Inter Quartile Range calculator for ungrouped data

Inter Quartile Range for ungrouped data

Inter quartile range is the difference between the third quartile $Q_3$ and first quartile $Q_1$. It is a good measure of spread to use for skewed distribution. Inter-quartile range (IQR) 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

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Inter Quartile Range Calculator for ungrouped data

Use this calculator to find the Inter Quartile Range (IQR) for ungrouped (raw) data.

How to calculate IQR for ungrouped data?

Step 1 - Enter the $x$ values separated by commas

Step 2 - Click on "Calculate" button to get inter quartile range for ungrouped data

Step 3 - Gives the output as number of observations $n$

Step 4 - Gives the output as ascending order data

Step 5 - Gives all the quartiles $Q_1$, $Q_2$ and $Q_3$

Step 6 - Gives the output of Inter-Quartile Range (IQR)

Inter quartile range for ungrouped data Example 1

A random sample of 15 patients yielded the following data on the length of stay (in days) in the hospital.

5, 6, 9, 10, 15, 10, 14, 12, 10, 13, 13, 9, 8, 10, 12.

Find quartiles.

Solution

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

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Arrange the data in ascending order

5, 6, 8, 9, 9, 10, 10, 10, 10, 12, 12, 13, 13, 14, 15

First Quartile $Q_1$

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

$$ \begin{aligned} Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{1(15+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{Value of }\big(4\big)^{th} \text{ observation}\\ &=9 \text{ days}. \end{aligned} $$

Thus, $25$ % of the patients had length of stay in the hospital less than or equal to $9$ days.

Third Quartile $Q_3$

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

$$ \begin{aligned} Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{3(15+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{Value of }\big(12\big)^{th} \text{ observation}\\ &=13 \text{ days}. \end{aligned} $$
Thus, $75$ % of the patients had length of stay in the hospital less than or equal to $13$ days.

Inter-Quartile Range

The inter-quartile range is

$$ \begin{aligned} IQR &= Q_3 - Q_1\\ &= 13 - 9\\ &= 4\text{ days}. \end{aligned} $$

Inter quartile range for ungrouped data Example 2

Blood sugar level (in mg/dl) of a sample of 20 patients admitted to the hospitals are as follows:

75,89,72,78,87, 85, 73, 75, 97, 87, 84, 76,73,79,99,86,83,76,78,73.

Find the value of $Q_1$, $Q_2$ and $Q_3$.

Solution

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

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Arrange the data in ascending order

72, 73, 73, 73, 75, 75, 76, 76, 78, 78, 79, 80, 82, 83, 84, 85, 86, 87, 97, 99

First Quartile $Q_1$

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

$$ \begin{aligned} Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{1(20+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(5.25\big)^{th} \text{ observation}\\ &= \text{Value of }\big(5\big)^{th} \text{ obs.}\\ &\quad +0.25 \big(\text{Value of } \big(6\big)^{th}\text{ obs.}-\text{Value of }\big(5\big)^{th} \text{ obs.}\big)\\ &=75+0.25\big(75 -75\big)\\ &=75 \text{ mg/dl}. \end{aligned} $$

Thus, $25$ % of the patients had blood sugar level less than or equal to $75$ mg/dl.

Third Quartile $Q_3$

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

$$ \begin{aligned} Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{3(20+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(15.75\big)^{th} \text{ observation}\\ &= \text{Value of }\big(15\big)^{th} \text{ obs.}\\ &\quad +0.75 \big(\text{Value of } \big(16\big)^{th}\text{ obs.}-\text{Value of }\big(15\big)^{th} \text{ obs.}\big)\\ &=84+0.75\big(85 -84\big)\\ &=84.75 \text{ mg/dl}. \end{aligned} $$

Thus, $75$ % of the patients had blood sugar level less than or equal to $84.75$ mg/dl.

Inter-Quartile Range

The inter-quartile range is

$$ \begin{aligned} IQR &= Q_3 - Q_1\\ &= 84.75 - 75\\ &= 9.75 \text{ mg/dl}. \end{aligned} $$

Inter quartile range for ungrouped data Example 3

The following data are the heights, correct to the nearest centimeters, for a group of children:

126, 129, 129, 132, 132, 133, 133, 135, 136, 137, 
137, 138, 141, 143, 144, 146, 147, 152, 154, 161 

Find inter quartile range for the above data.

Solution

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

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Arrange the data in ascending order

126, 129, 129, 132, 132, 133, 133, 135, 136, 137, 137, 138, 141, 143, 144, 146, 147, 152, 154, 161

First Quartile $Q_1$

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

$$ \begin{aligned} Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{1(20+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(5.25\big)^{th} \text{ observation}\\ &= \text{Value of }\big(5\big)^{th} \text{ obs.}\\ &\quad +0.25 \big(\text{Value of } \big(6\big)^{th}\text{ obs.}-\text{Value of }\big(5\big)^{th} \text{ obs.}\big)\\ &=132+0.25\big(133 -132\big)\\ &=132.25 \text{ cm}. \end{aligned} $$

Thus, $25$ % of the children had height less than or equal to $132.25$ cm.

Third Quartile $Q_3$

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

$$ \begin{aligned} Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{3(20+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(15.75\big)^{th} \text{ observation}\\ &= \text{Value of }\big(15\big)^{th} \text{ obs.}\\ &\quad +0.75 \big(\text{Value of } \big(16\big)^{th}\text{ obs.}-\text{Value of }\big(15\big)^{th} \text{ obs.}\big)\\ &=144+0.75\big(146 -144\big)\\ &=145.5 \text{ cm}. \end{aligned} $$

Thus, $75$ % of the children had height less than or equal to $145.5$ cm.

Inter-Quartile Range

The inter-quartile range is

$$ \begin{aligned} IQR &= Q_3 - Q_1\\ &= 145.5 - 132.25\\ &= 13.25 \text{ cm}. \end{aligned} $$

Inter quartile range for ungrouped data Example 4

The following measurement were recorded for the drying time in hours, of a certain brand of latex paint.

3.4 2.5 4.8 2.9 3.6 2.8 3.3 5.6 
3.7 2.8 4.4 4.0 5.2 3.0 4.8.

Compute inter quartile range for the above data.

Solution

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

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Arrange the data in ascending order

2.5, 2.8, 2.8, 2.9, 3, 3.3, 3.4, 3.6, 3.7, 4, 4.4, 4.8, 4.8, 5.2, 5.6

First Quartile $Q_1$

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

$$ \begin{aligned} Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{1(15+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(4\big)^{th} \text{ observation}\\ &=2.9 \text{ hours}. \end{aligned} $$

Thus, $25$ % of the drying time is less than or equal to $2.9$ hours.

Third Quartile $Q_3$

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

$$ \begin{aligned} Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{3(15+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(12\big)^{th} \text{ observation}\\ &=4.8 \text{ hours}. \end{aligned} $$

Thus, $75$ % of the drying time is less than or equal to $4.8$ hours.

Inter-Quartile Range

The inter-quartile range is

$$ \begin{aligned} IQR &= Q_3 - Q_1\\ &= 4.8 - 2.9\\ &= 1.9 \text{ hours}. \end{aligned} $$

Inter quartile range for ungrouped data Example 5

The rice production (in Kg) of 10 acres is given as: 1120, 1240, 1320, 1040, 1080, 1720, 1600, 1470, 1750, and 1885. Find the inter quartile range for the given data.

Solution

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

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

Arrange the data in ascending order

1040, 1080, 1120, 1240, 1320, 1470, 1600, 1720, 1750, 1885

First Quartile $Q_1$

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

$$ \begin{aligned} Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{1(10+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(2.75\big)^{th} \text{ observation}\\ &= \text{Value of }\big(2\big)^{th} \text{ obs.}\\ &\quad +0.75 \big(\text{Value of } \big(3\big)^{th}\text{ obs.}-\text{Value of }\big(2\big)^{th} \text{ obs.}\big)\\ &=1080+0.75\big(1120 -1080\big)\\ &=1110 \text{ Kg}. \end{aligned} $$

Thus, $25$ % of the plots had rice production less than or equal to $1110$ Kg.

Third Quartile $Q_3$

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

$$ \begin{aligned} Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ observation}\\ &=\text{Value of }\bigg(\dfrac{3(10+1)}{4}\bigg)^{th} \text{ observation}\\ &= \text{ Value of }\big(8.25\big)^{th} \text{ observation}\\ &= \text{Value of }\big(8\big)^{th} \text{ obs.}\\ &\quad +0.25 \big(\text{Value of } \big(9\big)^{th}\text{ obs.}-\text{Value of }\big(8\big)^{th} \text{ obs.}\big)\\ &=1720+0.25\big(1750 -1720\big)\\ &=1727.5 \text{ Kg}. \end{aligned} $$

Thus, $75$ % of the plots had rice production less than or equal to $1727.5$ Kg.

Inter-Quartile Range

The inter-quartile range is

$$ \begin{aligned} IQR &= Q_3 - Q_1\\ &= 1727.5 - 1110\\ &= 617.5 \text{ Kg}. \end{aligned} $$

Conclusion

In this tutorial, you learned about formula for Inter-quartile Range (IQR) for ungrouped data and how to calculate IQR for ungrouped data. You also learned about how to solve numerical problems based on IQR for ungrouped data.

To learn more about other descriptive statistics measures, please refer to the following tutorials:

Descriptive Statistics

Let me know in the comments if you have any questions on Inter-Quartile Range calculator for ungrouped data with examples and your thought on this article.