(II) Deciles

The deciles involve dividing a dataset into ten equal parts based on numerical values. There are therefore nine deciles altogether. Deciles are represented as follows: D₁, D₂, D₃, D₄,…………, D₉. One-tenth (1/10) of every given observation must be less than or equal to D₁, for D₁ to be the usual highest value. The remaining nine-tenths of the same observation, or 9/10, is, nevertheless, greater than or equal to D₁‘s value.

A decile is used to group big data sets in descriptive statistics either from highest to lowest values or vice versa. It takes place on a scale from one to ten, with each number after that representing a rise of ten percentage points. A decile is a type of quantile that, like the quartile and the percentile, divides a set of data into groups that are simple to measure and analyse. In the domains of finance and economics, this kind of data ranking is carried out as a part of several academic and statistical studies. The formulas for calculating deciles are:

[Tex]D_{1}=[\frac{N+1}{10}]^{th}~item [/Tex]

[Tex]D_{2}=[\frac{2(N+1)}{10}]^{th}~item [/Tex]

…………[Tex]D_{9}=[\frac{9(N+1)}{10}]^{th}~item[/Tex]

Where, n is the total number of observations, D₁ is First Decile, D₂ is Second Decile,……….D₉ is Ninth Decile.

Example 1: 

Calculate the D₁, D₅ from the following weights in a family: 25, 17, 32, 11, 40, 35, 13, 5, and 46.

Solution: 

First of all, organise the numbers in ascending order.

5, 11, 13, 17, 25, 32, 35, 40, 46

[Tex]D_{1}=[\frac{N+1}{10}]^{th}~item [/Tex]

[Tex]D_{1}=[\frac{9+1}{10}]^{th}~item [/Tex]

D₁ = 1^st item = 5

[Tex]D_{5}=[\frac{5(N+1)}{10}]^{th}~item [/Tex]

[Tex]D_{1}=[\frac{5(9+1)}{10}]^{th}~item [/Tex]

D₅ = 5^th item = 25

Example 2: 

Calculate D₂ and D₆ for the data related to the age (in years) of 99 members in a housing society.

Solution:

[Tex]D_{2}=[\frac{2(N+1)}{10}]^{th}~item [/Tex]

[Tex]D_{2}=[\frac{2(99+1)}{10}]^{th}~item [/Tex]

D₂ = 20^th item

Now, the 20^th item falls under the cumulative frequency of 25 and the age against this cf value is 18.

D₂ = 18 years

Similarly [Tex]D_{6}=[\frac{6(N+1)}{10}]^{th}~item [/Tex]

[Tex]D_{6}=[\frac{6(99+1)}{10}]^{th}~item [/Tex]

D₆ = 60^th item

Now, the 60^th item falls under the cumulative frequency of 65 and the age against this cf value is 35.

D₆ = 35 years

Example 3: 

Determine D₄ for the company’s salary listed below.

Solution:

In case N is an even number, the following formula is used:

[Tex]D_{4}=[\frac{4N}{10}]^{th}~item [/Tex]

[Tex]D_{4}=[\frac{4(60)}{10}]^{th}~item [/Tex]

D₄ = 24^th item

Now, the 24^th item falls under the cumulative frequency 22 and the salary against this cf value lies in the group 700-800.

[Tex]D_{4}=l+\frac{\frac{4(N)}{10}-m}{f}\times{c} [/Tex]

[Tex]D_{4}=700+\frac{\frac{4(60)}{10}-22}{16}\times{100} [/Tex]

D₄ =  ₹712.5