## Inter Quartile Range for grouped data

Inter quartile range is given by

$IQR = Q_3-Q_1$

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.

Inter-quartile range

The inter-quartile range is

\begin{aligned} IQR & = Q_3 - Q_1\\ &= 5 - 3\\ & = 2. \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.

Inter-quartile range

The inter-quartile range is

\begin{aligned} IQR & = Q_3 - Q_1\\ &= 20 - 15.25\\ & = 4.75 \text{ minutes}. \end{aligned}