Question

for the following data set, what is the first quartile? {45, 48, 54, 63, 67, 74, 77, 78, 87, 91}

Explanation:

Step1: Sort the data

The data set is already sorted: \( \{45, 48, 54, 63, 67, 74, 77, 78, 87, 91\} \)

Step2: Find the position of the first quartile

The formula for the position of the first quartile \( Q_1 \) is \( \frac{n + 1}{4} \), where \( n \) is the number of data points. Here \( n=10 \), so \( \frac{10 + 1}{4}=2.75 \)

Step3: Interpolate to find \( Q_1 \)

The 2nd value is \( 48 \) and the 3rd value is \( 54 \). Using the formula \( Q_1=x_{j}+(x_{j + 1}-x_{j})\times f \), where \( j = 2 \) (integer part of \( 2.75 \)), \( f=0.75 \) (fractional part of \( 2.75 \)).
\( Q_1=48+(54 - 48)\times0.75=48 + 4.5=52.5 \)

Answer:

\( 52.5 \)