Question

How do you find the range and interquartile range of 9, 10, 12, 13, 10, 14, 8, 10, 12,6, 8, 11, 12, 12, 9, 11, 10,15, 10, 8, 8, 12, 10, 14, 10, 9, 7, 5, 11, 15, 8, 9, 17, 12, 12, 13, 7, 14, 6, 17, 11, 15, 10, 13, 9, 7, 12, 13, 10, 12?

Answer 1

Hint: To obtain the range and interquartile use the data given. Firstly assign the data in ascending order then to find the range subtract the lowest value from the highest value. Then for finding the interquartile range get the value of the first second and third quartile. Finally subtract the first quartile from the third quartile to get the answer.

Complete step by step answer:
The data set is given as:
$\begin{align}
  & 9,10,12,13,10,14,8,10,12,6,8,11,12,12,9,11,10,15,10,8,8,12,10,14,10 \\
 & 9,7,5,11,15,8,9,17,12,12,13,7,14,6,17,11,15,10,13,9,7,12,13,10,12 \\
\end{align}$
Firstly we will arrange them in ascending order as,
$\begin{align}
  & 5,6,6,7,7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11 \\
 & 11,12,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14,15,15,15,17,17 \\
\end{align}$
Now for calculating the range we will subtract the lowest value from the highest value as:
Range $=17-5$
Range $=12$
Now, for finding the interquartile range $\left( IQR \right)$ we will find the Quartile 1 $\left( Q1 \right)$, Quartile 2 $\left( Q2 \right)$, and Quartile 3 $\left( Q3 \right)$ as below:
So as we know Quartile 2 is equal to the median of the data and median of the above data will be
$\begin{align}
  & 5,6,6,7,7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,10,10,10,\underline{10,11},11,11 \\
 & 11,12,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14,15,15,15,17,17 \\
\end{align}$
Median$=\dfrac{10+11}{2}$
Median $=10.5$
$Q2=10.5$…..$\left( 1 \right)$
Next the Q1 is the middle term of the first half of the data so the first half of the data is:
$\begin{align}
  & \underline{5,6,6,7,7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,10,10,10,10} \\
 & 11,11,11,11,12,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14,15,15,15,17,17 \\
\end{align}$
So from the first half the middle term is:
$\therefore Q1=9$……$\left( 2 \right)$
Next the Q3 is the middle term of the second half of the data so the second half of the data is:
$\begin{align}
  & 5,6,6,7,7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,10,10,10,10, \\
 & \underline{11,11,11,11,12,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14,15,15,15,17,17} \\
\end{align}$
So the middle term in the second half is:
$\therefore Q3=12$……$\left( 3 \right)$
On subtracting equation (2) from equation (3) we get,
$\begin{align}
  & IQR=12-9 \\
 & \therefore IQR=3 \\
\end{align}$

Hence the range and interquartile range of the data given is 12 and 3 respectively.

Note: The interquartile is obtained by getting the difference between the highest quartile and the lowest quartile of the data given whereas range is the difference between the highest and lowest value of the data.