**Percentiles : **

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

The points of sub-divisions being

P_{1}, P_{2}, P_{3}, P_{4}, .......................P_{99}

P_{1} is the value for which one-hundredth of the observations are less than are equal to P_{1} and the remaining ninety-nine hundredths observations are greater than or equal to P_{1}, once the values are arranged in ascending order of magnitude.

**Method 1 (Unclassified Data) : **

For unclassified data, formula to find p^{th} percentile is given by

**(n + 1)p ^{th} value**

Here, p = 1/100, p = 2/100, p = 3/100,......p = 99/100 for P_{1}, P_{2}, P_{3}, .......................P_{99 }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 P_{1}, P_{2}, P_{3}, .......................P_{99 }respectively.

Moreover,

l_{1} = Lower class boundary of the percentile class. That is, the class containing percentile.

N = Total frequency

N_{l} = Less than cumulative frequency corresponding to l_{1}. (Pre percentile class)

N_{u} = Less than cumulative frequency corresponding to l_{2}. (Post percentile class)

l_{2} being the upper class boundary of the percentile class.

C = l_{2} - l_{1} = length of the percentile class.

**Example 1 : **

Following are the wages of the laborers :

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

Find P_{82}.

**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)p^{th} value

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

(n + 1)p^{th} value = (10 + 1) ⋅ (82 / 100)^{th} value

Simplify.

(n + 1)p^{th} value = (11) ⋅ (82 / 100)^{th} value

(n + 1)p^{th} value = (902 / 100)^{th} value

(n + 1)p^{th} value = 9.02^{th} value

Find 9.02^{th} value in the ascending order of wages :

9.02^{th} value = 9th value + 0.02(10th value - 9th value)

9.02^{th} value = 120 + 0.02(130 - 120)

9.02^{th} value = 120 + 0.02(10)

9.02^{th} value = 120 + 0.2

9.02^{th} value = 120.20

So, P_{82} is $120.20.

**Example 2 : **

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

Compute P_{23}.

**Solution : **

The formula to find percentile for grouped frequency distribution is

Number of values given (N) = 100

For P_{23}, 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

N_{l} = 5

N_{u} = 28

l_{1} = 499.50

C = 699.50 - 499.50 = 200

To find P_{23}, substitute the above values in the formula.

P_{23} = 499.50 + [(23 - 5) / (28 - 5)] ⋅ 200

P_{23} = 499.50 + (18 / 23) ⋅ 200

P_{23} = 499.50 + (18 / 23) ⋅ 200

P_{23} = 499.50 + 156.52

P_{23} = 656.02

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

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