## Decile Calculator for grouped data

Use **Decile calculator** to find the Deciles for grouped data. Deciles for grouped data are the values which divide whole distribution in ten equal parts.These parts are 9 in numbers namely D1,D2..D9. here D1 refer first decile,D2 refer second decile and so on.

Given below Decile calculator for grouped data provides step by step guide procedure about how to calculate decile for grouped data with examples.

## Deciile Calculator

Decile Calculator for Grouped Data | |
---|---|

Type of Frequencies Distribution | DiscreteContinuous |

Enter the Classes for X (Separated by comma,) | |

Enter the Frequencies (f) (Separated by comma,) | |

Which Decile Octile? (Between 1 to 9) | |

Results | |

Number of Observation (N): | |

Required Decile : D{{index}} | |

### How to use Decile Calculator for Grouped Data (frequencies distribution)?

Step 1 - Select type of frequency distribution (Discrete or continuous)

Step 2 - Enter the Range or classes (X) seperated by comma (,)

Step 3 - Enter the Frequencies (f) seperated by comma

Step 4 - Enter the Decile Octile between 1 to 9

Step 5 - Click on Calculate for Decile Calculator for grouped data

Step 6 - Gives output as number of observation (n)

Step 7 - Calculate required decile

## Decile 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

## Example - 1 Decile Calculator for grouped data

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 $D_1$ (first decile) and $D_6$ (six decile).

### 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 for grouped data is

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

where $N$ is the total number of observations.

**$D_1$ First Decile** calculated as:

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

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

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

**$D_6$ Sixth Decile** calculated as:

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

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

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

## Example - 2 Decile for grouped data calculator

The following table gives the amount of time (in minutes) spent on the internet each evening by a group of 56 students.

Time spent on Internet ($x$) | 10-12 | 13-15 | 16-18 | 19-21 | 22-24 |
---|---|---|---|---|---|

No. of students ($f$) | 3 | 12 | 15 | 24 | 2 |

Calculate

a. the maximum time spent on the internet by lower 20 % of the students,

b. the maximum time spent on the internet by lower 50 % of the students,

c. the minimum time spent on the internet by upper 30 % of the students.

#### Solution

Let $X$ denote the amount of time (in minutes) spent on the internet.

Here the classes are inclusive. To make them exclusive type subtract 0.5 from the lower limit and add 0.5 to the upper limit of each class.

Class Interval | Class Boundries | $f_i$ | $cf$ | |
---|---|---|---|---|

10-12 | 9.5-12.5 | 3 | 3 | |

13-15 | 12.5-15.5 | 12 | 15 | |

16-18 | 15.5-18.5 | 15 | 30 | |

19-21 | 18.5-21.5 | 24 | 54 | |

22-24 | 21.5-24.5 | 2 | 56 | |

Total | 56 |

a. The maximum time spent on the internet by lower 20 % of the students is second decile $D_2$.

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

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

where $N$ is the total number of observations.

`$$ \begin{aligned} D_{2} &=\bigg(\dfrac{2(N)}{10}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{2(56)}{10}\bigg)^{th}\text{ value}\\ &=\big(11.2\big)^{th}\text{ value} \end{aligned} $$`

The cumulative frequency just greater than or equal to $11.2$ is $15$, the corresponding class $12.5-15.5$ is the $2^{st}$ decile class.

Thus

- $l = 12.5$, the lower limit of the $2^{st}$ decile class
- $N=56$, total number of observations
- $f =12$, frequency of the $2^{st}$ decile class
- $F_< = 3$, cumulative frequency of the class previous to $2^{st}$ decile class
- $h =3$, the class width

The **second decile** $D_2$ calculated as follows:

`$$ \begin{aligned} D_2 &= l + \bigg(\frac{\frac{2(N)}{10} - F_<}{f}\bigg)\times h\\ &= 12.5 + \bigg(\frac{\frac{2*56}{10} - 3}{12}\bigg)\times 3\\ &= 12.5 + \bigg(\frac{11.2 - 3}{12}\bigg)\times 3\\ &= 12.5 + \big(0.6833\big)\times 3\\ &= 12.5 + 2.05\\ &= 14.55 \text{ minutes} \end{aligned} $$`

The maximum time spent on the internet by lower $20$ % of the students is second decile $D_2 = 14.55$ minutes.

b. The maximum time spent on the internet by lower 50 % of the students is fifth decile $D_5$.

`$$ \begin{aligned} D_{5} &=\bigg(\dfrac{5(N)}{10}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{5(56)}{10}\bigg)^{th}\text{ value}\\ &=\big(28\big)^{th}\text{ value} \end{aligned} $$`

The cumulative frequency just greater than or equal to $28$ is $30$, the corresponding class $15.5-18.5$ is the $5^{th}$ decile class.

Thus

- $l = 15.5$, the lower limit of the $5^{th}$ decile class
- $N=56$, total number of observations
- $f =15$, frequency of the $5^{th}$ decile class
- $F_< = 15$, cumulative frequency of the class previous to $5^{th}$ decile class
- $h =3$, the class width

The **fifth decile** $D_5$ calculated as follows:

`$$ \begin{aligned} D_5 &= l + \bigg(\frac{\frac{5(N)}{10} - F_<}{f}\bigg)\times h\\ &= 15.5 + \bigg(\frac{\frac{5*56}{10} - 15}{15}\bigg)\times 3\\ &= 15.5 + \bigg(\frac{28 - 15}{15}\bigg)\times 3\\ &= 15.5 + \big(0.8667\big)\times 3\\ &= 15.5 + 2.6\\ &= 18.1 \text{ minutes} \end{aligned} $$`

The maximum time spent on the internet by lower $50$ % of the students is fifth decile $D_5 = 18.1$ minutes.

c. The minimum time spent on the internet by upper 30 % of the students is seventh decile $D_7$.

`$$ \begin{aligned} D_{7} &=\bigg(\dfrac{7(N)}{10}\bigg)^{th}\text{ value}\\ &= \bigg(\dfrac{7(56)}{10}\bigg)^{th}\text{ value}\\ &=\big(39.2\big)^{th}\text{ value} \end{aligned} $$`

The cumulative frequency just greater than or equal to $39.2$ is $54$, the corresponding class $18.5-21.5$ is the $7^{th}$ decile class.

Thus

- $l = 18.5$, the lower limit of the $7^{th}$ decile class
- $N=56$, total number of observations
- $f =24$, frequency of the $7^{th}$ decile class
- $F_< = 30$, cumulative frequency of the class previous to $7^{th}$ decile class
- $h =3$, the class width

The **seventh decile** $D_7$ calculated as follows:

`$$ \begin{aligned} D_7 &= l + \bigg(\frac{\frac{7(N)}{10} - F_<}{f}\bigg)\times h\\ &= 18.5 + \bigg(\frac{\frac{7*56}{10} - 30}{24}\bigg)\times 3\\ &= 18.5 + \bigg(\frac{39.2 - 30}{24}\bigg)\times 3\\ &= 18.5 + \big(0.3833\big)\times 3\\ &= 18.5 + 1.15\\ &= 19.65 \text{ minutes} \end{aligned} $$`

The minimum time spent on the internet by upper $30$ % of the students is seventh decile $D_7 = 19.65$ minutes.

## Conclusion

Hope you like **Decile for grouped data calculator** and step by step explaination with examples. Click on Theory to read more about decile for group data.

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