PERCENTILES

Percentiles : 

Percentiles are the values which divide a given set of values into 100 equal parts. 

The points of sub-divisions being 

P1, P2, P3, P4, .......................P99

P1 is the value for which one-hundredth of the observations are less than are equal to P1 and the remaining ninety-nine hundredths observations are greater than or equal to P1, once the values are arranged in ascending order of magnitude.  

Methods to Find Percentiles

Method 1 (Unclassified Data) : 

For unclassified data, formula to find pth percentile is given by 

(n + 1)pth value

Here, p  =  1/100, p  =  2/100, p  =  3/100,......p  =  99/100 for P1, P2, P3, .......................P99 respectively. 

Method 2 (Classified Data or Grouped Frequency Distributrion) : 

In case of grouped frequency distribution, we consider the following formula for computation of percentiles

Here, p  =  1/100, p  =  2/100, p  =  3/100,......p  =  99/100 for P1, P2, P3.......................P99 respectively. 

Moreover

l1  =  Lower class boundary of the percentile class. That is, the class containing percentile. 

N  =  Total frequency

Nl  =  Less than cumulative frequency corresponding to l1. (Pre percentile class)

Nu  =  Less than cumulative frequency corresponding to l2. (Post percentile class)

l2 being the upper class boundary of the percentile class.

C  =  l2 - l1  =  length of the percentile class. 

Percentiles - Examples

Example 1 : 

Following are the wages of the laborers : 

$82, $56, $90, $50, $120, $75, $75, $80, $130, $65

Find P82.

Solution : 

Number of values given (n)  =  10

Arrange the wages in ascending order : 

$50, $56, $65, $75, $75, $80, $82, $90, $120, $130

Formula to find percentile is   

(n + 1)pth value

To find P82, substitute 10 for n and 82/100 for p in the above formula.

(n + 1)pth value  =  (10 + 1) ⋅ (82 / 100)th value

Simplify. 

(n + 1)pth value  =  (11) ⋅ (82 / 100)th value

(n + 1)pth value  =  (902 / 100)th value

(n + 1)pth value  =  9.02th value

Find 9.02th value in the ascending order of wages :

9.02th value  =  9th value + 0.02(10th value - 9th value)

9.02th value  =  120 + 0.02(130 - 120)

9.02th value  =  120 + 0.02(10)

9.02th value  =  120 + 0.2

9.02th value  =  120.20

So, P82 is $120.20.

Example 2 : 

Following distribution relates to the distribution of monthly wages of 100 workers. 

Compute P23.

Solution : 

The formula to find percentile for grouped frequency distribution is 

Number of values given (N)  =  100

For P23, we have

p    =  23/100

Then, we have

Np  =  100 ⋅ (23 / 100)  =  23

In the cumulative frequency table above, 23 comes between 5 and 28.

Then, we have 

Nl  =  5

Nu  =  28

l1  =  499.50

C  =  699.50 - 499.50  =  200

To find P23, substitute the above values in the formula.

P23  =  499.50 + [(23 - 5) / (28 - 5)] ⋅ 200

P23  =  499.50 + (18 / 23) ⋅ 200

P23  =  499.50 + (18 / 23) ⋅ 200

P23  =  499.50 + 156.52

P23  =  656.02

After having gone through the stuff given above, we hope that the students would have understood percentiles.  

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