In this section, we will use some of the statistical information that you
have already learned to solve the practical problem of how to indicate the
relative standing of a person on a measure. As part of this section, you will
learn about the ** standard normal distribution**, which is a
distribution that is defined by a mathematical equation. Such mathematical distributions are a critical part of the inferential statistical process
that we will be covering later.

**Relative scores** indicate where a person stands within a specified
normative sample. In general, scores have little meaning unless you know how
other people scored. For example, on probably most of the exams that you have
taken in school, you need to be correct 70% or more of the time just to pass the
course, yet the standardized tests that you took as part of the college
admission process are designed so that less than half of the people taking them
get 70% or more of the questions correct. A professional baseball player who got
a hit only half the times he went up to bat would be by far the greatest hitter
of all time. In contrast, a driver who only reached his or her destination
without an accident half the time would be considered so bad that no insurance
company would cover the person. We are constantly seeking information about
relative scores, sometime even before the scores have been computed. How many
times have you walked out of a test and asked other students whether the test
seemed hard or easy. If you thought it was hard, and therefore are
worried that you did not do well, you are likely to feel a little better after
other students tell you that it seemed very hard to them as well.

The most basic relative score is the percentile rank, which specifies the percentage of people in the normative group who score lower on the measure than yourself. So if you scored at the 25th percentile, it means that 25% of the people score lower than you and 75% score higher than you. Percentile ranks can range from 0 (for the person with the lowest score) to 100 (for the person with the highest score).

Most often percentile ranks are computed from a frequency distribution. Let's again use the frequency distribution that we have used before for examples. That distribution is below. Suppose that we want to compute the percentile rank for a score of 15. From the table we can see that there are 333 people with a score below 15, but what do we do with the 33 people who have exactly 15? Do we count them as scoring above or below our person with a score of 15. The tradition is to assume that half of the people with the same score are below and half are above. That means that we have 33/2=16.5 plus 333 people with lower scores (349.5). There are a total of 394 people. To get the percentile rank, we divide the number of people below our score by the total number of people and multiply by 100 (to convert the proportion to a percent). In this case, the percentile rank is 89 [(394/349.5)*100]. We traditionally round percentile ranks to two significant digits. So we rounded 88.705584% to 89%.

Score |
Frequency |
CumulativeFrequency |

17 | 8 | 394 |

16 | 20 | 386 |

15 | 33 | 366 |

14 | 48 | 333 |

13 | 71 | 285 |

12 | 85 | 214 |

11 | 58 | 129 |

10 | 39 | 71 |

9 | 21 | 32 |

8 | 11 | 11 |

Many variables in psychology tend to show a distinctive shape when graphed
using a histogram or frequency polygon. The shape resembles a bell shaped curve
like the one shown below. This classic bell shaped curve is called a **normal
curve** or **normal distribution**. The normal curve is perfectly
symmetric. The right half and left half are mirror images of one another. The
curve also does not quite reaches zero, although it gets very close. The shape of the
normal curve is actually determined by a complex equation, with dictates the
height of the curve at every point. You need not know the details of this
equation, but you should know that the equation includes two variables. They are
the mean and the standard deviation. The mean dictates where the middle of the
distribution is, which is the highest point of the curve and the point that
separates the the area under the curve into two equal segments. The standard
deviation determines how spread out the curve is.

Because the normal curve is based on an equation, it is possible to know
exactly how high the curve is at every point and how much area is under the
curve between any two scores on the *X*-axis. The figure below marks off 1 and 2
standard deviations both above and below the mean. The area under the curve
between the mean and one standard deviation below the mean is approximately 34%,
as shown in the figure. More precisely, it is 34.13%. We will show you where
that number comes from shortly. Because the curve is symmetric, the area between
the mean and one standard deviation above the mean is also 34%. Similarly, the
area between 1 and 2 standard deviations, either above or below the mean, is
approximately 14%, and the area beyond 2 standard deviations is 2% on
either side of the distribution. All of these areas are determined by the
equation for the normal curve, but you do not need to use this equation, because
the values are computed for you and available in a table called the Area
under the Standard Normal Curve Table. If you click on the link to this
table, you can see what it looks like.

To use the Standard Normal Table, you need to know a little more about the
normal curve and you need to learn about the ** standard score**, also known as the
** Z-score**.
If you look at the two normal curves above, you might recognize that there are
no scores listed on the

Shown below is the equation that converts any score to a standard score using
the values of the score and the mean and standard deviation of the
distribution. A standard score shows where the person scores in a standard
normal distribution. It tells you instantly whether the score is above or below
the mean by the sign of the *Z*-score. If the *Z*-score is positive, the
person scored above the mean; if it is negative, the person scored below the
mean. The size of the *Z*-score indicates how far away from the mean the
person scored.

If you want to see exactly how the standard normal distribution and the *Z*-score can be used to compute a percentile rank, you can click on this link.
Besides walking you through the process, this link provides exercises to help
you master this concept and procedure.

The score on any measure could be converted to a *Z*-score, which would
tell you at a glance how a person scored relative to the reference group. For
example, if someone tells you that her Z-score on the exam was +1.55, you know
immediately that she scored above the mean and enough above the mean that she is
near the top of the class. Remember, most of the normal distribution is
contained between the boundaries of -2.0 and +2.0 standard deviations. There is
only about 2% of the area under the curve in each of the tails. If another
student tells you the his exam score was a *Z* of -.36, you know that he
scores a bit below the mean. With the standard normal table, you could compute
the percentile rank for each of these students in a few minutes. Although *Z*-scores
are very useful and allow people to judge the relative performance of an
individual quickly, many people get easily confused by the negative numbers that
are possible with *Z*-scores. Consequently, many tests compute Z-scores,
but then translate them mathematically to avoid negative numbers.

For example, the IQ test produces a distribution of scores that is very close
to normal. But the IQ test does not give a person's score as a Z-score, but
instead gives an IQ score. The IQ score is simply the Z-score multiplied by 15
and then added to 100, as shown in the equation below. The values of 15 and 100
are arbitrary, but the effect of this transformation is to produce a
normal distribution with a mean of 100 and a standard deviation of 15. So the IQ
distribution looks like the figure below. Note that this figure is identical in
shape to all the other figures in this section. The only difference is that the
scores on the *X*-axis are IQ scores. So just over 95% of people have IQ
scores between 70 and 130, and no one has a negative IQ.

Standardized tests often perform a similar transformation to avoid negative scores. For example, the Scholastic Aptitude Test (SAT) used for college admission and the Graduate Record Exam (GRE) used for admission to graduate school are both standardized so the the mean of the subtests is always 500, with a standard deviation of 100. So if you score 450 on the verbal section of the SAT, you are scoring .5 standard deviations below the mean, which puts you at the 31st percentile. (See if you can do the computations and use the standard normal table to verify this percentile rank. This link shows you the method to make this computation.)

*Z*-scores and transformed *Z*-scores, such as SAT scores, are very
handy and are used extensively in reporting test scores. But it is critical to
understand that the score is meaningful only if you take into account the
normative sample. A quick example will illustrate this point. Let's assume that
Dan took the SAT as a High School senior and scored 650 on each subtest. That is
150 points above the mean (1.5 standard deviations) and would place him at
approximately the 93rd percentile. Four years later, after doing well in
college, he decides to go to graduate school, and so he takes the GRE. This time he
only obtained a 550 on each of the subtests. What happened? Why did his
performance decrease despite the fact that he worked hard in college and did
very well?

You may already have guessed the answer to that question. In effect, we are trying to compare apples and oranges. The scores on the SAT and the GRE mean entirely different things, because they are based on entirely different normative samples. The SAT is taken by people who expect to complete high school and are considering going on to college. In contrast, the GRE is taken by people who expect to graduate from college and plan to go onto graduate school. Anyone who drops out of college or does poorly in college is unlikely to take the GRE. In other words, the normative sample for the GRE is much more exclusive than for the SAT. Dan's GRE score would place him in the 70th percentile of the people applying for graduate school, who are a pretty elite group academically. Most of the people who took the SAT did not take the GRE, and most of the people who take the GRE did very well on the SAT. The competition (i.e., the normative group) was tougher for the GRE than the SAT.

Whenever you are given a normative score, such as a *Z*-score,
percentile rank, or score on a standardized test, you should always consider the
nature of the normative sample. A person making $500,000 per year may be one of
the best paid people in the country (the normative sample including all
workers), but one of the lowest paid CEOs for a Fortune 500 company (a different
normative sample).