Decile : Decile Definition, How to calculate Decile, Decile Formulas.

Decile :
Decile
Decile Definition :
" All data divided Ten equal parts is known as Decile. "

It is clear that the items which divided the data into ten equal parts lie at the D1, D2, ...., D10 of the way down the arrangement.
Decile denoted by D1 means first decile, D2 means second decile, and so on D10 means tenth decile respectively.

             Mean        Median        Quartiles

Decile Formulas :

For Individual items and Discrete Distribution :
1) First Decile = D1 = Size of [ ( N + 1 ) / 10 ]th item.
2) Second Decile = D2 =  Size of 2 × [ ( N + 1 ) / 10 ]th item.
3) Third Decile = D3 = Size of 3 × [ ( N + 1 ) / 10 ] th item.
and so on,
10) Tenth Decile = D10 = Size of  ( N + 1 ) th item.

For Continues Distribution :
1) First Decile = D1 = Size of [ N / 10 ]th item.
2) Second Decile = D2 = Size of 2 × [ N / 10] th item
and so on,
10) Tenth Decile = D10 = Size N th item.

Deciles Examples :
Computation of Deciles for Individual Data :
1) Calculate the first, second and fifth decile 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.

First Decile = D1 = Size of ( N + 1 ) /10 th item
D1 = Size of ( 11 + 1 ) / 10 th item
D1 = Size of 12 / 10 th item
D1 = Size of 1.2 th item
D1 = Size of 1st item appro.
Therefore, D1 = 50 kg.

Second Decile = D2 = Size of 2 × [ ( N + 1 ) / 10 ] th item
D2 = size of ( 11 + 1 ) / 5 th item
D2 = size of ( 12 / 5 ) th item
D2 = Size of 2.4 th item
D2 = Size of 2 nd item appro.
Therefore, D2 = 50 kg.

Fifth Decile = D5 = Size of 5 × [ ( N + 1 ) /10 ]th item
D5 = Size of 5 × [ ( 11 + 1 ) / 10 ] th item
D5 = Size of 6 th item
Therefore, D5 = 51 kg.

Discrete Distribution :
Q2) From the following data , Calculate First Decile, Second Decile and Fifth  Decile.
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 D1, D2 and D5
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

First Decile = D1 = size of ( N + 1 ) / 10 th item
      = Size of ( 181 + 1 ) / 10 th item
      = Size of ( 182 / 10 ) th item
D1 = Size of 18.2 th item
The item 18.2 nearest to cumulative frequency 28 and it is weight is 50.
Therefore, D1 = 50 kg.

Second Decile = D2 = Size of 2 ×[( N + 1 ) /10] th item
D2 = Size of ( 181 + 1 ) / 5 th item
D2 = size of 182 / 5 th item
D2 = size of 36.4 th item
The item 36.4 lies between 28 and 52 and it is weight is 51.
Therefore, D2 = 51 kg.

Fifth Decile = D5 = Size of 5 [( N + 1 ) / 10] th item
D5 = size of  ( 182 / 2) th item
D5 = size of 91 th item
The item 91 lies between 77 and 99 and it is weight is 53.
Therefore, D5 = 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 :
D1 = Size of ( N / 10 ) th item
D1 = L1 + { [ ( N / 10 ) - c.f. ] / f } × i
Where,
L1 = lower limit of the decile class,
c.f. = cumulative frequency of the previous class,
f = frequency of the decile class,
 i= class interval of the decile class.

D2 = Size of ( N / 5 ) th item
D2 = L1 + { [ ( N / 5 ) - c.f. ] / f } × i
and so on,
D10 = Size of N th item
D10 = L1 + { [N - c.f. ] / f } 


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