## Quartile Deviation for grouped data

Quartile Deviation is given by

$QD = \dfrac{Q_3-Q_1}{2}$

where,

- $Q_1$ is the first quartile
- $Q_3$ is the third quartile

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

```
$$
\begin{aligned}
Q_i=l + \bigg(\frac{\frac{iN}{4} - F_<}{f}\bigg)\times h; \quad i=1,2,3
\end{aligned}
$$
```

where

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

## Example 1

A class teacher has the following data about the number of absences of 35 students of a class. Compute five number summary for the following frequency distribution.

No.of days ($x$) | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|

No. of Students ($f$) | 1 | 15 | 10 | 5 | 4 |

### Solution

$x_i$ | $f_i$ | $cf$ | |
---|---|---|---|

2 | 1 | 1 | |

3 | 15 | 16 | |

4 | 10 | 26 | |

5 | 5 | 31 | |

6 | 4 | 35 | |

Total | 35 |

**Inter-quartile range (IQR)**

The inter-quartile range is given by $IQR= Q_3-Q_1$.

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

$Q_i =\bigg(\dfrac{i(N)}{4}\bigg)^{th}$ value, $i=1,2,3$

where $N$ is the total number of observations.

**First Quartile $Q_1$**

```
$$
\begin{aligned}
Q_{1} &=\bigg(\dfrac{1(N)}{4}\bigg)^{th}\text{ value}\\
&= \bigg(\dfrac{1(35)}{4}\bigg)^{th}\text{ value}\\
&=\big(8.75\big)^{th}\text{ value}
\end{aligned}
$$
```

The cumulative frequency just greater than or equal to $8.75$ is $16$. The corresponding value of $X$ is the $1^{st}$ quartile. That is, $Q_1 =3$ days.

Thus, $25$ % of the students had absences less than or equal to $3$ days.

**Third Quartile $Q_3$**

```
$$
\begin{aligned}
Q_{3} &=\bigg(\dfrac{3(N)}{4}\bigg)^{th}\text{ value}\\
&= \bigg(\dfrac{3(35)}{4}\bigg)^{th}\text{ value}\\
&=\big(26.25\big)^{th}\text{ value}
\end{aligned}
$$
```

The cumulative frequency just greater than or equal to $26.25$ is $31$. The corresponding value of $X$ is the $3^{rd}$ quartile. That is, $Q_3 =5$ days.

Thus, $75$ % of the students had absences less than or equal to $5$ days.

**Quartile deviation**

The quartile deviation ($QD$) is

```
$$
\begin{aligned}
QD & = \frac{Q_3 - Q_1}{2}\\
&= \frac{5 - 3}{2}\\
& = 1.
\end{aligned}
$$
```

Coefficient of quartile deviation is

```
$$
\begin{aligned}
\text{Coeff. of }QD & = \frac{Q_3 - Q_1}{Q_3+Q_1}\\
&= \frac{5 - 3}{5 + 3}\\
& = 0.25.
\end{aligned}
$$
```

## Example 2

The following table gives the amount of time (in minutes) spent on the internet each evening by a group of 56 students. Compute Inter quartile range for the following frequency distribution.

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

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

### Solution

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 |

**Quartiles**

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

$Q_i =\bigg(\dfrac{i(N)}{4}\bigg)^{th}$ value, $i=1,2,3$

where $N$ is the total number of observations.

**First Quartile $Q_1$**

```
$$
\begin{aligned}
Q_{1} &=\bigg(\dfrac{1(N)}{4}\bigg)^{th}\text{ value}\\
&= \bigg(\dfrac{1(56)}{4}\bigg)^{th}\text{ value}\\
&=\big(14\big)^{th}\text{ value}
\end{aligned}
$$
```

The cumulative frequency just greater than or equal to $14$ is $15$. The corresponding class $12.5-15.5$ is the $1^{st}$ quartile class.

Thus

- $l = 12.5$, the lower limit of the $1^{st}$ quartile class
- $N=56$, total number of observations
- $f =12$, frequency of the $1^{st}$ quartile class
- $F_< = 3$, cumulative frequency of the class previous to $1^{st}$ quartile class
- $h =3$, the class width

The first quartile $Q_1$ can be computed as follows:

```
$$
\begin{aligned}
Q_1 &= l + \bigg(\frac{\frac{1(N)}{4} - F_<}{f}\bigg)\times h\\
&= 12.5 + \bigg(\frac{\frac{1*56}{4} - 3}{12}\bigg)\times 3\\
&= 12.5 + \bigg(\frac{14 - 3}{12}\bigg)\times 3\\
&= 12.5 + \big(0.9167\big)\times 3\\
&= 12.5 + 2.75\\
&= 15.25 \text{ minutes}
\end{aligned}
$$
```

Thus, $25$ % of the students spent less than or equal to $15.25$ minutes on the internet.

**Third Quartile $Q_3$**

```
$$
\begin{aligned}
Q_{3} &=\bigg(\dfrac{3(N)}{4}\bigg)^{th}\text{ value}\\
&= \bigg(\dfrac{3(56)}{4}\bigg)^{th}\text{ value}\\
&=\big(42\big)^{th}\text{ value}
\end{aligned}
$$
```

The cumulative frequency just greater than or equal to $42$ is $54$. The corresponding class $18.5-21.5$ is the $3^{rd}$ quartile class.

Thus

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

The third quartile $Q_3$ can be computed as follows:

```
$$
\begin{aligned}
Q_3 &= l + \bigg(\frac{\frac{3(N)}{4} - F_<}{f}\bigg)\times h\\
&= 18.5 + \bigg(\frac{\frac{3*56}{4} - 30}{24}\bigg)\times 3\\
&= 18.5 + \bigg(\frac{42 - 30}{24}\bigg)\times 3\\
&= 18.5 + \big(0.5\big)\times 3\\
&= 18.5 + 1.5\\
&= 20 \text{ minutes}
\end{aligned}
$$
```

Thus, $75$ % of the students spent less than or equal to $20$ minutes on the internet.

**Quartile Deviation**

The quartile deviation ($QD$) is

```
$$
\begin{aligned}
QD & = \frac{Q_3 - Q_1}{2}\\
&= \frac{20 - 15.25}{2}\\
& = 2.375\text{ minutes}.
\end{aligned}
$$
```

Coefficient of quartile deviation is

```
$$
\begin{aligned}
\text{Coeff. of }QD & = \frac{Q_3 - Q_1}{Q_3+Q_1}\\
&= \frac{20 - 15.25}{20 + 15.25}\\
& = 0.13475.
\end{aligned}
$$
```