DECILES

Deciles are the values which divide a given set of values into 10 equal parts. 

The points of sub-divisions being 

D₁, D₂, D₃, D₄, ................ D₉₉

D₁ is the value for which one-hundredth of the observations are less than are equal to D₁ and the remaining ninety-nine hundredths observations are greater than or equal to D₁, once the values are arranged in ascending order of magnitude.  

Example 1 : 

Following are the wages of the laborers : 

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

Find D₆.

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 decile is   

(n + 1)d^th value

To find D₆, substitute 10 for n and 6/10 for p in the above formula.

(n + 1)d^th value  =  (10 + 1) ⋅ (6/10)^th value

Simplify. 

(n + 1)d^th value  =  (11) ⋅ (6/10)^th value

(n + 1)d^th value  =  (66/10)^th value

(n + 1)d^th value  =  6.6^th value

Find 6.6^th value in the ascending order of wages :

6.6^th value  =  6^th value + 0.6(7^th value - 6^th value)

6.6^th value  =  80 + 0.6(82 - 80)

6.6^th value  =  80 + 0.6(2)

9.02^th value  =  80 + 1.2

9.02^th value  =  81.2

So, D_6.6 is $81.20.

Example 1 : 

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

Compute D₇.

Solution :

Computation of D₇

The formula to find decile for grouped frequency distribution is 

Number of values given (N)  =  100

For D₇, we have

d  =  7/10

Then, we have

Nd  =  100⋅ (7/10)  =  70

In the cumulative frequency table above, 70 comes between 57 and 84.

Then, we have 

N_l  =  57

N_u  =  84

l₁  =  899.50

C  =  1099.50 - 899.50  =  200

To find D7, substitute the above values in the formula.

D₇  =  899.50 + [(70 - 57)/(84 - 57)] ⋅ 200

D₇  =  899.50 + (13/27) ⋅ 200

D₇  =  899.50 + 2600/27

D₇  =  899.50 + 96.30

D₇  =  995.80

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