Question

in the data set below, what are the lower quartile, the median, and the upper
11 31 31 37 49 52 82 83 87 95 95
lower quartile =
median =
upper quartile =

Explanation:

Step1: Check data set order

The data set is already in ascending order: \(11, 31, 31, 37, 49, 52, 82, 83, 87, 95, 95\). The number of data points \(n = 11\).

Step2: Find the median

The median (second quartile, \(Q_2\)) is the middle value. For \(n = 11\) (odd), the median is the \(\frac{n + 1}{2}=\frac{11 + 1}{2}=6^\text{th}\) term. The \(6^\text{th}\) term is \(52\), so median \(= 52\).

Step3: Find the lower quartile (\(Q_1\))

The lower half of the data is the first \(5\) terms: \(11, 31, 31, 37, 49\). The median of this lower half (for \(n = 5\), middle term is \(\frac{5 + 1}{2}=3^\text{rd}\) term). The \(3^\text{rd}\) term is \(31\), so \(Q_1 = 31\).

Step4: Find the upper quartile (\(Q_3\))

The upper half of the data is the last \(5\) terms: \(82, 83, 87, 95, 95\). The median of this upper half (for \(n = 5\), middle term is \(\frac{5 + 1}{2}=3^\text{rd}\) term). The \(3^\text{rd}\) term is \(87\), so \(Q_3 = 87\).

Answer:

lower quartile \(= 31\)
median \(= 52\)
upper quartile \(= 87\)