Percentile Score and Percentile Rank

Percentile Score

Percentiles are the points which divide the entire scale of measurement into 100 equal parts.
These are denoted by P0, P1, P2, P3, P4, P5, ......, P99, P100.

According to -

Billford - A percentile is a value on the scoring scale below which lies any given percentage of the cases.
Odell - Percentiles are the points that divide a distribution into 100 equal parts each of which contains 1 percent of the total number of cases.

The first percentile may be defined as that point in a frequency distribution below which lie 1 percent of the total measures or scores.

P0 lies at the beginning of the distribution and P100 at the end of the distribution.

Importance of the Percentile Score

Percentile score of a person tells about his relative position in a group and according to this score we can compare his position with any other person in the group.
Percentile score is used in tests measuring aptitude, personal differences etc. where percentile scores are used instead of actually obtained scores during comparison of different individuals.
Percentile is used during the development of a profile of individual's characteristics.
Percentile scores are also used in the form of derived scores.
Median is the 50th percentile of a given data so it can be said that the percentile can be used in the measurement of central tendency.
First quartile (Q1) and third quartile (Q3) scores of a given data are its 25th (P25) and 75th (P75) percentile scores respectively.

Computation of Percentile

Px = L + [(xN/100 - cfb)/f] × i
where, Px = xth percentile of data
N is the total of all frequencies
cfb = total number of scores upto interval below the percentile class
f = frequency of percentile class
i = Class interval
L = Actual lower limit of the percentile class

Let's understand it with the help of an example -

i) 10th Percentile (P10)

10N/100 = (10×24)/100 = 2.4, it lies in the class 10-19
Thus, 10th percentile (P10) class is 10-19

Now, P10 = L + [(10N/100 - cfb)/f] × i
= 9.5 + [(2.4 - 2)/3] × 10
= 9.5 + 4/3
= 9.5 + 1.33
= 10.83

ii) 25th Percentile (P25)

25N/100 = (25×24)/100 = 6, it lies in the class 20-29
Thus, 25th percentile (P25) class is 20-29

Now, P25 = L + [(25N/100 - cfb)/f] × i
= 19.5 + [(6 - 5)/4] × 10
= 19.5 + 10/4
= 19.5 + 2.5
= 22

Percentile Rank

Percentile Rank (PR) is the point in the distribution below which a given percentage of scores falls.

According to -

Garrett - Percentile rank shows an individual's position on a scale of 100 in which his score entitles him.

If the 80th percentile rank is a score of 65, then it means that 80% of the scores falls below 65.

Importance of Percentile Ranks

Percentile rank of a person tells about his relative position in a group and according to this rank we can compare his position with any other person in the group.
Percentile ranks are also used in the form of derived scores.
Percentile rank is used during the development of a profile of individual's characteristics.
Percentile rank is used in tests measuring aptitude, personal differences etc. where percentile ranks are used instead of actually obtained scores during comparison of different individuals.
It is used in the educational and vocational guidance.

A. Computation of percentile rank in ungrouped data

PR = 100 - (100R - 50)/N
where, R = Rank/Position in the given data
N = Total number of cases.

Let's understand it with the help of an example -

In a class of 40 students, Tarun obtained 15th rank in English class test. Find out Tarun's percentile rank -
PR = 100 - (100R - 50)/N
= 100 - (100 × 15 - 50)/40
= 100 - (1500 - 50)/40
= 100 - 36.25
= 63.75

B. Computation of percentile rank in grouped data

PR = [cfb + (f / i)(x - L)]/N × 100
where, X = Given score
L = Actual lower limit of the score's class
f = frequency of score's class
cfb = total number of scores upto interval below the score's class
N = Total no. of frequencies
i = Class interval

Let's understand it with the help of an example -

i) Percentile Rank of the score 42

PR = [cfb + (f / i)(x - L)]/N × 100
= [5 + (5 / 5)(42 - 39.5)]/50 × 100
= [5 + 2.5]/50 × 100
= 7.5/50 × 100
= 7.5 × 2
= 15

ii) Percentile Rank of the score 62

PR = [cfb + (f / i)(x - L)]/N × 100
= [32 + (6 / 5)(62 - 59.5)]/50 × 100
= [32 + (6 / 5) × 2.5]/50 × 100
= [32 + 3]/50 × 100
= 35/50 × 100
= 35 × 2
= 70