Inter Quartile Range for grouped data Calculator

Inter Quartile Range for grouped data

Use this calculator to find the Inter Quartile Range for grouped (frequency distribution) data.

How to use Inter Quartile Range Calculator?

Step 1 - Select type of frequency distribution either 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 - Click on “Calculate” to find inter quartile range

Step 5 - Gives output as number of observation (n)

Step 6 - Calculate First Quartile, Second Quartile,Third Quartile

Step 7 - Calculate Inter Quartile Range (IQR)

Inter Quartile Range Formula

Inter quartile range is given by

$IQR =Q_3-Q1$

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 - Find Interquartile Range (IQR)

A class teacher has the following data about the number of absences of 35 students of a class. Compute inter quartile range 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 - Calculate Inter Quartile Range (IQR)

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

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

Conclusion

Hope you like above article on Inter Quartile Range Calculator for grouped data helpful and educational.

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