## Deciles for ungrouped data calculator

Use Decile calculator to find the deciles for ungrouped (raw) data.

Deciles are the values of arranged data which divide whole data into ten equal parts. They are 9 in numbers namely $D_1,D_2, \cdots, D_9$. Here $D_1$ is first decile, $D_2$ is second decile, $D_3$ is third decile and so on.

## Formula

The formula for $i^{th}$ decile is

### $D_i =$ Value of $\bigg(\dfrac{i(N+1)}{10}\bigg)^{th}$ observation, $i=1,2,3,\cdots, 7$

where $N$ is the total number of observations.

Decile Calculator for ungrouped data
Enter the X Values (Separated by comma,)
Which Decile? (Between 1 to 9)
Results
Number of Obs. (n):
Ascending order of X values :
Required Decile : D{{index}}

## Example 1

The marks obtained by a sample of 20 students in a class test are as follows:

20,30,21,29,10,17,18,15,27,25,16,15,19,22,13,17,14,18,12 and 9.

Find

a. the upper marks for the lowest 20 % of the students,,

b. the upper marks for the lowest 50 % of the students,

c. the lower marks for the upper 10 % of the students.

### Solution

The formula for $i^{th}$ decile is

$D_i =$ Value of $\bigg(\dfrac{i(n+1)}{10}\bigg)^{th}$ observation, $i=1,2,3,\cdots, 9$

where $n$ is the total number of observations.

Arrange the data in ascending order

9, 10, 12, 13, 14, 15, 15, 16, 17, 17, 18, 18, 19, 20, 21, 22, 25, 27, 29, 30

a. The upper marks for the lowest 20 % of the students is $D_2$.

The second decile $D_2$ can be computed as follows:

\begin{aligned} D_{2} &=\text{Value of }\bigg(\dfrac{2(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{2(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{Value of }\big(4.2\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(4\big)^{th} \text{ obs.}+0.2 \big(\text{Value of } \big(5\big)^{th}\text{ obs.}-\text{Value of }\big(4\big)^{th} \text{ obs.}\big)\\ &=13+0.2\big(14 -13\big)\\ &=13.2 \text{ marks} \end{aligned} Thus, the upper limit of marks obtained by the students for the lowest $20$ % of the students is $13.2$ marks.

b. The upper marks for the lowest 50 % of the students is $D_5$.

The fifth decile $D_5$ can be computed as follows:

\begin{aligned} D_{5} &=\text{Value of }\bigg(\dfrac{5(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{5(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{Value of }\big(10.5\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(10\big)^{th} \text{ obs.}+0.5 \big(\text{Value of } \big(11\big)^{th}\text{ obs.}-\text{Value of }\big(10\big)^{th} \text{ obs.}\big)\\ &=17+0.5\big(18 -17\big)\\ &=17.5 \text{ marks} \end{aligned} Thus, the upper limit of marks obtained by the students for the lowest $50$ % of the students is $17.5$ marks.

c. The lower marks for the upper 10 % of the students is $D_9$.

The nineth decile $D_9$ can be computed as follows:

\begin{aligned} D_{9} &=\text{Value of }\bigg(\dfrac{9(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{9(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{Value of }\big(18.9\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(18\big)^{th} \text{ obs.}+0.9 \big(\text{Value of } \big(19\big)^{th}\text{ obs.}-\text{Value of }\big(18\big)^{th} \text{ obs.}\big)\\ &=27+0.9\big(29 -27\big)\\ &=28.8 \text{ marks} \end{aligned} Thus, the lower marks obtained by the students for the upper $10$ % of the students is $28.8$ marks.

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