Percentiles and More Quartiles
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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 25th 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 84th percentile is as higher or higher than 84% of the graduates.  Good job, Jason!

Example:  If Jason graduated 25th 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 83rd 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% = 75th 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 92nd 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 45th 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, 8|6, 88, 88, 90, 94, 96, 98, 98, 99 

The score of 86 is at the 58th 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 25th percentile.
Median (or second quartile) to be at the 50th percentile.
Third quartile (upper quartile) to be a the 75th percentile.
 

Test Scores Frequency Cumulative Frequency
76-80

3

3

81-85

7

10

86-90

6

16

91-95

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 81-85.
 

In a similar fashion, the second quartile will be located  (50% 20 = 10) ten values up from the bottom in the interval 81-85.

The third quartile will be located (75% 20 = 15) fifteen values up from the bottom in the interval 86-90.

 

See how to use your
TI-83+/TI-84+ graphing calculator  to find quartiles.
Click calculator.