Question

What does the interquartile range tell us?

Answer 1

Hint: Before getting into the topic interquartile range, we need to know about quartiles.
Quartiles are the divided values that divide a list of numerical data into four equal parts. Thus, we can obtain three quartiles. The first quartile is called the lower quartile which is denoted by ${Q_1}$ . Then, the second quartile is denoted by ${Q_2}$ and ${Q_2}$ is nothing but the median of the given numerical data. And finally, the third quartile is denoted by ${Q_3}$ which is called the upper quartile.
In Statistics, the interquartile range is the difference between the first and third quartiles.
Formula to be used:
Formula to find the interquartile range is as follows.
$\operatorname{interquartile range} = {Q_3} - {Q_1}$
Where, ${Q_3}$ is the upper quartile
And${Q_1}$ is the lower quartile.

Complete step-by-step solution:
Let us consider an example to find quartiles.
Let us take data containing the first ten prime numbers.
That is, $1,3,5,7,11,13,17,19,23,29$.
First we need to arrange them in an ascending or descending order.
Already, our data is in ascending order.
Here, the total number of values is $10$.
Since ${Q_2}$ is the median, the median of $11$ and $13$ is
\[\dfrac{{11 + 13}}{2} = \dfrac{{24}}{2} = 12\]
Hence, ${Q_2} = 12$ .
Now, divide the data into parts to get lower and upper quartiles.
${Q_1}$ contains $1,3,5,7,11$ and ${Q_3}$ contains $13,17,19,23,29$.
Number of values in ${Q_1}$ is $5$ which is odd and the middle value is $5$.
Hence, ${Q_1} = 5$.
Similarly, the number of values in ${Q_3}$ is also $5$ and the middle value is $19$.
Hence, ${Q_3} = 19$.
Therefore, the interquartile range is found by using the above formula.
$\operatorname{interquartile range} = {Q_3} - {Q_1}$
$\begin{gathered}
   = 19 - 5 \\
   = 14 \\
\end{gathered} $
This is how we can calculate the interquartile range.
Both the range and the interquartile range is used to measure the spread of data. But the interquartile range is the range which measures how far apart the first and third quartiles are. That is, the interquartile range tells us how the data is spread from the middle $50\% $.

Note: Interquartile range is calculated by using the following steps:
> We need to arrange the given numerical data in an ascending or descending order. Then, count the number of values.
> If it is odd, the middle value will be median otherwise take the average of two middle values. It is denoted as ${Q_2}$.
> Now, the median will separate the set into two equal parts which are called ${Q_1}$ and ${Q_3}$
> The value of ${Q_1}$ will be obtained from the median of the first part and similarly ${Q_3}$ will be obtained from the median of the second part.
> Finally, subtract ${Q_3}$ from ${Q_1}$ to obtain the interquartile range.