Question

in the data set below, what is the interquartile range? 15 16 16 17 22 66 91

Explanation:

Step1: Order the data set

The data set is already ordered: \( 15, 16, 16, 17, 22, 66, 91 \)

Step2: Find the median (Q2)

The number of data points \( n = 7 \), which is odd. The median is the middle value, at position \( \frac{n + 1}{2}=\frac{7+1}{2} = 4 \). So the median (Q2) is the 4th value, which is \( 17 \).

Step3: Find Q1 (median of the lower half)

The lower half of the data (values below Q2) is \( 15, 16, 16 \). The number of values here is \( 3 \) (odd). The median of this lower half is the middle value, at position \( \frac{3 + 1}{2}=2 \). So Q1 is the 2nd value of the lower half, which is \( 16 \).

Step4: Find Q3 (median of the upper half)

The upper half of the data (values above Q2) is \( 22, 66, 91 \). The number of values here is \( 3 \) (odd). The median of this upper half is the middle value, at position \( \frac{3 + 1}{2}=2 \). So Q3 is the 2nd value of the upper half, which is \( 66 \).

Step5: Calculate the interquartile range (IQR)

The interquartile range is \( IQR=Q3 - Q1 \). Substituting the values of Q3 and Q1, we get \( IQR = 66-16=50 \).

Answer:

\( 50 \)