## Deciles for grouped data calculator

Use Decile calculator to find the Deciles for grouped (frequency distribution) data.

Deciles Calculator (Grouped Data)
Type of Freq. Dist. DiscreteContinuous
Enter the Classes for X (Separated by comma,)
Enter the frequencies (f) (Separated by comma,)
Which Octile? (Between 1 to 9)
Results
Number of Obs. (N):
Required Decile : D{{index}}

## Deciles for grouped data formula

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

For discrete frequency distribution, the formula for $i^{th}$ decile is

### $D_i =\bigg(\dfrac{i(N)}{10}\bigg)^{th}$ value, $i=1,2,\cdots, 9$

where,

• $N$ is total number of observations.

For continuous frequency distribution, the formula for $i^{th}$ quartile is

### $D_i=l + \bigg(\dfrac{\dfrac{iN}{10} - F_<}{f}\bigg)\times h$; $i=1,2,\cdots, 9$

where,

• $l$ is the lower limit of the $i^{th}$ decile class
• $N=\sum f$ total number of observations
• $f$ frequency of the $i^{th}$ decile class
• $F_<$ cumulative frequency of the class previous to $i^{th}$ decile class
• $h$ is the class width

## Decile for grouped data example

A librarian keeps the records about the amount of time spent (in minutes) in a library by college students. Data is as follows:

Time spent 30 32 35 38 40
No. of students 8 12 20 10 5

Calculate $D_1$ and $D_6$.

### Solution

$x_i$ $f_i$ $cf$
30 8 8
32 12 20
35 20 40
38 10 50
40 5 55
Total 55

Deciles

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

$D_i =\bigg(\dfrac{i(N)}{4}\bigg)^{th}$ value, $i=1,2,\cdots, 9$

where $N$ is the total number of observations.

First Decile $D_1$

\begin{aligned} D_{1} &=\bigg(\dfrac{1(N)}{10}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{1(55)}{10}\bigg)^{th}\text{ value}\\ &=\big(5.5\big)^{th}\text{ value} \end{aligned}

The cumulative frequency just greater than or equal to $5.5$ is $8$. The corresponding value of $X$ is the $1^{st}$ decile. That is, $D_1 =30$ minutes.

Thus, $10$ % of the students spent less than or equal to $30$ minutes.

Sixth Decile $D_6$

\begin{aligned} D_{6} &=\bigg(\dfrac{6(N)}{10}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{6(55)}{10}\bigg)^{th}\text{ value}\\ &=\big(33\big)^{th}\text{ value} \end{aligned}

The cumulative frequency just greater than or equal to $33$ is $40$. The corresponding value of $X$ is the $6^{th}$ decile. That is, $D_6 =35$ minutes.

Thus, $60$ % of the students spent less than or equal to $35$ minutes.

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