Question

in the data set below, what are the lower quartile, the median, and the upper quartile?
27 46 48 51 51 63 66 66 72 99 99
lower quartile =
median =
upper quartile =

Explanation:

Step1: Check Data Order

The data set is already in ascending order: \(27, 46, 48, 51, 51, 63, 66, 66, 72, 99, 99\). There are \(n = 11\) data points.

Step2: Find the Median

The median is the middle value of a sorted data set. For \(n = 11\) (odd), the median is at position \(\frac{n + 1}{2}=\frac{11+ 1}{2}=6\). The 6th value in the sorted data is \(63\), so median \(= 63\).

Step3: Split Data for Quartiles

Split the data into lower half (values below median) and upper half (values above median).
Lower half: \(27, 46, 48, 51, 51\) (5 values).
Upper half: \(66, 66, 72, 99, 99\) (5 values).

Step4: Find Lower Quartile (Q1)

Q1 is the median of the lower half. For the lower half (\(n = 5\), odd), median is at position \(\frac{5 + 1}{2}=3\). The 3rd value in lower half is \(48\), so Q1 (lower quartile) \(= 48\).

Step5: Find Upper Quartile (Q3)

Q3 is the median of the upper half. For the upper half (\(n = 5\), odd), median is at position \(\frac{5 + 1}{2}=3\). The 3rd value in upper half is \(72\), so Q3 (upper quartile) \(= 72\).

Answer:

lower quartile = \(48\)
median = \(63\)
upper quartile = \(72\)