Question

in the data set below, what is the interquartile range? 23 80 23 21 26 13 98 18 82 71 96 60

Explanation:

Step1: Sort the data

Sort the data set in ascending order: $13, 18, 21, 23, 23, 26, 60, 71, 80, 82, 96, 98$.

Step2: Find the median

Since there are $n = 12$ data points, the median is the average of the $\frac{n}{2}=6^{th}$ and $(\frac{n}{2}+1)=7^{th}$ values. The median $Q_2=\frac{26 + 60}{2}=43$.

Step3: Find $Q_1$ and $Q_3$

The lower half of the data is $13, 18, 21, 23, 23, 26$. The median of the lower half (first quartile $Q_1$) is the average of the $3^{rd}$ and $4^{th}$ values. $Q_1=\frac{21+23}{2} = 22$.
The upper half of the data is $60, 71, 80, 82, 96, 98$. The median of the upper half (third quartile $Q_3$) is the average of the $3^{rd}$ and $4^{th}$ values. $Q_3=\frac{80 + 82}{2}=81$.

Step4: Calculate the inter - quartile range (IQR)

The formula for the inter - quartile range is $IQR=Q_3 - Q_1$. Substitute $Q_1 = 22$ and $Q_3=81$ into the formula. $IQR=81-22$.

Answer:

$59$