Percentile, Define Percentile, How to calculate Percentile.

Percentile :
Percentile
Percentile Definition :
" All data divided into hundred equal parts is known as Percentile. "
It is clear that the items which divided the data into hundred equal parts lie at the P1, P2, P3, ..., P100 the way down the arrangement.
Percentile denoted by P1 means first Percentile, P2 means second Percentile, P3 means third Percentile and so on respectively.

      Quartiles        Deciles        Mean      
                
Percentile Formulas :
For Individual items and Discrete Distribution : 
First Percentile = P1 = Size of [ ( N + 1 ) / 100 ]th item.
Second Percentile = P2 =  Size of 2 × [ ( N + 1 ) / 100 ]th item
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Hundredth Percentile = P100 = Size of ( N + 1 ) th item.

For Continues Distribution :
First Percentile = P1 =Size of [N /100 ]th item.
Second Percentile = P2 =  Size of 2 × [ N / 100 ]th item
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Hundredth Percentile = P100 = Size of Nth item.

Percentiles Examples :
Computation of Percentiles for Individual Data :
1) Calculate the twentieth, twentyfifth and fiftieth Percentile from the following values weights in kg of a group of students.
50, 51, 52, 50, 51, 53, 55, 54, 52, 51, 50.
Ans. :
Arranging data in ascending order :
50, 50, 50, 51, 51, 51, 52, 52, 53, 54, 55.
Here, N = 11.

Twentieth Percentil = P20 = Size of 20 ×[ ( N + 1 ) ] / 100 th item
P20 = Size of ( 11 + 1 ) / 5 th item
P20 = Size of 12 / 5 th item
P20 = Size of 2.4 th item
P20 = Size of 2nd item appro.
Therefore, P20 = 50 kg.

 Twentyfifth Percentile = P25 
P25 = Size of 25× ( N + 1 ) / 100 th item
P25 = size of ( 11 + 1 ) / 4 th item
P25 = size of 3 th item
Therefore, P25 = 50 kg.

Fiftieth Percentile = P50= Size of 50 × [ ( N + 1 ) /100 ]th item
P50 = Size of  [ ( 11 + 1 ) / 2 ] th item
P50 = Size of 6th item
Therefore, P50= 51 kg.

Discrete Distribution :
Q2) From the following data , Calculate twentieth, twentyfifth and fiftieth Percentiles.
Weight in kg.  :   55,  56,  52,  51,  50, 53,  54.
No. of women :   20,  30,  25,  24,  28, 22,  32.
Ans. :
Step 1: Arrange the data in ascending order
Step 2: Prepare the column of cumulative frequency.
Calculations of  P20, P25 and P50.

Weight (X)  No. of Women (f).  Cumulative                                                                  frequency
     50                   28                        28
     51                   24                        52
     52                   25                        77
     53                   22                        99
     54                   32                      131
     55                   20                      151
     56                   30                      181
Twentieth Percentil = P20 = Size of 20 ×[ ( N + 1 ) ] / 100 th item
      = Size of ( 181 + 1 ) / 5th item
      = Size of ( 182 / 5 ) th item
P20 = Size of 36.4 th item
The item 36.4 i.e. 36 lies between 28 and 52 and it is weight is 51.
Therefore,  P20 = 51 kg.

Twentyfifth Percentile = P25 = Size of 25 × ( N + 1 ) / 100 th item
P25 = Me. = Size of ( 181 + 1 ) / 4th item
P25 = size of 182 / 4 th item
P25 = size of 45.5 th item
The item 45.5 i.e. 46 lies between 28 and 52 and it is weight is 51.
Therefore, P25 = 51 kg.

Fiftieth Percentile = P50 = Size of 50 [( N + 1 ) / 100 ] th item
P50 = Size of ( 181 + 1 ) / 2 th item
P50 = size of 182 / 2 th item
P50 = size of  91 th item
The item 91 lies between 77 and 99 and it is weight is 53.
Therefore, P50= 53 kg.

Continuous Distribution :
Step 1: Arrange the data in ascending order
Step 2: Prepare the column of cumulative frequency.
Step 3: Find the class in which this item lies.
Step 4: Formula :
P1 = Size of ( N / 100 ) th item
P1 = L1 + { [ ( N / 100 ) - c.f. ] / f } × i
Where,
L1 = lower limit of the Percentile class,
c.f. = cumulative frequency of the previous class,
f = frequency of the Percentile class,
 i= class interval of the Percentile class.

P2 = Size of 2 × ( N / 100) th item
P2 = L1 + { [ ( N / 50 ) - c.f. ] / f } × i

P3 = Size of 3( N / 100) th item
P3 = L1 + { [3 ( N / 100 ) - c.f. ] / f } 
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P100= Size of Nth item
P100 = L1 + { [N - c.f. ] / f } 



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