Percentile

A percentile or centile is the value of variable below which certain percent of observations fall (source: Wikipedia). Here, thirty percentile is value or score below which 30 percent of observations may be found.

The word percentile and related terminology percentile level used in expressive statistics in addition to in reporting of scores from norm-referenced tests. The thirty fifth percentile also known as first quartile Q1; seventieth percentile as median or second quartile Q2; 105th percentile as third quartile Q3.

There is no predefined definition for percentile. When the number of observation is large, all explanation yields the same results.

Nearest rank:

It says that the p-th percentile of N ordered values is acquired by first find the rank n= ((N/100)*p) + ½, rounding to the

adjacent integer and considering the value that consistent to that rank

Linear Interpolation between the closest ranks:

Linear interpolation between the closest ranks is another method. It is used to find the linear interpolation between the two

adjacent ranks rather than rounding.

For N values v1, v2, v3,...,vN , that is

ranked from the smallest amount to greatest then it  define the percentile corresponding to the n-th value as

Pn= (100/N)*(n-1/2)

For example if N=10 then the percentile corresponding to the third value is calculated as

P3= (100/10)*(3-1/2) = 25

Weighted Percentile:

Weighted percentile is defined as the percentage in the total weight is counted rather than the total number. For

example we have positive weights w1, w2, w3,...,wN , associated respectively with our N sample values

Sn = `sum_(k=1)^n` w k where k=1 to n. Let it be the nth partial sum of these weights.

Percentile Formula

Below is the percentile formula to complete your learning over percentile:

The percentile is used in calculation of individual population. We can find the percentile by using following two formulas:

We can find the percentile with 'x' values by using the formula as `(B + 0.5E)/n ` x 100
We can find the percentile without 'x' values by using the formula as `(Number of below x values)/n` x 100


Calculate percentile

Below you can see how to calculate percentile -

Problem 1: The scores for student are 21, 24, 26, 27, 28, 28, 31, 32, 34, 36, 39, 40, 42, 44, 45, 46, 49, 53, 57, 60. Find out the percentile for score 28.

Solution:

The number of scores for student is 20 and the number of below 28 is 4. The 28 is repeated in two times.

Formula for percentile with 'x' values is `(B + 0.5E)/n ` x 100 = `(4 + 0.5(2))/20 ` x 100 = `5/20 ` x 100 =25.

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The score 28 is in 25th percentile.

Problem 2: The scores for student are 40, 41, 42, 43, 45, 47, 49, 50, 51, 51, 54, 57, 59, 60, 61, 63, 67, 69, 70, 71.  Find out the percentile for score 51.

Solution:

The number of scores for student is 20 and the number of below 51 is 8. The number 51 is repeated in two times.

Formula for percentile with 'x' values is  `(B + 0.5E)/n ` x 100 = `(8 + 0.5(2))/20 ` x 100 = `9/20 ` x 100 = 45.

The score 51 is in 45th percentile.

Exercise problems for find percentile:

1. Michel scored 87 out of 150 students and 40 students scored below Michel. Find out the percentile of Michel score.

Solution: Percentile of Michel score is 26.

2. Scores are 11, 12, 13, 15, 15, 16, 18, 20, 22, 23. Find out the percentile for score 15.

Solution: Percentile of score 15 is 40.