## 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 | |
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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}} | |

## Deciles for ungrouped data example

## 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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