## Deciles

In this page 'Deciles' we are going to see the partitional values of the given data.

### Decile  :  Definition

Deciles are  nine partitional values of the data or the given set of observation into ten equal parts. These 9 values are represented by D₁, D₂, D₃, D₄, D₅, D₆, D₇, D₈ and D₉ .

They  shows the 10%, 20%.30%, 40%, 50%, 60%, 70%, 80% and 90%

For ungrouped data:

Example 1:

Given the series 3,5, 7, 4 6,2 and 9.

Calculate the 2nd and 4th decile.

Solution:

To find the decile first we have to arrange the data in order.

2,3, 4,5, 6, 7 and 9.

Here n = 7

D₂     =  value of  2[(n+1)/10]th item.

=  value of 2x[(7+1)/10]th item

=    value of 1.6th item.

=   1st  value + 0.6 of the distance between 1st and 2nd                                                         value

=       2  + 0.6(3-2)

D₂     =    2.6

Now let us find the value for D₄

Solution:

The ordered data is 2, 3, 4, 5, 6, 7 and 9.

Here n = 7

D₄          =     value of  4[(7+1)/10]th item

=      value of 4 x 8/10 th item.

=      value of 3.2 th item

=      3rd value + 0.2 of the distance between 3rd and                                  4th value

=        4   +   0.2(5-4)

=         4.2

For grouped data:

Where

Lᵢ      =   Lower limit of the decile class.

N      =    Sum of the absolute frequency

Fᵢ₋₁   =      Absolute frequency immediately below the                                                  decile class

aᵢ     =     Width of the class containing the decile                                                 class.

Note:     The decile is independent of the width of the classes.

Let us an example for the grouped data.

Example:

Calculate the decile D₁ and D₃ for the following table.

Solution:

Calculation for the first decile

D₁ = L₁  + { [(k.N)/10 - F₋₁]/f₁}.a

=  40 + {[(1.70)/10 - 0]/8}.10

=   40 +    [(70/10)/8) .10]

=    40 + 70/8

=     390/8

=      48.75

Calculation for 3rd decile

D₃    =   L₃   + { [(k.N)/10 - F₃₋₁]/f₃}.a

=   60   + {[3.70/10-F₂]/14}10

=   60  +  (210/10)-20]/14}.10

=   60  +    [(21-20)/14].10

=  60   +      10/14

=  60   +    0.71

=   60.71

The formula given below is also to find the deciles for the grouped data.

Where

l   =      lower boundary of the class containing the kth                                   decile

h  =    Width of that class

f   =    frequency of that class

n   =   total number of frequencies

c   =     cumulative frequency preceding to that class

We  will get the same answer on doing the above method also.

Students can choose the formula which is more convenient to them to find out the decile. If you are having any doubt you can contact us through mail, we will help you to clear your doubts.