve below
the
*Z*-score. Remember that you will use a different procedure depending on whether you are above or below the mean. - Finally, you convert that proportion (from the standard normal table) to a percentage by multiplying by 100, which just moves the decimal over two places.

For all of the exercises below, you can assume that the
distribution of scores is normal. You can access the
standard normal table
by clicking on this hyperlink during your exercises. Please note
that the standard normal table is formatted as two pages, with the
larger *Z*-scores on the second page.

- Score = 50; Mean = 35; Standard deviation = 10; Click here to see the correct answer.
- Score = 105; Mean = 80; Standard deviation = 8; Click here to see the correct answer.
- Score = 31; Mean = 40; Standard deviation = 6; Click here to see the correct answer.
- Score = 33; Mean = 36; Standard deviation = 4.5; Click here to see the correct answer.
- Score = 230; Mean = 310; Standard deviation = 150; Click here to see the correct answer.