Percentiles
are like quartiles, except that percentiles divide the
set of data into 100 equal parts while quartiles divide the set of data
into 4 equal parts. Percentiles measure position from the bottom.
Percentiles are most often used for determining the
relative standing of an individual in a population or the rank
position of the individual. Some of the most popular uses for
percentiles are connected with test scores and graduation standings.
Percentile ranks
are an easy way to convey an individual's standing at graduation relative to other
graduates.
Unfortunately, there is no
universally accepted definition of "percentile".
Consider the following two slightly different definitions:
Definition 1:
A percentile is a measure that tells us what percent of the total
frequency scored
at or below that measure. A
percentile rank is the percentage of scores that
fall at or below a given
score. 
Definition 2:
A percentile is a measure that tells us what percent of the total
frequency scored
below that measure. A
percentile rank is the percentage of scores that
fall below a given score. 
Formula:
To find the percentile rank of a
score, x, out of a set of n scores, where x is
included:
Where B = number of scores below x
E = number of
scores equal to x
n = number of
scores
See this formula in more detail in
the Examples section.

Formula:
To find the percentile rank of a
score, x, out of a set of n scores, where x is
not included:

Example: If Jason graduated 25^{th
}out of a class of 150 students, then 125 students were
ranked below Jason. Jason's percentile rank would
be:
Jason's standing in the class at the 84^{th }
percentile is as higher or higher than 84% of the
graduates. Good job, Jason! 
Example: If Jason graduated 25^{th
}out of a class of 150 students, then 125 students were
ranked below Jason. Jason's percentile rank would
be:
Jason's standing in the class at the 83^{rd }
percentile is higher than 83% of the
graduates. Good job, Jason! 
The slight difference in these two definitions can
lead to significantly different answers when dealing with small
amounts of data.
Note: We will be using Definition 1 for the
rest of this page.
(other interpretations are also possible 
check with your teacher)
About Percentile
Ranks:
• percentile rank is a number between 0 and 100
indicating the percent of cases
falling at or below that score.
• percentile ranks are usually written to the nearest whole percent:
74.5% = 75% = 75^{th} percentile
• scores are divided into 100 equally sized groups
• scores are arranged in rank order from lowest to highest
• there is no 0 percentile rank  the lowest score is
at the first percentile
• there is no 100th percentile  the highest score is
at the 99th percentile.
• you cannot perform the same mathematical
operations on percentiles that you can on raw scores. You
cannot, for example, compute the mean of percentile
scores, as the results may be misleading. 

Consider:
1. Karl takes the big Earth Science test and
his teacher tells him that he scored at the 92^{nd}
percentile. Is Karl pleased with his
performance on the test?
He should be. He scored
as high or higher than 92% of the people taking the test.
2. Sue takes the Chapter 4 math
test. If Sue's score is the same as "the mean"
score for
the math test, she scored at the 50th percentile and
she did "as well or better than" 50% of the
students taking the test. 

3. If
Ty scores at the 75th percentile on the Social
Studies test, he did "as well or better than" 75% of the students taking the
test.
Examples:
Finding Percentiles 
1. The
math test scores were: 50, 65, 70, 72, 72, 78, 80, 82, 84,
84, 85, 86, 88, 88, 90, 94, 96, 98, 98, 99. Find the
percentile rank for a score of 84 on this test.
Be sure the scores are ordered
from smallest to largest.
Locate the 84.
Solution Using Formula:
Solution Using Visualization:
Since
there are 2 values equal to 84, assign one to the group "above
84" and the other to the group "below 84".
50, 65, 70, 72, 72, 78, 80, 82, 84,
 84,
85, 86, 88, 88, 90, 94, 96, 98, 98, 99
The score of 84 is at the 45^{th}
percentile for this test.
2.
The math test scores were: 50, 65, 70, 72, 72,
78, 80, 82, 84, 84, 85, 86, 88, 88, 90, 94, 96, 98, 98, 99.
Find the percentile rank for a score of 86 on this test.
Be sure the scores are ordered
from smallest to largest.
Locate the 86.
Solution Using Formula:
Solution Using Visualization:
Since
there is only one value equal to 86, it will be counted as
"half" of a data value for the group "above 86" as well as the
group "below 86".
50, 65, 70, 72, 72, 78, 80, 82, 84, 84, 85,
86,
88, 88, 90, 94, 96, 98, 98, 99
The score of 86 is at the 58^{th}
percentile for this test.
3.
Quartiles can be thought of as percentile measure. Remember that
quartiles break the data set into 4 equal parts. If 100% is
broken into four equal parts, we have subdivisions at 25%, 50%, and
75% creating the:
First
quartile
(lower quartile) to be at the
25^{th}
percentile.
Median
(or
second quartile) to be at the
50^{th}
percentile.
Third
quartile
(upper
quartile) to be a the
75^{th}
percentile.

Test Scores 
Frequency 
Cumulative Frequency 
7680 
3 
3 
8185 
7 
10 
8690 
6 
16 
9195 
4 
20 

For the table at the left,
find the intervals in which the first, second and third
quartiles lie.
If there
are a total of 20 scores, the first quartile will be located
(25% · 20 = 5) five values up from the bottom. This
puts the first quartile in the interval 8185.

In a
similar fashion, the second quartile will be located
(50% · 20 = 10) ten values up from the bottom in the
interval 8185.
The third
quartile will be located (75% · 20 = 15) fifteen values up
from the bottom in the interval 8690. 

See how to use
your
TI83+/TI84+ graphing calculator to find
quartiles.
Click calculator. 


