Question

in the data set below, what is the interquartile range? 30 32 79 22 24 30 30 18 49 86 32 97

Explanation:

Step1: Sort the data set

Sort the data set in ascending order: $18,22,24,30,30,30,32,32,49,79,86,97$

Step2: Find the median

Since there are $n = 12$ data points, the median is the average of the $6^{th}$ and $7^{th}$ values. The median $Q_2=\frac{30 + 32}{2}=31$

Step3: Find $Q_1$ and $Q_3$

The lower half of the data is $18,22,24,30,30,30$. The median of the lower half ($Q_1$) is the average of the $3^{rd}$ and $4^{th}$ values. $Q_1=\frac{24+30}{2} = 27$
The upper half of the data is $32,32,49,79,86,97$. The median of the upper half ($Q_3$) is the average of the $3^{rd}$ and $4^{th}$ values. $Q_3=\frac{49 + 79}{2}=64$

Step4: Calculate the inter - quartile range (IQR)

The formula for the inter - quartile range is $IQR=Q_3 - Q_1$
Substitute $Q_1 = 27$ and $Q_3=64$ into the formula: $IQR=64 - 27=37$

Answer:

$37$