Question

in the data set below, what are the lower quartile, the median, and the upper quartile? 29 33 36 44 45 62 lower quartile = median = upper quartile =

Explanation:

Step1: Order the data (already ordered: 29, 33, 36, 44, 45, 62)

The data set has \( n = 6 \) values.

Step2: Find the median (second quartile, \( Q_2 \))

For \( n \) even, median is the average of the \( \frac{n}{2} \)-th and \( (\frac{n}{2}+1) \)-th values.
\( \frac{n}{2}=3 \), so average of 3rd (\( 36 \)) and 4th (\( 44 \)):
\( \text{median} = \frac{36 + 44}{2} = 40 \)

Step3: Split data into lower and upper halves

Lower half: \( 29, 33, 36 \) (first 3 values), Upper half: \( 44, 45, 62 \) (last 3 values)

Step4: Find lower quartile (\( Q_1 \))

Median of lower half (3 values, odd \( n \)): middle value (2nd value) \( = 33 \)

Step5: Find upper quartile (\( Q_3 \))

Median of upper half (3 values, odd \( n \)): middle value (2nd value) \( = 45 \)

Answer:

lower quartile = \( 33 \)
median = \( 40 \)
upper quartile = \( 45 \)